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Zeta and q-Zeta Functions and Associated
Series and Integrals
Zeta and q-Zeta Functions
and Associated Series and
Integrals
H. M. Srivastava
Department of Mathematics
University of Victoria
Victoria
Canada
Junesang Choi
Department of Mathematics
Dongguk University
Gyeongju
Republic of Korea
AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK • OXFORD
PARIS • SAN DIEGO • SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO
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First edition 2012
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Contents 
Preface 
Acknowledgements 
xi 
XV 
1 Introduction and Preliminaries 
1.1 Gamma and Beta Function 
1 
1 
The Gamma Function 1 
Pochhammcr's Symbol and the Factorial Function 4 
Multiplication Formulas of Legendre and Gauss 6 
Stirling's Formula for n! and its Generalizations 6 
The Beta Function 7 
The Incomplete Gamma Functions 10 
The Incomplete Beta Functions 10 
The Error Functions 11 
The Bohr-Mo llerup Theorem 12 
1.2 The Euler-Mascheroni Constant y 13 
A et of Known Integral Representations for y 15 
Further Integral Representations for y 18 
From an Application of the Residue Calculu 22 
1.3 Polygamma Functions 24 
The Psi (or Digamma) Function 24 
Integral Representations for 1/J(z) 25 
Gauss's fonnulas for lf!( �) 30 
Special Values of 1/r(z) 31 
The Polygamma Functions 33 
Special Value of t<"l (z) 34 
The Asymptotic Expansion for 1/r(z) 36 
1.4 The Multiple Gamma Functions 38 
The Double Gamma Function r2 38 
Integral Formulas involving the Double Gamma Function 45 
The Evaluation of an Integral Involving log G(z) 52 
The Multiple Gamma Functions 56 
The Triple Gamma Function r3 58 
1.5 The Gaussian Hypergeometric f'unction and its General ization 63 
The Gauss Hypergeometric Equation 63 
Gauss's Hypergeometric Series 64 
The Hypergeometric Series and Its Analytic Continuation 65 
vi 
2 
Contents 
Linear, Quadratic and Cubic Transfom1ations 67 
Hypergeometric Representations of Elementary Functions 67 
Hypcrgcomctric Representations of Other FuncLions 68 
The Conftuent Hypergeometric Function 69 
lmponam Properties of Kummer ' s Confluent Hypergeometric 
Function 70 
The Generalized (Gauss and Kummer) Hypergeometric Function 71 
Analytk Continuation of the Generalized HypergeomeLric 
Function 72 
Functions Expressible in Terms of the pFq Function 
J .6 Stirling umbers of the First and Second Kind 
Stirling Number of the First Kind 
Stirling Numbers of the Second Kind 
Relationships Among Stirl i ng Numbers of the Fir t and 
Second Kind and Bemoulli Numbers 
73 
76 
76 
78 
79 
1.7 Bernoulli, Euler and Gcnocchi Polynomials and Numbers 81 
Bernoulli Polynomials and· umbers 81 
The Generalized Bernoulli Polynomial and Number 83 
Euler Polynomial.. and umbers 86 
Fourier Series Expansions of Bernoulli and Euler Polynom ials 87 
Relations Between Bemou ll i and Euler Polynomial 88 
The Generalized Euler Polynomials and Numbers 88 
Gcnocchi Polynomials and umbers 90 
1.8 Apostol-Bemoulli, Apostol-Euler and Apostol-Genocchi 
Polynomials and Numbers 91 
Apostoi-Bernoulli Polynomials and Numbers 91 
Apostoi-Genocchi Polynomial and Number 98 
Important Remark and Ob ervations 99 
Generalizations and Unified Presentations of the Apostol Type 
Polynomials 100 
1.9 Inequalities for the Gamma Function and the Double Gamma 
Fun��n 105 
The Gamma Function and Its Relati es 105 
The Double Gamma Function 112 
Problem 112 
The Zeta and Related Functions 
2. I M ultiple Hurwitz Zeta Functions 
The Analytic Continuation of s, (s. a) 
Relation hip between Sn s, x) and B,�l (x) 
The Vardi-Barne Multiple Gamma Functions 
2.2 The Hurwitz (or Generalized) Zeta Function 
Hurwitz's Formula fors( , a) 
Hermite's Fom1llla for t(s, a) 
Further Integral Representations for s(s. a) 
141 
141 
142 
150 
153 
155 
156 
157 
'159 
Comems 
Some Applications of the Derivative Fom1ula ( 17) 
Another Form for f2(a) 
vii 
2.3 The Ricmann Zcta Function 
160 
162 
164 
166 Riemann's Functional Equation for �(s) 
Relationship between �(s) and the Mathematical 
Constants B and C 
Integral Representations for� (s) 
A Summation Identity for� (n) 
2.4 Polylogarithm Functions 
The Dilogarithm Function 
Clausen's Integral or Function) 
The Trilogarithm Function 
The Polylogarithm Functions 
The Log -Sine Integrals 
2.5 Hurwitz-Lerch Zeta Functions 
The Taylor Series Expansion of the Lipschitz-Lcrch 
Transcendent L(x, s, a) 
Evaluation of L(x. -n, a) 
2.6 Generalizations of the Hurwitz-Lerch Zeta Function 
2.7 Analytic Continuations of Multiple Zeta Functions 
Generalized Function of Gel'fand and Shilov 
Euler-Maclaurin Summation Fonnula 
Problems 
3 Series involving Zctu Functions 
3 .I Historical Introduction 
3.2 Use of the B inom ial Theorem 
Applications of Theorems 3.1 and 3.2 
167 
169 
172 
175 
176 
181 
183 
185 
191 
194 
198 
199 
200 
213 
213 
220 
224 
245 
245 
247 
257 
3.3 Use of Generating Functions 261 
Series Involving Polygamma Funct ions 266 
Series Involving Polylogarithm Functions 267 
3.4 Use of Multiple Gamma Functions 269 
Evaluation by Using the Gamma !'unction 269 
Evaluation in Terms of Catalan' Constant G 339 
f-urther Evaluation by Using the Triple Gamma !'unction 344 
Applications of Corollary 3.3 348 
3.5 Use of Hypergeometric Identities 350 
Series Derivable from Gauss's Summation Formula 1.4(7) 351 
Series Derivable from Kummer's Formula (3) 354 
Series Derivable from Other Hypergeometric Summation 
Fonnulas 358 
Further Summation Fom1ulas Related to Generalized Ham1onic 
Numbers 361 
3.6 Other Methods and their Applications 364 
The Weierstrass Canonical Product Form for the Gamm a Function 364 
viii 
Evaluation by Using Infinite Products 
Higher-OrderDeri vatives of the Gamma Funct ion 
3.7 Applications of Series Involving the Zcta Function 
The Multiple Gamma Functions 
Matbieu Serie 
Problems 
Contents 
366 
369 
375 
375 
382 
389 
4 Evaluations and Series Representations 399 
4.1 Evaluation of� 2n) 399 
l11e General Ca e of �(2n) 402 
4.2 Rapidly Convergent Series for �(2n + I) 405 
Remarks and Obser at ions 409 
4.3 Further Series Representations 415 
4.4 Computational Results 422 
Problem 433 
5 Determinants of the Laplacians 445 
5.1 The n-Dimensional Problem 445 
5.2 Computations Using the Simple and Multiple Gamma Function, 448 
Factorizations Into Simple and Multiple Gamma Functions 448 
Evaluations of det' !:i, (n = I. 2, 3) 452 
5.3 Computations Using eries of Zeta Functions 457 
5.4 Computations using Zeta Regularized Product 465 
A Lemma on Zeta Regularized Products and a Main Theorem 467 
Computations for small n 471 
5.5 Remark s and Observations 472 
Problems 473 
6 q -Extensions of Some Special Functions and Polynomials 479 
6.1 q-Shifted Factorials and q-Binomial Coefficients 
6.2 q-Derivative, q-Antiderivative and Jack son q-Integral 
q-Deri vative 
q-Antiderivative and Jackson q-lntegral 
6.3 q-Binomial Theorem 
6.4 q-Garnma Function and q-Beta Function 
q-Gamma Function 
q-Beta Function 
6.5 A q-Extension of the Mu lt iple Gamma Functions 
6.6 q-Bernoulli umbers and q-Bernoulli Polynomials 
q-Stirl ing umbers of the Second Kind 
The Polynomial /3* (x) = f3t:q(.:r) 
6.7 q-Euler umbers and q-Euler Polynomials 
6.8 The q-Apostol-Bernoulli Polynomials !3/>(x; A) of Order 11 
6.9 The q-Apostoi-Eulcr Polynomials E}">(x; A) of Order 11 
479 
483 
484 
484 
487 
490 
490 
495 
497 
499 
504 
506 
509 
513 
518 
Comems ix 
6.10 A Generalized q-Zeta Function 519 
An Auxiliary Function Defining Generalized q-Zeta Function 519 
Application of Eulcr-Maclaurin Summation Fom1tda 524 
6.1 1 Multiple q-Zeta Functions 530 
Analytic Continuation of gq and {q 530 
Analytic Continuation of Mul tip l e Zeta Functions 533 
Special Values of l;q (s1, s2) 541 
Problems 542 
7 Miscellaneous Result'> 
7.1 A Set of Useful Mathematical Constants 
Euler-Mascherotti Constant y 
Series Representations for y 
A Class of Constants Analogous to {Dk} 
Other Classes of Mathematical Constants 
7.2 Log-Sine Integrals Involving Series Associated with the Zeta 
555 
555 
555 
556 
560 
563 
Function and Polylogarithms 568 
Analogous Log-Sine Integral 571 
Remark on Cl,(O) and GI,(O) 575 
f-urther Remarks and Observations 
7.3 Applications of the Gamma and Polygamma Functions Involving 
578 
Convolutions of the Rayleigh Functions 581 
Ser i e s Expressible in Terms of the 1ft-Function 582 
Convolutions of the Rayleigh Functions 584 
7.4 Bemoulli and Euler Polynomial at Rational Argument 587 
The Cvijovit-Kiinowsld Summation Formulas 588 
Srivastava's Shorter Proofs of Theorem 7.3 and Theorem 7.4 589 
Fonnulas Involving the Hurwitz-Lerch Zeta Function 
An Application of Lerch's Functional Equation 2.5(29) 
7.5 Closed-Form ummation of Trigonometric erie 
Problems 
Bibliography 
591 
593 
594 
597 
603 
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Preface
This book is essentially a thoroughly revised, enlarged and updated version of the
authors’ work: Series Associated with the Zeta and Related Functions (Kluwer
Academic Publishers, Dordrecht, Boston and London, 2001). It aims at presenting
a state-of-the-art account of the theories and applications of the various methods and
techniques which are used in dealing with many different families of series associated
with the Riemann Zeta function and its numerous generalizations and basic (or q-)
extensions. Systematic accounts of only some of these methods and techniques, which
are widely scattered in journal articles and book chapters, were included in the above-
mentioned book.
In recent years, there has been an increasing interest in problems involving closed-
form evaluations of (and representations of the Riemann Zeta function at positive
integer arguments as) various families of series associated with the Riemann Zeta
function ζ(s), the Hurwitz Zeta function ζ(s,a), and their such extensions and gener-
alizations as (for example) Lerch’s transcendent (or the Hurwitz-Lerch Zeta function)
8(z,s,a). Some of these developments have apparently stemmed from an over two-
century-old theorem of Christian Goldbach (1690−1764), which was stated in a letter
dated 1729 from Goldbach to Daniel Bernoulli (1700−1782), from recent rediscov-
eries of a fairly rapidly convergent series representation for ζ(3), which is actually
contained in a 1772 paper by Leonhard Euler (1707−1783), and from another known
series representation for ζ(3), which was used by Roger Apéry (1916−1994) in 1978
in his celebrated proof of the irrationality of ζ(3).
This revised, enlarged and updated version of our 2001 book is motivated essen-
tially by the fact that the theories and applications of the various methods and tech-
niques used in dealing with many different families of series associated with the
Riemann Zeta function, its aforementioned relatives and its many different basic
(or q-) extensions are to be found so far only in widely scattered journal articles pub-
lished during the last decade or so. Thus, our systematic (and unified) presentation of
these results on the evaluation and representation of the various families of Zeta and
q-Zeta functions is expected to fill a conspicuous gap in the existing books dealing
exclusively with these Zeta and q-Zeta functions.
The main objective of this revised, enlarged and updated version is to provide a
systematic collection of various families of series associated with the Riemann and
Hurwitz Zeta functions, as well as with many other higher transcendental functions,
which are closely related to these functions (including especially the q-Zeta and related
functions). It, therefore, aims at presenting a state-of-the-art account of the theory and
applications of many different methods (which are available in the rather scattered
xii Preface
literature on this subject, especially since the publication of our aforementioned 2001
book) for the derivation of the types of results considered here.
In our attempt to make this book as self-contained as possible within the obvi-
ous constraints, we include in Chapter 1 (Introduction and Preliminaries) a reason-
ably detailed account of such useful functions as the Gamma and Beta functions,
the Polygamma and related functions, multiple Gamma functions, the Gauss hyper-
geometric function and its familiar generalization, the Stirling numbers of the first
and second kind, the Bernoulli, Euler and Genocchi polynomials and numbers, the
Apostol-Bernoulli, the Apostol-Euler and the Apostol-Genocchi polynomials and
numbers, as well as some interesting inequalities for the Gamma function and the
double Gamma function. In Chapter 2 (The Zeta and Related Functions), we present
the definitions and various potentially useful properties (and characteristics) of the
Riemann, Hurwitz and Hurwitz-Lerch Zeta functions and their generalizations, the
Polylogarithm and related functions and the multiple Zeta functions, together with
their analytic continuations.
In Chapter 3 (Series Involving Zeta Functions), we begin by providing a brief his-
torical introduction to the main subject of this book. We then describe and illustrate
some of the most effective methods of evaluating series associated with the Zeta and
related functions. Further developments on the evaluations and (rapidly convergent)
series representations of ζ(s) when s ∈ N \ {1} are presented in Chapter 4 (Evaluations
and Series Representations), which also deals with various computational results on
this subject.
Chapter 5 (Determinants of the Laplacians) considers the problem involving com-
putations of the determinants of the Laplacians for the n-dimensional sphere Sn
(n ∈ N). It is here in this chapter that we show how fruitfully some of theseries eval-
uations (which are presented in the earlier chapters) can be applied in the solution of
the aforementioned problem.
In a brand new Chapter 6 (q-Extensions of Some Special Functions and Polyno-
mials), we first introduce the concepts of the basic (or q-) numbers, the basic (or q-)
series and the basic (or q-) polynomials. We then proceed to apply these concepts
and present a reasonably detailed theory of the various basic (or q-) extensions of the
Gamma and Beta functions, the derivatives, antiderivatives and integrals, the bino-
mial theorem, the multiple Gamma functions, the Bernoulli numbers and polynomials,
the Euler numbers and polynomials, the Apostol-Bernoulli polynomials, the Apostol-
Euler polynomials and so on.
The last chapter (Chapter 7) contains a wide variety of miscellaneous results deal-
ing with (for example) the analysis of several useful mathematical constants, a variety
of Log-Sine integrals involving series associated with the Zeta function and Polyloga-
rithms, applications of the Gamma and Polygamma functions involving convolutions
of the Rayleigh functions, evaluations of the Bernoulli and Euler polynomials at ratio-
nal arguments, and the closed-form summation of several classes of trigonometric
series.
Each chapter in this book begins with a brief outline summarizing the material pre-
sented in the chapter and is then divided into a number of sections. Equations in every
section are numbered separately. While referring to an equation in another section of
Preface xiii
the book, we use numbers like 3.2(18) to represent Equation (18) in Section 3.2 (that
is, the second section of Chapter 3).
At the close of each chapter, we have provided a set of carefully-selected problems,
which are based essentially upon the material presented in the chapter. Many of these
problems are taken from recent research publications, and (in all such instances) we
have chosen to include the precise references for further investigation (if necessary).
Another valuable feature of this book is the extensive and up-to-date bibliography on
the subject dealt with in the book.
Just as its predecessor (that is, the 2001 edition), this book is written primarily as
a reference work for various seemingly diverse groups of research workers and other
users of series associated with the Zeta and related functions. In particular, teachers,
researchers and postgraduate students in the fields of mathematical and applied sci-
ences will find this book especially useful, not only for its detailed and systematic
presentations of the theory and applications of the various methods and techniques
used in dealing with many different classes of series associated with the Zeta and
related functions, or for its stimulating historical accounts of a large number of prob-
lems considered here, but also for its well-classified tables of series (and integrals) and
its well-motivated presentation of many sets of closely related problems with their pre-
cise bibliographical references (if any).
This page intentionally left blank 
Acknowledgements
Many persons have contributed rather significantly to this thoroughly revised, enlarged
and updated version, just as to its predecessor (that is, the 2001 edition), both directly
and indirectly. Contribution of subject matter is duly acknowledged throughout the
text and in the bibliography. Indeed, we are greatly indebted to the various authors
whose works we have freely consulted and who occasionally provided invaluable ref-
erences and advice serving for the enrichment of the matter presented in this book.
The first-named author wishes to express his deep sense of gratitude to his wife and
colleague, Professor Rekha Srivastava, for her cooperation and support throughout the
preparation of this thoroughly revised, enlarged and updated version of the 2001 book.
The collaboration of the authors on the 2001 book project was conceptualized as
long ago as August 1995, and the preparation of a preliminary outline was initiated
in December 1997, during the first-named author’s visits to Dongguk University at
Gyeongju. The first drafts of some of the chapters in this book were written during sev-
eral subsequent visits of the first-named author to Dongguk University at Gyeongju.
The final drafts of most of the chapters in the 2001 book were prepared during the
second-named author’s visit to the University of Victoria from August 1999 to August
2000, while he was on Study Leave from Dongguk University at Gyeongju. The prepa-
ration of this thoroughly revised, enlarged and updated version was carried out, in
most part, during the period from January 2008 to January 2009, during the second-
named author’s visit to the University of Victoria, while he was on Study Leave from
Dongguk University at Gyeongju for the second time. Our sincere thanks are due to the
appropriate authorities of each of these universities, to the Korea Research Foundation
(Support for Faculty Research Abroad under its Research Fund Program) and to the
Natural Sciences and Engineering Research Council of Canada, for providing financial
support and other facilities for the completion of each of the projects leading eventu-
ally to the 2001 edition and this thoroughly revised, enlarged and updated version. We
especially acknowledge and appreciate the financial support that was received under
the Basic Science Research Program through the National Research Foundation of the
Republic of Korea.
We take this opportunity to express our thanks to the editorial (and technical) staff
of the Elsevier Science Publishers B.V. (especially the Publisher, Ms. Lisa Tickner, for
Serials and Elsevier Insights) for their continued interest in this book and for their pro-
ficient (and impeccable) handling of its publication. Springer’s permission to publish
this thoroughly revised, enlarged and updated edition of the 2001 book is also greatly
appreciated.
Finally, we should like to record our indebtedness to the members of our respec-
tive families for their understanding, cooperation and support throughout this project.
xvi Acknowledgements
The second-named author and his family would, especially, like to express their appre-
ciation for the first-named author and his family’s hospitality and every prudent con-
sideration during their stay in Victoria for over one year, first from August 1999 to
August 2000 and then again from January 2008 to January 2009, while the second-
named author was on Study Leave from Dongguk University at Gyeongju.
H. M. Srivastava
University of Victoria
Canada
Junesang Choi
Dongguk University
Republic of Korea
February 2011
1 Introduction and Preliminaries
In this introductory chapter, we present the definitions and notations (and some of the
important properties and characteristics) of the various special functions, polynomials
and numbers, which are potentially useful in the remainder of the book. The special
functions considered here include (for example) the Gamma, Beta and related func-
tions, the Polygamma functions, the multiple Gamma functions, the Gaussian hyper-
geometric function and the generalized hypergeometric function. We also consider the
Stirling numbers of the first and second kind, the Bernoulli, Euler and Genocchi poly-
nomials and numbers and the various families of the generalized Bernoulli, Euler and
Genocchi polynomials and numbers. Relevant connections of some of these functions
with other special functions and polynomials, which are not listed above, are also
presented here.
1.1 Gamma and Beta Functions
The Gamma Function
The origin of the Gamma function can be traced back to two letters from Leonhard
Euler (1707–1783) to Christian Goldbach (1690–1764), just as a simple desire to
extend factorials to values between the integers. The first letter (dated October 13,
1729) dealt with the interpolation problem, whereas the second letter (dated January
8, 1730) dealt with integration and tied the two together.
The Gamma function 0(z) developed byEuler is usually defined by
0(z) :=
∞∫
0
e−t tz−1 dt (<(z) > 0). (1)
We also present here several equivalent forms of the Gamma function 0(z), one by
Weierstrass:
0(z)=
e−γ z
z
∞∏
k=1
{(
1+
z
k
)−1
ez/k
}
(
z ∈ C \Z−0 ; Z
−
0 := {0,−1,−2, . . .}
)
,
(2)
Zeta and q-Zeta Functions and Associated Series and Integrals. DOI: 10.1016/B978-0-12-385218-2.00001-3
c© 2012 Elsevier Inc. All rights reserved.
http://dx.doi.org/10.1016/B978-0-12-385218-2.00001-3
2 Zeta and q-Zeta Functions and Associated Series and Integrals
where γ denotes the Euler-Mascheroni constant defined by
γ := lim
n→∞
(
n∑
k=1
1
k
− logn
)
∼= 0.577215664901532860606512 . . . , (3)
and the other by Gauss:
0(z)= lim
n→∞
{
(n− 1)!nz
z(z+ 1) · · ·(z+ n− 1)
}
= lim
n→∞
{
n!(n+ 1)z
z(z+ 1) · · ·(z+ n)
}
= lim
n→∞
{
n!nz
z(z+ 1) · · ·(z+ n)
}
(z ∈ C \Z−0 ),
(4)
since
lim
n→∞
n
z+ n
= 1= lim
n→∞
nz
(n+ 1)z
.
In terms of the Pochhammer symbol (λ)n defined (for λ ∈ C) by
(λ)n :=
{
1 (n= 0)
λ(λ+ 1) · · ·(λ+ n− 1) (n ∈ N := {1, 2, 3, . . .}),
(5)
the definition (4) can easily be written in an equivalent form:
0(z)= lim
n→∞
(n− 1)!nz
(z)n
(z ∈ C \Z−0 ). (6)
By taking the reciprocal of (2) and applying the definition (3), we have
1
0(z)
= z
[
lim
n→∞
exp
{(
1+
1
2
+ ·· ·+
1
n
− logn
)
z
}][
lim
n→∞
n∏
k=1
{(
1+
z
k
)
e−z/k
}]
= z lim
n→∞
[
exp
{(
1+
1
2
+ ·· ·+
1
n
− logn
)
z
}
·
n∏
k=1
{(
1+
z
k
)
e−z/k
}]
= z lim
n→∞
{
n−z
n∏
k=1
(
1+
z
k
)}
= z lim
n→∞
[{
n−1∏
k=1
(
1+
1
k
)−z}{ n∏
k=1
(
1+
z
k
)}]
= z
∞∏
k=1
{(
1+
z
k
)(
1+
1
k
)−z}
,
Introduction and Preliminaries 3
which yields Euler’s product form of the Gamma function:
0(z)=
1
z
∞∏
k=1
{(
1+
1
k
)z(
1+
z
k
)−1}
. (7)
When t in (1) is replaced by − log t, (1) is also written in an equivalent form:
0(z)=
1∫
0
(
log
1
t
)z−1
dt (<(z) > 0). (8)
This representation of the Gamma function as well as the symbol 0 are attributed to
Legendre.
Integration of (1) by parts easily yields the functional relation:
0(z+ 1)= z0(z), (9)
so that, obviously,
0(z)=
0(z+ n)
z(z+ 1) · · ·(z+ n− 1)
(n ∈ N0 := N∪ {0}), (10)
which enables us to define 0(z) for <(z) >−n(n ∈ N0) as an analytic function except
for z= 0,−1,−2, . . . ,−n+ 1. Thus, 0(z) can be continued analytically to the whole
complex z-plane except for simple poles at z ∈ Z−0 .
The representation (2) in conjunction with the well-known product formula:
sin πz= πz
∞∏
n=1
(
1−
z2
n2
)
(11)
also yields the following useful relationship between the Gamma and circular func-
tions:
0(z)0(1− z)=
π
sin πz
(z 6∈ Z := {0,±1,±2, . . .}), (12)
which incidentally provides an immediate analytic continuation of 0(z) from right to
the left half of the complex z-plane.
Several special values of 0(x), when x is real, are worthy of note. Indeed, from (1)
we note that
0(1)=
∞∫
0
e−t dt = 1, (13)
4 Zeta and q-Zeta Functions and Associated Series and Integrals
and that 0(x) > 0 for all x in the open interval (0,∞). Thus, (12) with z= 12 immedi-
ately yields
0
(
1
2
)
=
√
π, (14)
which, in view of (1), implies
∞∫
0
e−t
√
t
dt =
√
π (15)
or, equivalently,
∞∫
0
exp(−t2)dt =
√
π
2
. (16)
By making use of the relation (10), we obtain
0(n+ 1)= n!; 0
(
n+
1
2
)
=
(2n)!
√
π
22n n!
;
0
(
−n+
1
2
)
= (−1)n
√
π
22n n!
(2n)!
; (n ∈ N0).
(17)
The last two results in (17) also require the use of (14). The formulas listed under
(17) enable us to compute 0(x) when x is a positive integer and when x is half an odd
integer, positive or negative.
From (2) or (4), it also follows that 0(z) is a meromorphic function on the whole
complex z-plane with simple poles at z=−n(n ∈ N0) with their respective residues
given by
Res
z=−n
0(z)=
(−1)n
n!
(n ∈ N0). (18)
Since the function 0(1/w) has simple poles at w=−1, − 12 , −
1
3 , . . . , which implies
that w= 0 is an accumulation point of the poles of 0(1/w), the Gamma function 0(z)
has an essential singularity at infinity. Furthermore, it follows immediately from (2)
that 1/0(z) has no poles, and, therefore, 0(z) is never zero.
Pochhammer’s Symbol and the Factorial Function
Since (1)n = n!, the Pochhammer symbol (λ)n defined by (5) may be looked upon as
a generalization of the elementary factorial; hence, the symbol (λ)n is also referred to
as the shifted factorial.
Introduction and Preliminaries 5
In terms of the Gamma function, we have (cf. Definition (5))
(λ)n =
0(λ+ n)
0(λ)
(
λ ∈ C \Z−0
)
, (19)
which can easily be verified. Furthermore, the binomial coefficient may now be
expressed as(
λ
n
)
=
λ(λ− 1) · · ·(λ− n+ 1)
n!
=
(−1)n (−λ)n
n!
(20)
or, equivalently, as(
λ
n
)
=
0(λ+ 1)
n!0(λ− n+ 1)
. (21)
It follows from (20) and (21) that
0(λ+ 1)
0(λ− n+ 1)
= (−1)n (−λ)n,
which, for λ= α− 1, yields
0(α− n)
0(α)
=
(−1)n
(1−α)n
(α 6∈ Z). (22)
Equations (19) and (22) suggest the definition:
(λ)−n =
(−1)n
(1− λ)n
(n ∈ N; λ 6∈ Z). (23)
Equation (19) also yields
(λ)m+n = (λ)m(λ+m)n, (24)
which, in conjunction with (23), gives
(λ)n−k =
(−1)k (λ)n
(1− λ− n)k
(0 5 k 5 n). (25)
For λ= 1, we have
(n− k)!=
(−1)k n!
(−n)k
(0 5 k 5 n), (26)
which may alternatively be written in the form:
(−n)k =

(−1)k n!
(n− k)!
(0 5 k 5 n),
0 (k > n).
(27)
6 Zeta and q-Zeta Functions and Associated Series and Integrals
Multiplication Formulas of Legendre and Gauss
In view of the definition (5), it is not difficult to show that
(λ)2n = 2
2n
(
1
2
λ
)
n
(
1
2
λ+
1
2
)
n
(n ∈ N0), (28)
which follows also from Legendre’s duplication formula for the Gamma function, viz
√
π 0(2z)= 22z−10(z)0
(
z+
1
2
) (
z 6= 0,−
1
2
,−1,−
3
2
, . . .
)
. (29)
For every positive integer m, we have
(λ)mn = m
mn
m∏
j=1
(
λ+ j− 1
m
)
n
(m ∈ N; n ∈ N0), (30)
which reduces to (28) when m= 2. Starting from (30) with λ= mz, it can be proved
that
0(mz)= (2π)
1
2 (1−m)mmz−
1
2
m∏
j=1
0
(
z+
j− 1
m
)
(
z 6= 0,−
1
m
,−
2
m
, . . . ; m ∈ N
)
,
(31)
which is known in the literature as Gauss’s multiplication theorem for the Gamma
function.
Stirling’s Formula for n! and its Generalizations
For a large positive integer n, it naturally becomes tedious to compute n!.An easy way
of computing an approximate value of n! for large positive integer n was initiated by
Stirling in 1730 and modified subsequently by De Moivre, who showed that
n!∼
(n
e
)n √
2πn (n→∞) (32)
or, more generally, that
0(x+ 1)∼
(x
e
)x √
2πx (x→∞; x ∈ R), (33)
where e is the base of the natural logarithm.
Introduction and Preliminaries 7
For a complex number z, we have the following asymptotic expansion:
log0(z)=
(
z−
1
2
)
logz− z+
1
2
log(2π)+
n∑
k=1
B2k
2k(2k− 1)z2k−1
+O
(
z−2n−1
)
(|z| →∞; |arg(z)|5 π − � (0< � < π); n ∈ N0),
(34)
which, upon taking exponentials, yields an asymptotic formula for the Gamma
function:
0(z)= zz e−z
√
2π
z
[
1+
1
12z
+
1
288z2
−
139
51840z3
−
571
2488320z4
+
163879
209018880z5
+
50043869
75246796800z6
+O
(
z−7
)]
(35)
(|z| →∞; |arg(z)|5 π − � (0< � < π)).
The asymptotic formula (35), in conjunction with the recurrence relation (9), is useful
in computing the numerical values of 0(z) for large real values of z.
Some useful consequences of (34) or (35) include the asymptotic expansions:
log0(z+α)=
(
z+α−
1
2
)
logz− z+
1
2
log(2π)+O
(
z−1
)
(|z| →∞; |arg(z)|5 π − �; |arg(z+α)|5 π − �; 0< � < π),
(36)
and
0(z+α)
0(z+β)
= zα−β
[
1+
(α−β)(α+β − 1)
2z
+O
(
z−2
)]
(|z| →∞; |arg(z)|5 π − �; |arg(z+α)|5 π − �; 0< � < π),
(37)
where α and β are bounded complex numbers.
Yet another interesting consequence of (35) is the following asymptotic expansion
of |0(x+ iy)|:
|0(x+ iy)| ∼
√
2π |y|x−
1
2 e−
1
2π |y| (|x|<∞; |y| →∞), (38)
where x and y take on real values.
The Beta Function
The Beta function B(α, β) is a function of two complex variables α and β, defined by
B(α, β) :=
1∫
0
tα−1(1− t)β−1 dt = B(β, α) (<(α) > 0; <(β) > 0) (39)
8 Zeta and q-Zeta Functions and AssociatedSeries and Integrals
or, equivalently, by
B(α, β)= 2
π/2∫
0
(sin θ)2α−1 (cos θ)2β−1 dθ (<(α) > 0; <(β) > 0), (40)
which follows from (39) on setting t = sin 2 θ.
The integrals in (39) and (1) are known as the Eulerian integrals of the first and
second kind, respectively.
Putting t = u/(1+ u) in (39), we obtain the following representation of B(α, β) as
an infinite integral:
B(α, β)=
∞∫
0
uα−1
(1+ u)α+β
du (<(α) > 0; <(β) > 0). (41)
The Beta function is closely related to the Gamma function; in fact, we have
B(α, β)=
0(α)0(β)
0(α+β)
(
α, β 6∈ Z−0
)
, (42)
which not only confirms the symmetry property in (39), but also continues the Beta
function analytically for all complex values of α and β, except when α, β ∈ Z−0 . Thus,
we may write
B(α, β)=

1∫
0
tα−1(1− t)β−1 dt (<(α) > 0; <(β) > 0)
0(α)0(β)
0(α+β)
(
<(α) < 0; <(β) < 0; α, β 6∈ Z−0
)
.
(43)
Next we combine the relationship (42) with (40) and (41), and we obtain the fol-
lowing useful integral formulas:
π/2∫
0
sin µθ cos νθ dθ =
0
(
1
2µ+
1
2
)
0
(
1
2ν+
1
2
)
20
(
1
2µ+
1
2ν+ 1
) (<(µ) >−1; <(ν) >−1)
(44)
and
∞∫
0
uλ−1
(1+ u)µ
du=
0(λ)0(µ− λ)
0(µ)
(0< <(λ) < <(µ)). (45)
Introduction and Preliminaries 9
It should be remarked in passing that the integral (44) provides a generalization of
Wallis’s formula of elementary calculus and that (45), with µ= 1, yields the familiar
infinite integral:
∞∫
0
uλ−1
1+ u
du= 0(λ)0(1− λ)=
π
sin πλ
(0< <(λ) < 1), (46)
which is usually evaluated in the literature by contour integration (see, e.g., Copson
[341, p. 139, Example 1]).
In addition to (41) and (45), by means of suitable substitutions, a number of definite
integrals are expressible in terms of the Beta function:
1∫
0
tα−1(1− t)β−1 (1+ at)−α−β dt = (1+ a)−α B(α, β)
(47)
(a>−1; <(α) > 0; <(β) > 0);
∞∫
0
tβ−1 (1+ at)−α−β dt = a−β B(α, β)
(48)
(a>0; <(α) > 0; <(β) > 0);
a∫
b
(t− b)α−1 (a− t)β−1 dt = (a− b)α+β−1 B(α, β)
(49)
(a> b; <(α) > 0; <(β) > 0);
a∫
b
(t− b)α−1 (a− t)β−1
(t− c)α+β
dt =
(a− b)α+β−1
(a− c)α (b− c)β
B(α, β)
(a> b> c; <(α) > 0; <(β) > 0).
(50)
The following functional equations for the Beta function can be deduced easily
from (39) and (42):
B(α, β + 1)=
β
α
B(α+ 1, β)=
β
α+β
B(α, β); (51)
B(α, β)B(α+β, γ )= B(β, γ )B(β + γ, α)= B(γ, α)B(α+ γ, β); (52)
B(α, β)B(α+β, γ )B(α+β + γ, δ)=
0(α)0(β)0(γ )0(δ)
0(α+β + γ + δ)
, (53)
10 Zeta and q-Zeta Functions and Associated Series and Integrals
or, more generally,
n∏
k=1
B
 k∑
j=1
αj, αk+1
= 0(α1) · · ·0(αn+1)
0(α1+ ·· ·+αn+1)
(n ∈ N); (54)
1
B(n, m)
= m
(
n+m− 1
n− 1
)
= n
(
n+m− 1
m− 1
)
(n, m ∈ N). (55)
The Incomplete Gamma Functions
The incomplete Gamma function γ (z, α) and its complement 0(z, α) (also known as
Prym’s function) are defined by
γ (z, α) :=
α∫
0
tz−1 e−t dt (<(z) > 0; |arg(α)|< π), (56)
0(z, α) :=
∞∫
α
tz−1 e−t dt (|arg(α)|< π), (57)
so that
γ (z, α)+0(z, α)= 0(z). (58)
For fixed α, 0(z, α) is an entire (integral) function of z, whereas γ (z, α) is a mero-
morphic function of z, with simple poles at the points z ∈ Z−0 .
The following recursion formulas are worthy of note:
γ (z+ 1, α)= zγ(z, α)−αz e−α, (59)
0(z+ 1, α)= z0(z, α)+αz e−α. (60)
The Incomplete Beta Functions
The incomplete Beta function Bx(α, β) is defined by
Bx(α, β) :=
x∫
0
tα−1(1− t)β−1 dt (<(α) > 0). (61)
For the associated function:
Ix(α, β)=
Bx(α, β)
B(α, β)
, (62)
Introduction and Preliminaries 11
we note here the following properties that are easily verifiable:
Ix(α, β)= 1− I1−x(β, α), (63)
Ix(k, n− k+ 1)=
n∑
j=k
(
n
j
)
xj (1− x)n−j (1≤ k ≤ n), (64)
Ix(α, β)= xIx(α− 1, β)+ (1− x) Ix(α, β − 1), (65)
(α+β −αx) Ix(α, β)= α(1− x) Ix(α+ 1, β − 1)+βIx(α, β + 1), (66)
(α+β) Ix(α, β)= α Ix(α+ 1, β)+βIx(α, β + 1). (67)
The Error Functions
The error function erf(z), also known as the probability integral 8(z), is defined for
any complex z by
erf(z) :=
2
√
π
z∫
0
exp(−t2)dt =8(z), (68)
and its complement by
erfc(z) := 1− erf(z)=
2
√
π
∞∫
z
exp(−t2)dt. (69)
Clearly, we have
erf(0)= 0 and erfc(0)= 1, (70)
and, in view of the well-known result (16), we also have
erf(∞)= 1 and erfc(∞)= 0. (71)
The following alternative notations:
Erf(z)=
√
π
2
erf(z) and Erfc(z)=
√
π
2
erfc(z) (72)
are sometimes used for the error functions. Many authors use the notations Erf(z) and
Êrf(z) for the error functions erf(z) and Erf(z), respectively, defined by (68) and (72),
and the notations Erfc(z) and Êrfc(z) for their complements.
In terms of the incomplete Gamma functions, it is easily verified that
erf(z)=
1
√
π
γ
(
1
2
, z2
)
and erfc(z)=
1
√
π
0
(
1
2
, z2
)
. (73)
12 Zeta and q-Zeta Functions and Associated Series and Integrals
The Bohr-Mollerup Theorem
We have already observed that Euler’s definition (1) and its such consequences as
(9) and (13) enable us to compute all the real values of the Gamma function from
the knowledge merely of its values in the interval (0,1), as noted in conjunction
with (35). Since the solution to the interpolation problem is not determined uniquely,
it makes sense to add more conditions to the problem. After various trials to find
those conditions to guarantee the uniqueness of the Gamma function, in 1922, Bohr
and Mollerup were able to show the remarkable fact that the Gamma function is the
only function that satisfies the recurrence relationship and is logarithmically convex.
The original proof was simplified, several years later, by Emil Artin, and the theo-
rem, together with Artin’s method of proof, now constitute the Bohr-Mollerup-Artin
theorem:
Theorem 1.1 Let f : R+→ R+ satisfy each of the following properties:
(a) log f (x) is a convex function;
(b) f (x+ 1)= x f (x) for all x ∈ R+;
(c) f (1)= 1.
Then f (x)= 0(x) for all x ∈ R+.
Instead of giving here the proof of Theorem 1.1 (see Conway [339, p. 179] and
Artin [72, p. 14]), we simply state the necessary and sufficient condition for the loga-
rithmic convexity of a given function.
Theorem 1.2 Let f : [a,b]→ R, and suppose that f (x) > 0 for all x ∈ [a,b] and that
f has a continuous second derivative f ′′(x) for x ∈ [a,b]. Then f is logarithmically
convex, if and only if
f ′′(x) f (x)−
[
f ′(x)
]2
= 0 (x ∈ [a,b]).
Remmert [973] admires the following Wielandt’s uniqueness theorem for the
Gamma function: It is hardly known that there is also an elegant function theo-
retic characterization of 0(z). This uniqueness theorem was discovered by Helmut
Wielandt in 1939. A function theorist ought to be as much fascinated by Wielandt’s
complex-analytic characterization as by the Bohr-Mollerup theorem. For further com-
ment and applications for Wielandt’s theorem, see [675, pp. 47–49], [973], and [1065].
Here, without proof, we present
Theorem 1.3 (Wielandt’s Theorem) Let F(z) be an analytic function in the right
half plane A := {z ∈ C | <(z) > 0} having the following two properties:
(a) F(z+ 1)= zF(z) for all z ∈A;
(b) F(z) is bounded in the strip S := {z ∈ C | 1 5 <(z) < 2}.
Then F(z)= a0(z) in A with a := F(1).
Introduction and Preliminaries 13
1.2 The Euler-Mascheroni Constant γ
The divergence of the harmonic series:
∞∑
k=1
1
k
= 1+
1
2
+
1
3
+ ·· · = lim
n→∞
n∑
k=1
1
k
=∞ (1)
was attributed by James Bernoulli to his brother (see [482]). Yet, the connection
between 1+ 12 + ·· ·+
1
n and logn was first established in 1735 by Euler [427] (see
Walfisz [1203]), who used the notation C for it and stated that it was worthy of seri-
ous consideration. We are surprised at Euler’s foresight that there is a huge amount of
literature on this famous mathematical constant γ among which we just refer to the
book [543] and the references therein.
The Euler (or, more precisely, the Euler-Mascheroni) constant γ is defined as fol-
lows (see Eq. 1.1(3)):
γ := lim
n→∞
(Hn− logn)
∼= 0.577215664901532860606512090082402431042 · · · ,
(2)
where Hn are called harmonic numbers defined by
Hn :=
n∑
k=1
1
k
(n ∈ N). (3)
The symbol γ was first used by Mascheroni in1790 (see [802]) and the notation C
has still been used for the notation γ in (2) (see, e.g., [505]). In fact, an := Hn− logn
(n ∈ N) is a decreasing sequence and 0< an < 1 for all n ∈ N. Thus, the convergence
of the sequence in (2) follows by the monotone convergence theorem (see, e.g., [1202,
p. 45]).
It is noted that γ is a constant so chosen that 0(1)= 1 in the Weierstrass product
form 1.1(2) of the Gamma function 0(z), and the constant γ is the very Euler constant
in (2).
The relatively more familiar constants are π and e, whose transcendence was
shown by Ferdinand Lindemann in 1882 and Charles Hermite in 1873, respectively.
The true nature of the Euler constant γ (whether an algebraic or a transcendental num-
ber) has not yet been known. This was a part of the famous Hilbert’s seventh problem.
David Hilbert [557] announced 23 problems for the twentieth century in the Second
International Congress of Mathematicians at Paris in 1900. We introduce here only his
seventh problem: Irrationality and Transcendence of Certain Numbers. G. H. Hardy
was alleged to have offered to give up his Savilian Chair at Oxford to anyone who
proved γ to be irrational (see [543, p. 52]). The degree of possible rationality of γ
has been tried (see [1222], [543, p. 97]). Appell [69] gave a proof that γ is irrational.
Appell himself’s finding an error, quickly he published a retraction, within a couple of
weeks, of his original announcement (see Ayoub [82]).
14 Zeta and q-Zeta Functions and Associated Series and Integrals
Euler [427] gave the formula
1+
1
2
+ ·· ·+
1
x
= γ + logx+
1
2x
−
B2
2x2
−
B4
4x4
−
B6
6x6
+ ·· · ,
Bn being Bernoulli numbers, in which, by putting x= 10, he calculated
γ = 0.57721 56649 01532 5 · · · .
Mascheroni [802] evaluated the value of Euler constant with 32 figures as follows:
γ = 0.57721 56649 01532 86061 811209008239 · · ·
Soldner (see [482]) computed the value of γ as
γ = 0.57721 56649 01532 86060 6065 · · · ,
which differs from Mascheroni’s value in the twentieth place. In fact, Mascheroni’s
value turned out to be incorrect. Nonetheless, since Mascheroni’s error has led to eight
additional calculations of this celebrated mathematical constant, so γ is often called
the Euler-Mascheroni constant. Knuth [678] computed the first 1271 decimals. Gour-
don and Demichel [503] computed a record 108 million digits of γ . Kondo [693] has
computed γ to 2 billion digits, which is apparently the current world record.
The Euler (or, more popularly, the Euler-Mascheroni) constant γ is considered the
third important mathematical constant next to π and e. The mathematical constants
π , e and γ are often referred to as the holy trinity. The constant γ has been involved
in a variety of mathematical formulas and results. For instance, the book [505] con-
tains about 160 formulas involving γ . Conversely, Wilf [1228] posed as a problem
the following elegant infinite product formula, which contains all of the three most
important mathematical constants π , e and γ (see Eq. 3.6(19)):
∞∏
j=1
[
e−
1
j
(
1+
1
j
+
1
2 j2
)]
=
e
π
2 + e−
π
2
π eγ
. (4)
Choi et al. [275] presented several general infinite product formulas, which include, as
their special cases, the product formula (4) of Wilf [1228] and other product formulas
given by Choi and Seo [285]. The function d(n) is the number of divisors of n ∈ N,
including 1 and n. The average order of d(n) was proved by Dirichlet in 1838 that
1
n
n∑
k=1
d(k)= lnn+ 2γ − 1+O
(
1
√
n
)
(n→∞), (5)
where γ is the Euler-Mascheroni constant (see [538, p. 264,Theorem 320]). Very
recently, Pillichshammer [893] treated this subject in a more general way. A lot of
Introduction and Preliminaries 15
series representations of γ have been presented (see, e.g., [12, 208, 264, 274, 286,
480, 531, 963, 964, 1094, 1179]), such as (see Eq. 3.4(23))
γ =
∞∑
k=2
(−1)k
ζ(k)
k
, (6)
where ζ(s) is the Riemann Zeta function given in 2.3(1).
Sondow [1051] presented an elegant double integral representation:
γ =
1∫
0
1∫
0
x− 1
(1− xy) ln(xy)
dxdy. (7)
In fact, many integral representations of γ have been developed. Here, in what follows,
several further integral representations of γ with mild generalizations of some known
formulas of γ are presented.
A Set of Known Integral Representations for γ
Among a variety of known integral representations for γ , here we choose to recall
some of them (see, e.g., [53, 54, 220, 264, 284, 365, 505, 1094, 1225]):
γ =
1∫
0
1− e−x
x
dx−
∞∫
1
e−x
x
dx. (8)
γ =
∞∫
0
e−x
(
1
1− e−x
−
1
x
)
dx. (9)
γ = 1+
∞∫
0
(
e−x− 1
x
+
1
1+ x
)
dx
x
. (10)
γ =
1∫
0
(
1− e−x− e−
1
x
x
)
dx. (11)
γ =
∞∫
0
(
1
1+ x
− e−x
)
dx
x
. (12)
γ = 2
∞∫
0
(
1
1+ x2
− e−x
2
)
dx
x
. (13)
16 Zeta and q-Zeta Functions and Associated Series and Integrals
γ =
∞∫
0
(
1
1+ x2
− e−x
)
dx
x
. (14)
γ = 2
∞∫
0
(
e−x
2
− e−x
)dx
x
. (15)
γ =
4
3
∞∫
0
(
e−x
4
− e−x
)dx
x
. (16)
γ = 4
∞∫
0
(
e−x
4
− e−x
2
)dx
x
. (17)
γ =
1
1− 2−n
∞∫
0
[
exp
(
−x2
n
)
− e−x
] dx
x
(n ∈ N). (18)
γ = 2n
∞∫
0
(
1
1+ x2n+1
− e−x
2
)
dx
x
(n ∈ Z), (19)
where Z denotes the set of integers.
γ = 2n
∞∫
0
(
1
1+ x2
− exp
(
−x2
n
))dx
x
(n ∈ N). (20)
γ =
∞∫
0
(
1
1+ x
− cos x
)
dx
x
. (21)
γ =
∞∫
0
(
1
1+ x2
− cos x
)
dx
x
. (22)
γ = 1+
∞∫
0
(
1
1+ x
−
sin x
x
)
dx
x
. (23)
γ = 2
∞∫
0
(
e−x
2
− cos x
)dx
x
. (24)
γ =− lnp+
∞∫
0
(
2
π
arccotx− e−px
)
dx
x
(p> 0). (25)
Introduction and Preliminaries 17
γ =
3
2
+ 2
∞∫
0
(
cos x− 1
x2
+
1
2(1+ x)
)
dx
x
. (26)
γ =
π/2∫
0
[
1− sec 2x cos (tanx)
] dx
tanx
. (27)
γ =− lnp−
2
π
∞∫
0
sin (px) lnx
dx
x
(p> 0). (28)
γ = 1− ln(2p)−
2
pπ
∞∫
0
(
sin 2(px)
x2
)
lnxdx (p> 0). (29)
γ = 1+
1
p− q
ln
(
qq
pp
)
+
2
π(p− q)
∞∫
0
(
cos (px)− cos (qx)
x2
)
lnxdx
(p> 0; q> 0; p 6= q).
(30)
γ =
1
2
+ 2
∞∫
0
x
1+ x2
dx
e2πx− 1
. (31)
γ =
1∫
0
(
x−
1
1− logx
)
dx
x logx
. (32)
γ =−
∞∫
0
e−x logxdx=−
1∫
0
log
(
log
1
x
)
dx. (33)
γ = log2−π
1∫
0
1
2∫
0
tan
(πx
2
) ( sin πxu
sin πu
− x
)
dudx. (34)
γ =
1
2
+ 2
∞∫
0
sin (tan−1 x)(
e2πx− 1
)√
1+ x2
dx. (35)
γ = log2− 2
∞∫
0
sin (tan−1 x)(
e2πx+ 1
)√
1+ x2
dx. (36)
γ = 1+
∞∫
0
(
cos x−
sin x
x
)
logx
x
dx. (37)
18 Zeta and q-Zeta Functions and Associated Series and Integrals
γ =
∞∫
0
(
1
2
− cos x
)
dx
x
. (38)
γ =
1
2
+
B2
2
+
B4
4
+ ·· ·+
B2n
2n
− (2n+ 1)!
∞∫
1
Q2n+1(x)
x2n+2
dx, (39)
where the functions Qn(x) are defined by
Qn(x) :=

x−
1
2
(n= 1; 0< x< 1),
1
n!
Bn(x− [x]) (n ∈ N \ {1}; 0 5 x<∞),
Bn := Bn(0) and Bn(x) being the Bernoulli numbers and polynomials, respectively
(see [1094, Section 1.6]). As observed by Knopp [676] by an explicit example with
n= 3 in (2.61), the approximate value of γ can easily be calculated with much greater
accuracy than before (and, theoretically, to any degree of accuracy whatever) by means
of the formula (39).
Further Integral Representations for γ
Very recently, Choi and Srivastava [302] presented several further integral rep-
resentations for γ by making use of some formulas in the previous subsection
and other known formulas for log0(z), ψ(z) (Section 1.3) and the Hurwitz (or
generalized) Zeta function ζ(s,a) (Section 2.2) or the Riemann Zeta function
ζ(s) (Section 2.3) in conjunction with the residue calculus. Here, we choose to
record some of them: We begin by recalling an integral formula for logz (see
[1225, p. 248]).
The following integral formula holds true for logz:
∞∫
0
(
e−t− e−t z
) dt
t
=
1∫
0
(
tz−1− 1
) dt
log t
= logz
(
<(z) > 0
)
, (40)
where the logz is an appropriate branch of the multiple-valued function logz,
such as
logz= ln |z| + i argz
(
|z|> 0; α < arg(z) < α+ 2π
)
for some real α ∈ R with possibly −π 5 α 5−π2 .
Introduction and Preliminaries 19
If (40) is used in the formulas (25), (28), (29) and (30) and tanx is replaced by x in
(27), the following integral formulas for γ are obtained:Each of the following integral representations holds true for γ :
γ =
∞∫
0
(
2
π
arccotx− e−x
)
dx
x
, (41)
γ =−
2
π
∞∫
0
[
e−x− e−p
π
2 x
+ lnx sin (px)
]
dx
x
(p> 0), (42)
γ = 1−
2
pπ
∞∫
0
(
e−x− exp
[
−(2p)
pπ
2 x
]
+ lnx
sin 2(px)
x
)
dx
x
(p> 0), (43)
γ = 1+
2
π(p− q)
∞∫
0
(
e−x− exp
[
−
(
qq
pp
) π
2
x
]
+ lnx
cos (px)− cos (qx)
x
)
dx
x
(p> 0; q> 0) (44)
and
γ =
∞∫
0
[
cos 2(arctanx)− cos x
] dx
x
. (45)
If x is replaced by xp in (10), (12), (21), (23) and (26), the following mildly more
general formulas for γ are obtained.
Each of the following integral representations holds true for γ :
γ = p
∞∫
0
(
1
1+ xp
− exp
(
−xp
)) dx
x
(p> 0), (46)
γ = p
∞∫
0
(
1
1+ xp
− cos
(
xp
)) dx
x
(p> 0), (47)
γ = 1+ p
∞∫
0
(
1
1+ xp
+
exp(−xp)− 1
xp
)
dx
x
(p> 0), (48)
γ = 1+ p
∞∫
0
(
1
1+ xp
−
sin (xp)
xp
)
dx
x
(p> 0) (49)
20 Zeta and q-Zeta Functions and Associated Series and Integrals
and
γ =
3
2
+ 2p
∞∫
0
(
cos (xp)− 1
x2p
+
1
2(1+ xp)
)
dx
x
(p> 0). (50)
It is noted that the case p= 2 of (46) would obviously reduce to (13).
A class of vanishing integrals is provided just below.
The following vanishing integral formula holds true:
∞∫
0
(
1
1+ xp
−
1
1+ xq
)
dx
x
=
∞∫
0
(
xq−1− xp−1
(1+ xp)(1+ xq)
)
dx= 0 (51)
(p> 0; q> 0).
Proof. The integral (51) is separated into two parts as follows:
∞∫
0
(
xq−1− xp−1
(1+ xp)(1+ xq)
)
dx=
 1∫
0
+
∞∫
1
 ( xq−1− xp−1
(1+ xp)(1+ xq)
)
dx,
which, upon replacing x by 1/x in the second integral, is seen to vanish to 0. �
Applying Eq. (51) to Eqs. (46)–(50), Choi and Srivastava [302] derived much more
general integral representations for γ , which are recorded here.
γ = p
∞∫
0
(
1
1+ xq
− exp
(
−xp
))dx
x
(p> 0; q> 0), (52)
γ = p
∞∫
0
(
1
1+ xq
− cos
(
xp
))dx
x
(p> 0; q> 0), (53)
γ = 1+ p
∞∫
0
(
1
1+ xq
+
exp(−xp)− 1
xp
)
dx
x
(p> 0; q> 0), (54)
γ = 1+ p
∞∫
0
(
1
1+ xq
−
sin (xp)
xp
)
dx
x
(p> 0; q> 0), (55)
γ =
3
2
+ 2p
∞∫
0
(
cos (xp)− 1
x2p
+
1
2(1+ xq)
)
dx
x
(p> 0; q> 0), (56)
γ =
pq
q− p
∞∫
0
[
exp
(
−xq
)
− exp
(
−xp
)]dx
x
(p> 0; q> 0; p 6= q), (57)
Introduction and Preliminaries 21
γ =
pq
q− p
∞∫
0
[
cos
(
xq
)
− cos
(
xp
)] dx
x
(p> 0; q> 0; p 6= q), (58)
γ = 1+
pq
q− p
∞∫
0
(
sin (xq)
xq
−
sin (xp)
xp
)
dx
x
(p> 0; q> 0; p 6= q), (59)
γ =
3
2
+
2pq
q− p
∞∫
0
(
cos (xp)− 1
x2p
−
cos (xq)− 1
x2q
)
dx
x
(p> 0; q> 0; p 6= q),
(60)
γ =
pq
q− p
∞∫
0
[
cos (xp)− exp(−xp)
] dx
x
(p> 0; q> 0; p 6= q), (61)
γ =
p
p− q
+
pq
p− q
∞∫
0
(
exp(−xp)−
sin (xp)
xp
)
dx
x
(p> 0; q> 0; p 6= q),
(62)
γ =
3p
2(p− q)
+
pq
p− q
∞∫
0
(
exp(−xp)+
2[cos (xp)− 1]
x2q
)
dx
x
(p> 0; q> 0; p 6= q),
(63)
γ =
p
p− q
+
pq
p− q
∞∫
0
(
cos (xp)−
sin (xp)
xp
)
dx
x
(p> 0; q> 0; p 6= q), (64)
γ =
3p
2(p− q)
+
pq
p− q
∞∫
0
(
cos (xp)+
2[cos (xp)− 1]
x2q
)
dx
x
(p> 0; q> 0; p 6= q),
(65)
γ =
3p− 2q
2(p− q)
+
pq
p− q
∞∫
0
(
sin (xp)
xp
+
2[cos (xp)− 1]
x2q
)
dx
x
(p> 0; q> 0; p 6= q),
(66)
γ =
p
p− q
+
pq
p− q
∞∫
0
(
exp(−xp)− 1
xp
+ exp(−xp)
)
dx
x
(p> 0; q> 0; p 6= q),
(67)
22 Zeta and q-Zeta Functions and Associated Series and Integrals
γ =
p
p− q
+
pq
p− q
∞∫
0
(
exp(−xp)− 1
xp
+ cos
(
xp
)) dx
x
(p> 0; q> 0; p 6= q),
(68)
γ = 1+
pq
p− q
∞∫
0
(
exp(−xp)− 1
xp
+
sin (xp)
xp
)
dx
x
(p> 0; q> 0; p 6= q),
(69)
γ =
2p− 3q
2(p− q)
+
pq
p− q
∞∫
0
(
exp(−xp)− 1
xp
+
2[1− cos (xp)]
x2p
)
dx
x
(p> 0; q> 0; p 6= q).
(70)
It is noted that the integral formula (57) is recorded in [505, p. 364, Entry 3.476-2]
and many (if not all) of the integral formulas in the previous subsection can be seen to
be special cases of the corresponding integral formulas asserted in this subsection.
From an Application of the Residue Calculus
Consider a function f (z) given by
f (z)=
1
z
(
1
1+ zn
− ei z
n
)
(n ∈ N).
Since
lim
z→0
f (z)=
−1− i (n= 1)0 (n ∈ N \ {1}),
the function f (z) has a removable singularity at z= 0 and simple poles at
z= exp
(
(2k+ 1)π i
n
)
(k = 0, 1, . . . , n− 1).
We now consider a counterclockwise-oriented simple closed contour:
C := Cδ ∪L1 ∪CR ∪L2 (0< δ < 1< R),
where
Cδ : z= δ e
iθ
(
θ varies from
π
2n
to 0
)
,
Introduction and Preliminaries 23
L1 a line segment from δ to R on the positive real axis,
CR : z= Re
iθ
(
θ varies from 0 to
π
2n
)
and
L2 : z= xexp
(
iπ
2n
)
(x varies from R to δ) ,
that is, a line segment on the half-line beginning at the origin with the argument π2n .
Since f (z) is analytic throughout the domain interior to and on the closed contour C,
it follows from the Cauchy-Goursat theorem that∫
Cδ
+
∫
L1
+
∫
CR
+
∫
L2
 f (z)dz= 0,
which, upon taking the limits as
δ→ 0+ and R→∞
and equating the real and imaginary parts of the last resulting equation, yields the
following two interesting integral identities:
∞∫
0
(
1
1+ xn
− cos
(
xn
)) dx
x
=
∞∫
0
(
1
1+ x2n
− exp
(
−xn
))dx
x
(n ∈ N) (71)
and
∞∫
0
xn−1
1+ x2n
dx=
∞∫
0
sin (xn)
x
dx=
π
2n
(n ∈ N). (72)
It is noted that the integral identity (71) is a special case of (52) or (53). Moreover,
(72) can be evaluated, as above, by applying the residue calculus to another function
f(z)=
exp(izn)
z
(n ∈ N)
and a counterclockwise-oriented simple closed contour
C := Cδ ∪L1 ∪CR ∪L2 (0< δ < R),
24 Zeta and q-Zeta Functions and Associated Series and Integrals
where Cδ and L1 are the same as above,
CR : z= Re
iθ
(
θ varies from 0 to
π
n
)
and
L2 : z= xexp
(
iπ
n
)
(x varies from R to δ).
We conclude this section by remarking that more integral representations for γ
can be obtained by applying the same techniques employed here (see [302]) or other
methods (if any) to some other known formulas that have not been used (see [572]).
1.3 Polygamma Functions
The Psi (or Digamma) Function
The Psi (or Digamma) function ψ(z) defined by
ψ(z) :=
d
dz
{log 0(z)} =
0′(z)
0(z)
or log 0(z)=
z∫
1
ψ(t)dt (1)
possesses the following properties:
ψ(z)= lim
n→∞
(
logn−
n∑
k=0
1
z+ k
)
;
ψ(z)=−γ −
1
z
+
∞∑
n=1
z
n(z+ n)
(2)
=−γ + (z− 1)
∞∑
n=0
1
(n+ 1)(z+ n)
, (3)
where γ is the Euler-Mascheroni constant defined by 1.1(3) (or 1.2(2)).
These results clearly imply that ψ(z) is meromorphic (that is, analytic every-
where in the bounded complex z–plane, except for poles) with simple poles at
z=−n(n ∈ N0) with its residue −1. Also we have
ψ(1)=−γ, (4)
which follows at once from (3). It is noted that, very recently, Bagby [83] proved (4)
in another way.
Introduction and Preliminaries 25
The following additional properties of ψ(z) can be deduced from known results
for 0(z) :
log
0(z+ 1)
0(z)
= logz=−γ +
∞∑
n=0
[
1
n+ 1
− log
(
1+
1
n+ z
)]
; (5)
ψ(z)= logz−
∞∑
n=0
[
1
n+ z
− log
(
1+
1
n+ z
)]
, (6)
which follows from (3) and (5);
ψ(z+ n)= ψ(z)+
n∑
k=1
1
z+ k− 1
(n ∈ N); (7)
ψ(z)−ψ(−z)=−πcot πz−
1
z
; (8)
ψ(1+ z)−ψ(1− z)=
1
z
−πcot πz; (9)
ψ(z)−ψ(1− z)=−πcot πz; (10)
ψ
(
1
2
+ z
)
−ψ
(
1
2
− z
)
= π tanπz; (11)
ψ(mz)= logm+
1
m
m−1∑
k=0
ψ
(
z+
k
m
)
(m ∈ N). (12)
Integral Representations for ψ(z)
Expanding (1− t)−1 into a series, integrating term by term and using (3), we get
ψ(z)=−γ +
1∫
0
(
1− tz−1
)
(1− t)−1 dt (<(z) > 0), (13)
which, upon replacing t by e−t, yields
ψ(z)=−γ +
∞∫
0
(
e−t− e−tz
)(
1− e−t
)−1
dt (<(z) > 0). (14)
Making use of (10), it follows from (13) and (14) that
ψ(z)=−γ −πcot πz+
1∫
0
(
1− t−z
)
(1− t)−1 dt (<(z) < 1) (15)
26 Zeta and q-Zeta Functions and Associated Series and Integrals
and
ψ(z)=−γ −πcot πz+
∞∫
0
(
1− etz
) (
et− 1
)−1
dt (<(z) < 1). (16)
It is not difficult to derive each of the following integral representations for ψ(z).
ψ(z)=
∞∫
0
[
t−1e−t−
(
1− e−t
)−1
e−tz
]
dt (<(z) > 0), (17)
which is due to Gauss and reduces, when z= 1, to the integral expression 1.2(9) for
the Euler-Mascheroni constant γ ;
ψ(z)=
∞∫
0
[
e−t− (1+ t)−z
]t−1 dt (<(z) > 0), (18)
which is due to Dirichlet and reduces, when z= 1, to 1.2(11);
ψ(z)=−γ +
∞∫
0
[
(1+ t)−1− (1+ t)−z
]
t−1 dt (<(z) > 0); (19)
ψ(z)= logz+
∞∫
0
[
t−1−
(
1− e−t
)−1]
e−tz dt (<(z) > 0), (20)
ψ(z)= logz−
1
2z
−
∞∫
0
[(
1− e−t
)−1
− t−1−
1
2
]
e−tz (<(z) > 0), (21)
ψ(z)= logz+
∞∫
0
[(
1− e−t
)−1
+ t−1− 1
]
e−tz dt (<(z) > 0), (22)
and
ψ(z)= logz−
1
2z
−
∞∫
0
[(
et− 1
)−1
− t−1+
1
2
]
e−tz dt (<(z) > 0), (23)
all four being attributed to Binet;
ψ(z)= logz−
1
2z
− 2
∞∫
0
(
t2+ z2
)−1(
e2π t− 1
)−1
t dt (<(z) > 0), (24)
which, in the special case when z= 1, yields 1.2(31).
Introduction and Preliminaries 27
Gauss’s integral representation (17) also yields Malmstén’s formula:
log0(z)=
z∫
1
ψ(t)dt
=
∞∫
0
[
(z− 1)−
1− e−(z−1)t
1− e−t
]
e−t
t
dt (<(z) > 0).
(25)
From (23), we can deduce that
log0(z)=
(
z−
1
2
)
logz− z+ 1+
∞∫
0
(
1
2
−
1
t
+
1
et− 1
)
e−tz
t
dt
−
∞∫
0
(
1
2
−
1
t
+
1
et− 1
)
e−t
t
dt (<(z) > 0).
(26)
The second integral in (26) can readily be evaluated:
∞∫
0
(
1
2
−
1
t
+
1
et− 1
)
e−t
t
dt = 1−
1
2
log(2π) (27)
by using an elementary (yet technical) separation and recalling a formula for log z:
logz=
∞∫
0
(
e−t− e−zt
) dt
t
(<(z) > 0). (28)
Now, putting (27) into (26) yields Binet’s first expression for log 0(z):
log 0(z)=
(
z−
1
2
)
logz− z+
1
2
log(2π)
+
∞∫
0
(
1
2
−
1
t
+
1
et− 1
)
e−zt
t
dt (<(z) > 0),
(29)
which may also give an approximate expression for log 0(z), just as the special case
of 1.1(34), when n= 0.
28 Zeta and q-Zeta Functions and Associated Series and Integrals
Binet’s second expression for log0(z) may be written as follows:
log0(z)=
(
z−
1
2
)
logz− z+
1
2
log(2π)
+ 2
∞∫
0
arctan(t/z)
e2π t− 1
dt (<(z) > 0).
(30)
From 1.1(12) and (25), we obtain Kummer’s expression for log0(z):
log0(z)=
1
2
logπ −
1
2
log(sin πz)
(31)
+
1
2
∞∫
0
 sinh
[(
1
2 − z
)
t
]
sinh
(
1
2 t
) − (1− 2z)e−t
 dt
t
(0< <(z) < 1).
By applying (31), Kummer derived the following Fourier series for log0(x):
log0(x)=
1
2
logπ −
1
2
log(sin πx)+ 2
∞∑
n=1
an sin(2nπx) (0< x< 1), (32)
where
an =
∞∫
0
(
2nπ
t2+ 4n2π2
−
e−t
2nπ
)
dt
t
=
1
2nπ
 ∞∫
0
(
1
1+ t2
− cos t
)
dt
t
+
∞∫
0
e−t− e−2πnt
t
dt+
∞∫
0
(
cos t− e−t
) dt
t

=
1
2nπ
(γ + log2π + logn). (33)
Since
log(sin πx)=− log2−
∞∑
n=1
1
n
cos(2πnx) (0< x< 1), (34)
by rewriting (32) in the form:
log0(x)=
(
1
2
− x
)
(γ + log2)+ (1− x) logπ −
1
2
log(sin πx)
+
∞∑
n=1
logn
πn
sin(2πnx) (0< x< 1), (35)
Introduction and Preliminaries 29
we finally obtain
log0(x)=
1
2
log(2π)
+
∞∑
n=1
(
1
2n
cos(2πnx)+
γ + log(2πn)
πn
sin(2πnx)
)
(0< x< 1).
(36)
The Fourier series (36) readily implies each of the integral formulas:
1∫
0
sin(2πnx) log0(x)dx=
γ + log(2πn)
2πn
(n ∈ N), (37)
1∫
0
cos(2πnx)log0(x)dx=
1
4n
(n ∈ N), (38)
and
1∫
0
log0(x)dx=
1
2
log(2π). (39)
Considering 1.1(31) and 1.1(34), when n= 0, we readily obtain a more general
integral formula than (39) above:
x+1∫
x
log0(t)dt = x logx− x+
1
2
log(2π) (x = 0), (40)
by observing the following relation:
lim
m→∞
m−1∑
k=0
1
m
log0
(
x+
k
m
)
=
1∫
0
log0(x+ u)du=
x+1∫
x
log0(t)dt.
The formula (40) can be extended, without difficulty, to the case, when x ∈ C \Z−(
Z− := {−1,−2, . . .}
)
and the principal values of the involved logarithms are taken,
and can also be further generalized (by iteration) as follows:
x+n∫
x
log0(t)dt =
n−1∑
k=0
(x+ k) log(x+ k)− nx−
1
2
n(n− 1)
(41)
+
1
2
n log(2π) (n ∈ N).
30 Zeta and q-Zeta Functions and Associated Series and Integrals
Gauss’s Formulas for ψ
(
p
q
)
Taking z= pq (0< p< q; p, q ∈ N) in (13) and t = x
p, we obtain
γ +ψ
(
p
q
)
= q
1∫
0
xq−1− xp−1
1− xp
dx, (42)
which, upon noting that
q
xp−1− xq−1
xp− 1
=
q−1∑
k=1
ω
p
k − 1
x−ωk
(
ωk := exp
(
2π ik
q
))
, (43)
readily yields
γ +ψ
(
p
q
)
=
π
2q
q−1∑
k=1
(2k− q) sin
2kpπ
q
−
q−1∑
k=1
(
1− cos
2kpπ
q
)
log
(
2sin
kπ
q
)
.
(44)
By applying the known trigonometric identities:
q−1∑
k=0
cos2kα =
1
2
+
sin (2q− 1)α
2sin α
(45)
and
q−1∑
k=1
sin 2kα =
cosα− cos(2q− 1)α
2sinα
(46)
in (44), we obtain Gauss’s formula:
ψ
(
p
q
)
=− γ −
π
2
cot
pπ
q
− logq+
q−1∑
k=1
cos
2kpπ
q
log
(
2sin
kπ
q
)
(0< p< q; p, q ∈ N),
(47)
which implies that, for a positive proper fraction z, the value of ψ(z) can be expressed
as a finite combination of elementary functions, and, yet, by means of 1.1(10), may be
extended to every rational value of z.
Introduction and Preliminaries 31
By reversing the order of summation, Gauss’s formula (47) can be rewritten in the
following form:
ψ
(
p
q
)
=− γ −
π
2
cot
pπ
q
− logq
+ 2
[(q−1)/2]∑
k=1
cos
(
2kpπ
q
)
log
(
2sin
kπ
q
)
+ rq(p),
(48)
where [x] denotes the greatest integer ≤ x, and
rq(p)=
{
(−1)p log2 if q is even,
0 if q is odd.
If we rewrite (42) as
γ − q
1∫
0
xq−1
1− xp
dx=−ψ
(
p
q
)
− q
1∫
0
xp−1
1− xp
dx,
where the left-hand side is independent of p and only the right-hand side is dependent
on p, we can derive Gauss’s second formula:
q∑
p=1
ψ
(
p
q
)
exp
(
2pkπ i
q
)
= q log
[
1− exp
(
2kπ i
q
)]
(q ∈ N; k ∈ Z). (49)
Special Values of ψ(z)
Setting z= 1 and z= 12 in (7), we obtain
ψ(n)=−γ +
n−1∑
k=1
1
k
(n ∈ N) (50)
and
ψ
(
n+
1
2
)
=−γ − 2log2+ 2
n−1∑
k=0
1
2k+ 1
(n ∈ N0), (51)
it being understood (here as well as throughout this book) that an empty sum is nil.
By suitably applying the various formulas for theψ–function, we [1094, pp. 20–22]
could derive the following special values of ψ(z), some of which were corrected and
32 Zeta and q-Zeta Functions and Associated Series and Integrals
simplified by Tee [1146] as follows:
ψ
(
1
2
)
=−γ − 2log2;
ψ
(
1
3
)
=−γ −
1
6
π
√
3−
3
2
log3;
ψ
(
2
3
)
=−γ +
1
6
π
√
3−
3
2
log3;
ψ
(
1
4
)
=−γ −
1
2
π − 3log2;
ψ
(
3
4
)
=−γ +
1
2
π − 3log2;
ψ
(
1
5
)
=−γ −
π
2
√
1+
2
5
√
5−
5
4
log5−
√
5
2
log
1+
√
5
2
;
ψ
(
2
5
)
=−γ −
π
2
√
1−
2
5
√
5−
5
4
log5+
√
5
2
log
1+
√
5
2
;
ψ
(
3
5
)
=−γ +
π
2
√
1−
2
5
√
5−
5
4
log5+
√
5
2
log
1+
√
5
2
;
ψ
(
4
5
)
=−γ +
π
2
√
1+
2
5
√
5−
5
4
log5−
√
5
2
log
1+
√
5
2
;
ψ
(
1
6
)
=−γ −
1
2
π
√
3−
3
2
log3− 2log2;
ψ
(
5
6
)
=−γ +
1
2
π
√
3−
3
2
log3− 2log2;
ψ
(
1
8
)
=−γ −
π
2
(
√
2+ 1)− 4log2−
√
2 log(
√
2+ 1);
ψ
(
3
8
)
=−γ −
π
2
(
√
2− 1)− 4log2+
√
2 log(
√
2+ 1);
ψ
(
5
8
)
=−γ +
π
2
(
√
2− 1)− 4log2+
√
2 log(
√
2+ 1);
ψ
(
7
8
)
=−γ +
π
2
(
√
2+ 1)− 4log2−
√
2 log(
√
2+ 1);
ψ
(
1
10
)
=−γ −
π
2
√
5+ 2
√
5− 2log2−
5
4
log5−
√
5
2
log
(
2+
√
5
)
;
ψ
(
3
10
)
=−γ −
π
2
√
5− 2
√
5− 2log2−
5
4
log5+
√
5
2
log
(
2+
√
5
)
;
ψ
(
7
10
)
=−γ +
π
2
√
5− 2
√
5− 2log2−
5
4
log5+
√
5
2
log
(
2+
√
5
)
;
Introduction and Preliminaries 33
ψ
(
9
10
)
=−γ +
π
2
√
5+ 2
√
5− 2log2−
5
4
log5−
√
5
2
log
(
2+
√
5
)
;
ψ
(
1
12
)
=−γ −
π
2
(2+
√
3)−
√
3log(2+
√
3)−
3
2
log3− 3log2;
ψ
(
5
12
)
=−γ −
π
2
(2−
√
3)+
√
3log(2+
√
3)−
3
2
log3− 3log2;
ψ
(
7
12
)
=−γ +
π
2
(2−
√
3)+
√
3log(2+
√
3)−
3
2
log3− 3log2;
ψ
(
11
12
)
=−γ +
π
2
(2+
√
3)−
√
3log(2+
√
3)−
3
2
log3− 3log2;
ψ
(
−
1
2
)
= 2− γ − 2log2;
ψ
(
−
1
3
)
= 3− γ +
√
3
6
π −
3
2
log3;
ψ
(
−
5
6
)
=
6
5
− γ −
√
3
2
π −
3
2
log3− 2log2.
The Polygamma Functions
The Polygamma functions ψ (n)(z) (n ∈ N) are defined by
ψ (n)(z) :=
dn+1
dzn+1
log0(z)=
dn
dzn
ψ(z) (n ∈ N0; z 6∈ Z−0 ). (52)
In terms of the generalized (or Hurwitz) Zeta function ζ(s,a) (see Section 2.2), we
can write
ψ (n)(z)= (−1)n+1 n!
∞∑
k=0
1
(k+ z)n+1
= (−1)n+1 n!ζ(n+ 1,z) (n ∈ N; z 6∈ Z−0 ),
(53)
which may be used to deduce the properties of ψ (n)(z) (n ∈ N) from those of ζ(s,z)
(s= n+ 1; n ∈ N).
It is also easy to have the following expressions:
ψ(n)(z+m)−ψ (n)(z)= (−1)n n!
m∑
k=1
1
(z+ k− 1)n+1
(m, n ∈ N0) (54)
and
ψ (n)(z)− (−1)nψ (n)(1− z)=−π
dn
dzn
{cot πz} (n ∈ N0), (55)
34 Zeta and q-Zeta Functions and Associated Series and Integrals
it being understood (as usual) that
ψ (0)(z)= ψ(z).
By using Gauss’s multiplication formula for 0(z) in 1.1(31), it is easy to get the
multiplication formula for ψ(n)(z):
ψ(n) (mz)=
m∑
j=1
ψ(n)
(
z+
j− 1
m
)
(n, m ∈ N). (56)
Davis [368] extended Gauss’s result (47) (or (48)) to the Polygamma functions
ψ(n)(z) (n ∈ N), by using a known series representation of ψ(n)(z) in an elementary
(yet technical) way. Kölbig [687, 688], in his CERN technical report, also gave two
extensions of Gauss’s result to ψ(n)(z), by using the series definition of Polyloga-
rithm function and the above-known series representation of ψ(n)(z). Recently, Choi
and Cvijović [271] presented a unified formula of ψ(n)(p/q) (p,q,n ∈ N;1 5 p< q),
which can be specialized in those formulas of Davis [368] and Kölbig [687, 688],
expressed in terms of the Bernoulli polynomials given in 1.7 and the generalized Zeta
functions in 2.2, which was shown in the following two ways:
ψ(n)
(
p
q
)
=(−1)n+1 n!qn
·
q−1∑
s=0
[
En(s;p ;q)(−1)
1+
[
1
2 (n+1)
]
(2π)n+1
2 ·(n+ 1)!
Bn+1
(
s
q
)
+
1
qn+1
Fn(s;p ;q)
q∑
k=1
ζ
(
n+ 1,
k
q
)
En+1(k;s ;q)
]
(p, q, n ∈ N; 1 5 p< q),
(57)
where [x] denotes the greatest integer 5 x, and
En(s;p ;q) :=
1+ (−1)n
2
sin
(
2πsp
q
)
+
1− (−1)n
2
cos
(
2πsp
q
)
and
Fn(s;p ;q) :=
1+ (−1)n
2
cos
(
2πsp
q
)
+
1− (−1)n
2
sin
(
2πsp
q
)
.
Special Values of ψ(n)(z)
Since ψ(n)
(
p
q
)
in (57) is expressed in terms of Bn(x) and ζ(s,a), in order to give
special cases of (57) (see [688] and [9]), it is natural to know some of their properties.
We demonstrate to give the value of the simple case of (57) when p= 1 and q= 2:
ψ(n)
(
1
2
)
= (−1)n+1 n!
(
2n+1− 1
)
ζ(n+ 1) (n ∈ N), (58)
Introduction and Preliminaries 35
which is obtained from the aid of the formulas 1.7(19), 2.3(3) and 2.3(18). It is noted
that (58) is easily derived from (47) and 2.3(3). Likewise, we have (see [272])
ψ(2n)
(
1
3
)
ψ(2n)
(
2
3
)
=± (−1)n
√
332n
(2π)2n+1
2(2n+ 1)
B2n+1
(
1
3
)
+
(2n)!
2
(
1− 32n+1
)
ζ(2n+ 1) (n ∈ N)
(59)
and
ψ(2n)
(
1
4
)
ψ(2n)
(
3
4
)
=± (−1)n 42n
(2π)2n+1
2n+ 1
B2n+1
(
1
4
)
+ (2n)!
(
1− 22n+1
)
22n ζ(2n+ 1) (n ∈ N).
(60)
We give the relations
ψ(2n)
(
1
6
)
= 2ψ(2n)
(
1
3
)
(n ∈ N) (61)
and
ψ(2n)
(
5
6
)
=−2ψ(2n)
(
1
3
)
−ψ(2n)
(
1
2
)
(n ∈ N), (62)
which is easily obtained by recalling the multiplication formula (56) for ψ(n)(z).
Finally, several further special values are:
ψ(2n−1)
(
1
4
)
ψ(2n−1)
(
3
4
)
=±(2n− 1)!2
4n−1β(2n) (63)
+ (−1)n−122n−2(22n− 1)B2n
(2π)2n
2n
,
where
β(s) :=
ζ
(
s, 14
)
− ζ
(
s, 14
)
4s
. (64)
36 Zeta and q-Zeta Functions and Associated Series and Integrals
The Asymptotic Expansion for ψ(z)
From 1.1(34), we obtain the following asymptotic expansion for ψ(z):
ψ(z)= logz−
1
2z
−
n∑
k=1
B2k
2k z2k
+O
(
z−2n−2
)
(|z| →∞; |arg(z)|5 π − � (0< � < π); n ∈ N0).
(65)
Now we shall show (see Barnes [94]) that
n∑
k=1
ψ(k)= n logn− n+
1
2
+O
(
n−1
)
(n→∞) (66)
and
n∑
k=1
ψ ′(k)= logn+ 1+ γ +O
(
n−1
)
(n→∞), (67)
where γ denotes the Euler-Mascheroni constant given in 1.1(3).
Indeed, setting f (x)= 1/x and f (x)= 1/x2 with a= 1 in the Euler-Maclaurin sum-
mation formula 1.4??, we obtain
1+
1
2
+ ·· ·+
1
n
= γ + logn+
1
2n
−
1
8n2
+
15
2n4
− ·· · (68)
and
1+
1
22
+ ·· ·+
1
n2
=
π2
6
−
1
n
+
1
2n2
−
1
6n3
+
1
30n5
− ·· · , (69)
respectively. We can readily deduce from (69) that
1
n2
+
1
(n+ 1)2
+ ·· · =
1
n
+
1
2n2
+O
(
n−3
)
(n→∞). (70)
Since
ψ(m)=
1
1
+
1
2
+ ·· ·+
1
m− 1
− γ (m ∈ N \ {1}), (71)
we can also write (66) in the form:
n∑
k=1
ψ(k)=−nγ +
n− 1
1
+
n− 2
2
+ ·· ·+
1
n− 1
.
Introduction and Preliminaries 37
We, thus, obtain
n∑
k=1
ψ(k)= n
(
−γ +
1
1
+
1
2
+ ·· ·+
1
n− 1
)
−
1
1
−
2
2
− ·· ·−
n− 1
n− 1
= n
(
−γ +
n−1∑
k=1
1
k
)
− n+ 1,
which, in view of (68), yields
n∑
k=1
ψ(k)= n
[
−γ + γ + log(n− 1)+
1
2(n− 1)
+O
(
n−2
)]
− n+ 1 (n→∞),
which implies the result (66).
In order to prove (67), we consider (54) with m= n= 1:
ψ ′(z+ 1)= ψ ′(z)−
1
z2
, (72)
the repeated application of which gives
ψ ′(m)=−
[
1
12
+
1
22
+ ·· ·+
1
(m− 1)2
]
+ψ ′(1),
which may also follow directly from (54). Hence, we have
n∑
k=1
ψ ′(k)= nψ ′(1)−
n− 1
12
−
n− 2
22
− ·· ·−
1
(n− 1)2
. (73)
Now apply the following special case of (53) when n= z= 1:
ψ ′(1)= ζ(2)=
π2
6
, (74)
in conjunction with (68), and (73) readily yields
n∑
k=1
ψ ′(k)= n
π2
6
−
[
1
12
+
1
22
+ ·· ·+
1
(n− 1)2
]
+
1
12
+
2
22
+ ·· ·+
n− 1
(n− 1)2
= n
[
1
n2
+
1
(n+ 1)2
+ ·· ·
]
+ γ + log(n− 1)+O
(
n−1
)
(n→∞),
(75)
which, by applying (70), produces the desired result (67).
38 Zeta and q-Zeta Functions and Associated Series and Integrals
1.4 The Multiple Gamma Functions
The double Gamma function 02 and the multiple Gamma functions 0n were defined
and studied systematically by Barns [94–97] in about 1900. Before their investigation
by Barnes, these functions had been introduced in a different form by (for example)
Hölder [563], Alexeiewsky [18] and Kinkelin [666]. Although these functions did not
appear in the tables of the most well-known special functions, the double Gamma
function was cited in the exercises by Whittaker and Watson [1225, p. 264] and
recorded also by Gradshteyn and Ryzhik [505, p. 661, Entry 6.441(4); p. 937, Entry
8.333]. In about the middle of the 1980s, these functions were revived in the study
of the determinants of the Laplacians on the n–dimensional unit sphere Sn (see [260],
[706], [881], [954], [1190], [1201]). Shintani [1026] also used the double Gamma
function to prove the classical Kronecker limit formula. Friedman and Ruijsenaars
[463] showed that Shintani’s work on multiple Zeta and Gamma functions can be sim-
plified and extended by making use of difference equations. Its p-adic analytic exten-
sion appeared in a formula of Cassou-Noguès [221] for the p-adic L-functions at the
point 0. Choi et al. (see [269, 291, 292]) used these functions to evaluate some families
of series involving the Riemann Zeta function, as well as to compute the determinants
of the Laplacians. Choi et al. [269] addressed the converse problem and applied var-
ious formulas for series associated with the Zeta and related functions with a view
to developing the corresponding theory of multiple Gamma functions. Adamchik [8]
discussed some theoretical aspects of the multiple Gamma functions and their appli-
cations to summation of series and infinite products. Matsumoto [804] proved sev-
eral asymptotic expansions of the Barnes double Zeta function and the double Gamma
function and presented an application to the Hecke L-functions of real quadratic fields.
Ruijsenaars [990] showed how various known results concerning the Barnes multiple
Zeta and Gamma functions can be obtained as specializations of the simple features
shared by a quite remarkably extensive class of functions.
The Double Gamma Function 02
Here, we summarize some properties of the double Gamma and related functions.
We also introduce some mathematical constants associated with the double and triple
Gamma functions.
Barnes [94] defined the double Gamma function 02 = 1/G, satisfying each of the
following properties:
(a) G(z+ 1)= 0(z)G(z) (z ∈ C);
(b) G(1)= 1;
(c) Asymptotically,
logG(z+ n+ 2)=
n+ 1+ z
2
log(2π)+
[
n2
2
+ n+
5
12
+
z2
2
+ (n+ 1)z
]
logn
−
3n2
4
− n− nz− logA+
1
12
+O
(
n−1
)
(n→∞),
(1)
Introduction and Preliminaries 39
where 0 is the familiar Gamma function introduced in Section 1.1 and A is called the
Glaisher-Kinkelin constant, defined by
logA= lim
n→∞
{
n∑
k=1
k logk−
(
n2
2
+
n
2
+
1
12
)
logn+
n2
4
}
, (2)
the numericalvalue of A being given by
A∼= 1.282427130 · · · .
From this definition, Barnes [94] deduced several explicit Weierstrass canonical
product forms of the double Gamma function 02, one of which is recalled here in the
form:
{02(z+ 1)}
−1
= G(z+ 1)
= (2π)
1
2 zexp
(
−
1
2
z−
1
2
(γ + 1)z2
) ∞∏
k=1
{(
1+
z
k
)k
exp
(
−z+
z2
2k
)}
,
(3)
where γ denotes the Euler-Mascheroni constant given by 1.1(3).
Barnes [94] also gave the following two more equivalent forms of the double
Gamma function 02:
{02(z+ 1)}
−1
= G(z+ 1)
= (2π)
1
2 z exp
(
−
1
2
z(z+ 1)−
1
2
γ z2
) ∞∏
k=1
0(k)
0(z+ k)
exp
[
zψ(k)+
1
2
z2ψ′(k)
]
;
(4)
{02(z+ 1)}
−1
= G(z+ 1)= (2π)
1
2 z exp
[(
γ −
1
2
)
z−
(
π2
6
+ 1+ γ
)
z2
2
]
0(z+ 1)
·
∞∏
m=0
∞∏
n=0
′(
1+
z
m+ n
)
exp
(
−
z
m+ n
+
z2
2(m+ n)2
)
,
(5)
where the prime denotes the exclusion of the case n= m= 0 and the Psi (or Digamma)
function ψ is given by 1.2(1). Each form of these products is convergent for all finite
values of |z|, by the Weierstrass factorization theorem (see Conway [339, p. 170]).
The double Gamma function satisfies the following relations:
G(1)= 1 and G(z+ 1)= 0(z)G(z) (z ∈ C). (6)
40 Zeta and q-Zeta Functions and Associated Series and Integrals
For sufficiently large real x and a ∈ C, we have the Stirling formula for the
G-function:
logG(x+ a+ 1)=
x+ a
2
log(2π)− logA+
1
12
−
3x2
4
− ax (7)
+
(
x2
2
−
1
12
+
a2
2
+ ax
)
logx+O
(
x−1
)
(x→∞).
The following special values of G (see Barnes [94]) may be recalled here:
G
(
1
2
)
= 2
1
24 ·π−
1
4 ·e
1
8 ·A−
3
2 ; (8)
G(n+ 2)= 1!2! · · · n! and G(n+ 1)=
(n!)n
1 ·2 ·32 ·43 · · ·nn−1
(n ∈ N). (9)
We shall deduce only the expression (4) (see Barnes [94]). Indeed, taking the log-
arithmic derivative on both sides of the fundamental functional relation in (6), with
respect to z, we obtain
G′(z+ 1)
G(z+ 1)
=
0′(z)
0(z)
+
G′(z)
G(z)
,
from which
G′(z+ n+ 2)
G(z+ n+ 2)
=
n∑
k=0
0′(z+ k+ 1)
0(z+ k+ 1)
+
G′(z+ 1)
G(z+ 1)
.
For sufficiently small values of |z|, by Taylor’s theorem, we have
0′(z+ 1+ k)
0(z+ 1+ k)
=
0′(1+ k)
0(1+ k)
+ z
d
dk
0′(1+ k)
0(1+ k)
+
z2
2!
d2
dk2
0′(1+ k)
0(1+ k)
+ ·· · ,
provided that the coefficients in the expansion are finite.
Conversely, it follows from 1.1(2) or 1.3(3) that
−
0′(1+ z)
0(1+ z)
= γ +
∞∑
m=1
(
1
z+m
−
1
m
)
,
which, upon differentiating r times with respect to z and setting z= k in the resulting
equation, yields
dr
dkr
0′(1+ k)
0(1+ k)
= (−1)r−1 r!
∞∑
m=1
1
(m+ k)r+1
(r ∈ N\{1}),
Introduction and Preliminaries 41
so that
0′(z+ 1+ k)
0(z+ 1+ k)
−
0′(1+ k)
0(1+ k)
− z
d
dk
0′(1+ k)
0(1+ k)
=
∞∑
r=2
{
(−1)r−1
∞∑
m=1
1
(m+ k)r+1
zr
}
,
from which we find that
n∑
k=0
{
0′(z+ 1+ k)
0(z+ 1+ k)
−
0′(1+ k)
0(1+ k)
− z
d
dk
0′(1+ k)
0(1+ k)
}
=
∞∑
r=2
(−1)r−1
{
n∑
k=0
∞∑
m=1
1
(m+ k)r+1
zr
}
,
where we are not rearranging a double series, if n is not actually infinite.
Now it is not difficult to see from Eisenstein’s theorem (see Forsyth [457, p. 87])
that
lim
n→∞
n∑
k=0
∞∑
m=1
1
(m+ k)r+1
is convergent when r = 2. Hence,
lim
n→∞
n∑
k=0
{
0′(z+ 1+ k)
0(z+ 1+ k)
−
0′(1+ k)
0(1+ k)
− z
d
dk
0′(1+ k)
0(1+ k)
}
is finite, when |z|< 1 (z ∈ C).
We also have the following relation:
−
G′(z+ 1)
G(z+ 1)
=
n∑
k=0
{
0′(z+ 1+ k)
0(z+ 1+ k)
−
0′(1+ k)
0(1+ k)
− z
d
dk
0′(1+ k)
0(1+ k)
}
−
G′(z+ n+ 2)
G(z+ n+ 2)
+
n∑
k=0
{
0′(1+ k)
0(1+ k)
+ z
d
dk
0′(1+ k)
0(1+ k)
}
.
Therefore, for sufficiently small values of |z|, we may take
−
G′(z+ 1)
G(z+ 1)
= α+ 2βz
+
d
dz
log
[
∞∏
k=0
{
0(z+ 1+ k)
0(1+ k)
exp
(
−z
0′(1+ k)
0(1+ k)
−
z2
2
d
dk
0′(1+ k)
0(1+ k)
)}]
,
(10)
42 Zeta and q-Zeta Functions and Associated Series and Integrals
provided that
− lim
n→∞
G′(z+ n+ 2)
G(z+ n+ 2)
=−α− 2βz+ lim
n→∞
n∑
k=0
{
0′(1+ k)
0(1+ k)
+ z
d
dk
0′(1+ k)
0(1+ k)
}
,
and provided also that the expression on the right-hand side tends to the same value as
that obtainable from (1). Thus, we have
G′(z+ n+ 2)
G(z+ n+ 2)
= (n+ 1+ z) logn+
1
2
log(2π)− n+O
(
n−1
)
(n→∞).
If, then, we can prove that, for suitable values of α and β,
−α− 2βz+
n∑
k=0
(
0′(1+ k)
0(1+ k)
+ z
d
dk
0′(1+ k)
0(1+ k)
)
= (n+ 1+ z) logn+
1
2
log(2π)− n+O
(
n−1
)
(n→∞),
(11)
we shall have shown that, for suitable values of α and β, (10) holds true from the fact
(see Barnes [94, p. 269]) that G′(z+ 1)/G(z+ 1) is the only solution of
f (z+ 1)= f (z)+
0′(z)
0(z)
,
which has such an asymptotic expansion near at infinity.
Now the left-hand side of (11) may be written as
−α− 2βz+
n+1∑
k=1
{
ψ(k)+ zψ ′(k)
}
,
which, in view of 1.3(57) and 1.3(58), reduces at once to
(n+ 1) logn− n+ z(logn+ 1+ γ − 2β)+
1
2
−α+O
(
n−1
)
(n→∞),
which, upon putting
α =
1
2
−
1
2
log(2π) and β =
1+ γ
2
, (12)
becomes the right-hand side of (7).
Introduction and Preliminaries 43
If we now substitute the values of α and β from (12) into (10), we obtain the
following expression:
−
G′(z+ 1)
G(z+ 1)
=
1
2
−
1
2
log(2π)+ (1+ γ )z
+
d
dz
log
[
∞∏
k=0
{
0(z+ 1+ k)
0(1+ k)
exp
(
−zψ(1+ k)−
1
2
z2ψ ′(1+ k)
)}]
(|z|< 1; z ∈ C),
which, upon integrating with respect to z and obtaining the value of the constant of
integration by making z= 0, finally yields the desired expression (4) valid for |z|< 1.
In fact, (4) holds true for all finite values of |z| by the principle of analytic continuation.
From (4), we may also evaluate a series involving the ψ-function:
∞∑
k=1
{
ψ(k)+
1
2
ψ ′(k)− logk
}
= 1+
γ
2
−
1
2
log(2π) (13)
by taking logarithms on both sides of (4) and setting z= 1 in the resulting equation.
In view of 1.2(53), the left-hand side of (13) can be written in its equivalent form:
∞∑
k=1
{
ψ(k)+
1
2
ζ(2,k)− logk
}
= 1+
γ
2
−
1
2
log(2π). (14)
In view of their need in evaluating integrals involving the double Gamma function,
it may be important to know other special values of the G-function. Here, we give two
special values of the G-function as an illustration. The following special value of the
0-function is known (see Spiegel [1058, p. 1]):
0
(
1
4
)
∼= 3.625609908221908 · · · . (15)
The Catalan constant G is defined by
G :=
1
2
1∫
0
K(k)dk =
∞∑
m=0
(−1)m
(2m+ 1)2
∼= 0.915965594177219015 · · · , (16)
where K is the complete elliptic integral of the first kind, given by (cf. Equation
1.5(33))
K(k) :=
π/2∫
0
dt
√
1− k2 sin 2t
(|k|< 1). (17)
44 Zeta and q-Zeta Functions and Associated Series and Integrals
The following integral is known (see Gradshteyn and Ryzhik [505, p. 526, Entry
4.224]):
π/4∫
0
logsin t dt =−
π
4
log2−
1
2
G. (18)
Considering (6), (18) and (28), we obtain
G
(
3
4
)
= 2−
1
8 ·π−
1
4 ·e
G
2π ·0
(
1
4
)
G
(
1
4
)
. (19)
We also recall a duplication formula for the G-function (see Choi [263, p. 290]):
G(z)
{
G
(
z +
1
2
)}2
G(z+ 1)= e
1
4 ·A−3 ·2−2z
2
+3z− 1112 ·π z−
1
2 ·G(2z), (20)
which is an obvious special case (n= 2) of the following multiplication formula given
by Barnes [94, p. 291]:
n−1∏
i=0
n−1∏
j=0
G
(
z+
i+ j
n
)
=K(n)(2π)
1
2 n(n−1)z nnz−
1
2 n
2 z2 G(nz), (21)
where, for convenience,
K(n) := A1−n
2
e
1
12 (n
2
−1) (2π)−
1
2 (n−1) n−
5
12 .
Setting z= 14 in (20), and using (6) and (8), we obtain
0
(
1
4
){
G
(
1
4
)
G
(
3
4
)}2
= 2−
1
4 ·π−
1
2 ·e
3
8 ·A−
9
2 . (22)
Combining (19) and (22), we obtain
G
(
1
4
)
= e
3
32−
G
4π ·A−
9
8
{
0
(
1
4
)}− 34
∼= 0.293756 · · · (23)
or, equivalently,
G
(
3
4
)
= 2−
1
8 ·π−
1
4 ·e
3
32+
G
π ·A−
9
8
{
0
(
1
4
)} 1
4
∼= 0.848718 · · · . (24)
It follows from (23), (24) and (6) that
G
(
3
4
)
G
( 5
4
) = 2− 18 ·π− 14 ·e G2π . (25)
Introduction and Preliminaries 45
Integral Formulas Involving the Double Gamma Function
We begin by recalling the following integral formula:
z∫
0
π t cot π t dt = z log(2π)+ log
G(1− z)
G(1+ z)
, (26)
which is due, originally, to Kinkelin [666]. Indeed, in view of (3), ifwe set
8(z) :=
G(1+ z)
G(1− z)
= (2π)ze−z
∞∏
k=1
{(
1+ zk
1− zk
)k
e−2z
}
and differentiate logarithmically with respect to z, and apply the known expansion (see
Ahlfors [13, p. 188]):
πzcot πz= 1+ 2
∞∑
n=1
z2
z2− n2
(z 6∈ Z), (27)
we shall readily obtain
d
dz
log8(z)= log(2π)−πzcot πz (z 6∈ Z),
which, upon integration, yields (26), since 8(0)= 1.
Some simple consequences of Kinkelin’s formula (26) are worthy of note here (see
also Barnes [94, p. 279]). First, by using integration by parts in (26), we have
z∫
0
logsin π t dt = z log
(
sin πz
2π
)
+ log
G(1+ z)
G(1− z)
, (28)
which, upon setting t = 12 − u and replacing z by
1
2 − z, yields
z∫
0
logcos π t dt =
(
z−
1
2
)
log
(cos πz
2π
)
−
1
2
log2− log0
(
1
2
− z
)
+ log
G
(
1
2 + z
)
G
(
1
2 − z
) .
(29)
Making use of (29), we obtain the following analogue of (26):
z∫
0
π t tanπ t dt =−
1
2
log
(cos πz
π
)
− z log(2π)− log0
(
1
2
− z
)
+ log
G
(
1
2 + z
)
G
(
1
2 − z
) ,
(30)
46 Zeta and q-Zeta Functions and Associated Series and Integrals
which would follow also from (26) by setting t = 12 − u and replacing z by
1
2 − z.
Combining (28) and (29), we readily have the integral formula:
z∫
0
log tanπ t dt = z log tanπz+
1
2
log
cos πz
π
+ log0
(
1
2
− z
)
+ log
G(1+ z)
G(1− z)
− log
G
(
1
2 + z
)
G
(
1
2 − z
) . (31)
Similarly, by using various trigonometric identities, we can obtain the following
integral formulas:
z∫
0
(
π t
cos π t
)2
dt = πz2 tanπz+ log
cos πz
π
+ 2z log(2π)
+ 2log0
(
1
2
− z
)
− 2log
G
(
1
2 + z
)
G
(
1
2 − z
) ;
(32)
z∫
0
(
π t
sin π t
)2
dt =−πz2 cot πz+ 2z log(2π)− 2log
G(1− z)
G(1+ z)
. (33)
z∫
0
π t tan π t dt = log
G(1− z)
G(1+ z)
−
1
2
log
G(1− 2z)
G(1+ 2z)
, (34)
which, in view of the known duplication formula (20) for the G-function, is the same
as (31);
z∫
0
π t
sin π t
dt = z log(2π)+ 4log
G
(
1− 12 z
)
G
(
1+ 12 z
) − log G(1− z)
G(1+ z)
; (35)
z∫
0
(π t tanπ t)2 dt =−
π2z3
3
+πz2 tanπz+ log
cos πz
π
+ 2z log(2π)+ 2log
G
(
3
2 − z
)
G
(
1
2 + z
) ; (36)
z∫
0
(π t cot π t)2 dt =−
π2z3
3
−πz2 cot πz+ 2z log(2π)+ 2log
G(1− z)
G(1+ z)
; (37)
Introduction and Preliminaries 47
z∫
0
(
π t
sin π t
)2
cos π t dt = 2
z/2∫
0
(
π t
sin π t
)2
dt− 2
z/2∫
0
(
π t
cos π t
)2
dt
=−
πz2
sin πz
− 2log
cos πz2
π
+ 4log
G
(
1− z2
)
G
(
1+ z2
) + 4log G
(
1
2 +
z
2
)
G
(
3
2 −
z
2
) ; (38)
z∫
0
π t
sin π t cos π t
dt =−
1
2
log
cos πz
π
+ log
G(1− z)
G(1+ z)
+ log
G
(
1
2 + z
)
G
(
3
2 − z
) . (39)
Replacing t by at/π in (28), we obtain
z∫
0
logsin at dt = z log
sin az
2π
+
π
a
log
G
(
1+ a
π
z
)
G
(
1− a
π
z
) , (40)
which, in view of the trigonometric identity:
cos bt− cos at = 2sin
[
1
2
(a+ b)t
]
sin
[
1
2
(a− b)t
]
,
yields
z∫
0
log(cos bt− cos at)dt = z log
sin
[
1
2 (a+ b)z
]
sin
[
1
2 (a− b)z
]
2π2
+
2π
a+ b
log
G
(
1+ a+b2π z
)
G
(
1− a+b2π z
) + 2π
a− b
log
G
(
1+ a−b2π z
)
G
(
1− a−b2π z
) .
(41)
By differentiating Alexeiewsky’s theorem (see Barnes [94, p. 281]):
z∫
0
log0(t+ 1)dt =
1
2
[log(2π)− 1]z−
z2
2
+ z log0(z+ 1)− logG(z+ 1) (42)
with respect to z, we obtain
d
dz
logG(z)=
1
2
[log(2π)+ 1]− z+ (z− 1)ψ(z), (43)
and so we have
∂
∂a
logG(h(az))
=
{
1
2
[log(2π)+ 1]− h(az)+ [h(az)− 1]ψ(h(az))
}
∂
∂a
h(az),
(44)
48 Zeta and q-Zeta Functions and Associated Series and Integrals
where h(z) is a function of z and a is a constant.
Differentiating each side of (41) with respect to a and using (44) and 1.3(9), we
obtain
z∫
0
tsin at
cos bt− cos at
dt =
2az
a2− b2
log(2π)
−
2π
(a+ b)2
log
G
(
1− a+b2π z
)
G
(
1+ a+b2π z
) − 2π
(a− b)2
log
G
(
1− a−b2π z
)
G
(
1+ a−b2π z
) .
(45)
Replacing t by 1
π
arctanat in (31), we obtain
z∫
0
log t
1+ a2t2
dt =
(
logz
a
)
arctanaz+
π
2a
log
(
cos arctanaz
π
)
+
π
a
log
G
(
1+ 1
π
arctanaz
)
G
(
3
2 −
1
π
arctanaz
)
G
(
1− 1
π
arctanaz
)
G
(
1
2 +
1
π
arctanaz
) .
(46)
Setting a= 1 and z= 1 in (46) and using 1.3(4), we obtain
1∫
0
log t
1+ t2
dt =−G. (47)
It is easy to verify that
z∫
0
a
1+ a2+ 2acos t
dt =
2a
1− a2
arctan
(
1− a
1+ a
tan
1
2
z
)
and
z∫
0
cos t
1+ a2+ 2acos t
dt =
z
2a
−
1+ a2
a(1− a2)
arctan
(
1− a
1+ a
tan
1
2
z
)
.
Using these identities and applying the Leibniz rule (see De Lillo [372, p. 665]) to
the following integral and again integrating the resulting equation with respect to a
Introduction and Preliminaries 49
from 1 to a, we obtain
z∫
0
log
(
1+ a2+ 2acos t
2+ 2cos t
)
dt = z loga− 2
a∫
1
arctan
(
1− t
1+ t
tan
1
2
z
)
dt
t
= (2z−π) loga+ 2
a∫
1
arctan
(
t+ cos z
sin z
)
dt
t
.
Equivalently, from (29) and the following integral:
z∫
0
log(2+ 2cos t)dt = (2z− 2π) log
cos
(
1
2 z
)
π
+ 4π log G
(
1
2 +
z
2π
)
G
(
3
2 −
z
2π
) ,
we have
z∫
0
log(1+ a2+ 2acos t)dt = π loga+ (2z− 2π) log
acos
(
1
2 z
)
π

+ 4π log
G
(
1
2 +
z
2π
)
G
(
3
2 −
z
2π
) + 2 a∫
1
arctan
(
t+ cos z
sin z
)
dt
t
.
Applying integration by parts to the last integral gives us the following equivalent
form:
z∫
0
log(1+ a2+ 2acos t)dt = π loga+ (z−π) log
a2 cot
(
1
2 z
)
2π2

+ 2arctan
(
a+ cos z
sin z
)
log
(
a
sin z
)
+ 4π log
G
(
1
2 +
z
2π
)
G
(
3
2 −
z
2π
)
− 2
a+cos z
sin z∫
cot 12 z
log(t− cot z)
1+ t2
dt.
(48)
50 Zeta and q-Zeta Functions and Associated Series and Integrals
Setting z= 12 π in (48) and using 1.3(4), (46) and (47), we obtain
π/2∫
0
log(1+ a2+ 2acos t)dt =−π log
(
cos (arctana)
π
)
− 2π log
G
(
1+ 1
π
arctana
)
G
(
3
2 −
1
π
arctana
)
G
(
1− 1
π
arctana
)
G
(
1
2 +
1
π
arctana
) .
(49)
Integrating by parts and using (46), we have
z∫
0
arctanat
t
dt =−
π
2
log
(
cos (arctanaz)
π
)
−π log
G
(
1+ 1
π
arctanaz
)
G
(
3
2 −
1
π
arctanaz
)
G
(
1− 1
π
arctanaz
)
G
(
1
2 +
1
π
arctanaz
) .
(50)
Replacing t by 1
π
arcsinat in (28), we get
z∫
0
log t dt
√
1− a2t2
=
(
arcsinaz
a
)
log
( z
2π
)
+
π
a
log
G
(
1+ 1
π
arcsinaz
)
G
(
1− 1
π
arcsinaz
) . (51)
Using (51) and the following relation:
z∫
0
arcsinat
t
dt = (arcsinaz) logz− a
z∫
0
log t dt
√
1− a2t2
,
we obtain
z∫
0
arcsinat
t
dt = log(2π)(arcsinaz) +π log
G
(
1− 1
π
arcsinaz
)
G
(
1+ 1
π
arcsinaz
) . (52)
Introduction and Preliminaries 51
Integrating by parts and using (46), we find that
z∫
0
(arctanat)2dt
t2
=−
(arctanaz)2
z
−πa log
(
cos (arctanaz)
π
)
− a log
(
1+ a2z2
)
arctan az
+ 2πa log
G
(
1− 1
π
arctanaz
)
G
(
1
2 +
1
π
arctanaz
)
G
(
1+ 1
π
arctanaz
)
G
(
3
2 −
1
π
arctanaz
)
+ a2
z∫
0
log(1+ a2t2)
1+ a2t2
dt. (53)
Numerous special cases of the integral formulas (considered in this section) include
the following results:
π/2∫
0
( x
sin x
)2
cos xdx=−
π2
4
+ 4G; (54)
1/2∫
0
log0(t+ 1)dt =−
1
2
−
7
24
log2+
1
4
logπ +
3
2
logA; (55)
1/4∫
0
log0(t+ 1)dt =−
1
4
−
3
8
log2+
1
8
logπ +
9
8
logA+
G
π
; (56)
1∫
0
arctanx
x
dx= G; (57)
1/
√
2∫
0
arcsinx
x
dx=
π
8
log2+
G
2
; (58)
π/4∫
0
xcot xdx=
π
8
log2+
G
2
; (59)
π/4∫
0
logcos xdx=−
π
4
log2+
G
2
; (60)
52 Zeta and q-Zeta Functions and Associated Series and Integrals
π/4∫
0
x tanxdx=−
π
8
log2+
G
2
; (61)
π/4∫
0
log tanxdx=−G; (62)
π/4∫
0
( x
cos x
)2
dx=
π2
16
+
π
4
log2−G; (63)
π/4∫
0
( x
sin x
)2
dx=−
π2
16
+
π
4
log2+G; (64)
π/4∫
0
x2 tan2 xdx=−
π3
192
+
π2
16
+
π
4
log2−G; (65)
π/4∫
0
x2 cot 2xdx=−
π3
192
−
π2
16
+
π
4
log2+G; (66)
π/2∫
0
x
sin x
dx= 2G. (67)
The Evaluation of an Integral Involving logG(z)
First, we shall introduce two interesting mathematical constants, in addition to the
Glaisher-Kinkelin constant A, by recalling the Euler-Maclaurin summation formula
(cf. Hardy [537, p. 318]):
n∑
k=1
f (k)∼ C0+
n∫
a
f (x)dx+
1
2
f (n)+
∞∑
r=1
B2r
(2r)!
f (2r−1)(n), (68)
where C0 is an arbitrary constant to be determined in each special case and
B0 = 1, B1 =−
1
2
, B2 =
1
6
,B4 =−
1
30
, B6 =
1
42
, B8 =−
1
30
, B10 =
5
66
, · · · ,
and B2n+1 = 0 (n ∈ N)
are the Bernoulli numbers (see Section 1.7). For another useful summation formula,
see Edwards [399, p. 117].
Introduction and Preliminaries 53
Letting f (x)= x2 logx and f (x)= x3 logx in (68) with a= 1, we obtain
logB= lim
n→∞
[
n∑
k=1
k2 logk−
(
n3
3
+
n2
2
+
n
6
)
logn+
n3
9
−
n
12
]
(69)
and
logC = lim
n→∞
[
n∑
k=1
k3 logk−
(
n4
4
+
n3
2
+
n2
4
−
1
120
)
logn+
n4
16
−
n2
12
]
, (70)
respectively; here, B and C are constants, whose approximate numerical values are
given by
B∼= 1.03091 675 · · · (71)
and
C ∼= 0.97955 746 · · · . (72)
The constants B and C were considered, recently, by Choi and Srivastava [289, 292].
See also Adamchik [6, p. 199]. Bendersky [114] presented a set of constants, including
B and C.
It is known that (cf. Barnes [94, p. 288])
1∫
0
logG(t+ 1)dt =
1
12
+
1
4
log(2π)− 2logA. (73)
Setting z= t in the logarithmic form of (3) and integrating both sides of the result-
ing equation from t = 0 to t = 12 , we obtain
1
2∫
0
logG(t+ 1)dt =
1
16
log(2π)−
γ
48
−
1
12
+
1
4
∞∑
k=1
[(
4k2+ 2k
)
log(2k+ 1)−
(
4k2+ 2k
)
log(2k)− 2k−
1
2
+
1
12k
]
.
(74)
54 Zeta and q-Zeta Functions and Associated Series and Integrals
Consider the sum Sn given by
Sn :=
n∑
k=1
[(
4k2+ 2k
)
log(2k+ 1)−
(
4k2+ 2k
)
log(2k)
]
=
n∑
k=1
[
(2k+ 1)2 log(2k+ 1)− (2k+ 1) log(2k+ 1)− (2k)2 log(2k)− 2k log(2k)
]
=
[
2n+1∑
k=1
k2 logk− 2
n∑
k=1
(2k)2 log(2k)−
2n+1∑
k=1
k logk
]
=
[
2n+1∑
k=1
k2 logk− 8
n∑
k=1
k2 logk−
2n+1∑
k=1
k logk− 8log2
n∑
k=1
k2
]
,
which, in view of (2) and (69), yields
Sn =
[
logB+
{
(2n+ 1)3
3
+
(2n+ 1)2
2
+
(2n+ 1)
6
}
log(2n+ 1)
−
(2n+ 1)3
9
+
(2n+ 1)
12
− 8logB− 8
(
n3
3
+
n2
2
+
n
6
)
logn+
8
9
n3−
8
12
n
(75)
− logA−
{
(2n+ 1)2
2
+
(2n+ 1)
2
+
1
12
}
log(2n+ 1)
+
(2n+ 1)2
4
− 8log2
n(n+ 1)(2n+ 1)
6
]
+O
(
n−1
)
(n→∞)
=− logA− 7logB+
2
9
+ lim
n→∞
[(
8
3
n3+ 4n2+
4
3
n−
1
12
)
log(2n+ 1)
−
(
8
3
n3+ 4n2+
4
3
n
)
logn−
1
3
n2−
1
6
n
−
(
8
3
log2
)
n3− (4log2)n2−
(
4
3
log2
)
n
]
+O
(
n−1
)
(n→∞).
Now, substituting from (75) and recalling 1.1(3) and applying the following
expansion:
log
(
1+
1
2n
)
=
1
2n
−
1
8n2
+
1
24n3
−
dn
64n4
(0< dn < 1; n ∈ N), (76)
Introduction and Preliminaries 55
we evaluate the summation part of (74) as follows:
Tn :=− logA− 7logB+
2
9
+
[(
8
3
n3+ 4n2+
4
3
n−
1
12
)
log(2n+ 1)
−
(
8
3
n3+ 4n2+
4
3
n
)
logn−
1
3
n2−
1
6
n
−
(
8
3
log2
)
n3− (4log2)n2−
(
4
3
log2
)
n
−2
n(n+ 1)
2
−
n
2
+
1
12
(γ + logn)
]
+O
(
n−1
)
(n→∞)
=
γ
12
− logA− 7logB−
1
12
+
[(
8
3
n3+ 4n2+
4
3
n−
1
12
)
log2
+
(
8
3
n3+ 4n2+
4
3
n−
1
12
)
log
(
1+
1
2n
)
−
(
8
3
log2
)
n3−
(
4log2+
4
3
)
n2−
(
4
3
log2+
5
3
)
n
]
+O
(
n−1
)
(n→∞).
Thus, we have
Tn =
γ
12
− logA− 7logB−
1
12
+
[(
8
3
n3+ 4n2+
4
3
n−
1
12
)
log2+
4
3
n2+
5
3
n+
5
18
−
(
8
3
log2
)
n3−
(
4log2+
4
3
)
n2−
(
4
3
log2+
5
3
)
n
]
+O
(
n−1
)
(n→∞),
which, upon taking the limit as n→∞ and using (74), yields our desired result:
1
2∫
0
logG(t+ 1)dt =
1
24
(log2+ 1)+
1
16
logπ −
1
4
logA−
7
4
logB, (77)
in terms of the mathematical constant B defined by (69).
Vignéras [1199] gave an integral representation for the double Gamma function 02:
log02(z+ 1)=−
∞∫
0
e−t
t
(
1− e−t
)2 (1− zt− z2 t22 − e−zt
)
dt
+ (1+ γ )
z2
2
−
3
2
logπ (<(z) >−1).
(78)
56 Zeta and q-Zeta Functions and Associated Series and Integrals
The Multiple Gamma Functions
There are two known ways to define an n-ple Gamma functions 0n: Barnes [97] (see
also Vardi [1190]) defined 0n by using the n-ple Hurwitz Zeta functions (see Choi and
Quine [278]; also Seo et al. [1019]); a recurrence formula of the Weierstrass canon-
ical product forms of the n-ple Gamma functions 0n was given by Vignéras [1199],
who used the theorem of Dufresnoy and Pisot [395], which provides the existence,
uniqueness and expansion of the series of Weierstrass satisfying a certain functional
equation.
By making use of the aforementioned Dufresnoy-Pisot theorem and starting with
f1(x)=−γ x+
∞∑
n=1
{ x
n
− log
(
1+
x
n
)}
, (79)
Vignéras [1199] obtained a recurrence formula of 0n (n ∈ N), which is given by
Theorem 1.4 The n-ple Gamma functions 0n are defined by
0n(z)= {Gn(z)}
(−1)n−1 (n ∈ N), (80)
where
Gn(z+ 1)= exp( fn(z)) (81)
and the functions fn(z) are given by
fn(z)=−zAn(1)+
n−1∑
k=1
pk(z)
k!
[
f (k)n−1(0)−A
(k)
n (1)
]
+An(z), (82)
with
An(z)=
∑
m∈N0n−1×N
[
1
n
(
z
L(m)
)n
−
1
n− 1
(
z
L(m)
)n−1
+ ·· ·
+(−1)n−1
z
L(m)
+ (−1)n log
(
1+
z
L(m)
)]
,
(83)
where
L(m)= m1+m2+ ·· ·+mn,
if
m= (m1, m2, . . . , mn) ∈ N0n−1×N
Introduction and Preliminaries 57
and the polynomials pn(z) given by
pn(z) :=

1n+ 2n+ 3n+ ·· ·+ (N− 1)n (z= N; N ∈ N \ {1})
Bn+1(z)−Bn+1
n+ 1
(z ∈ C)
(84)
satisfy the following relations:
p′n(z)=
B′n+1(z)
n+ 1
= Bn(z) and pn(0)= 0,
Bn(z) being the Bernoulli polynomial of degree n in z.
Clearly, we find from the definition (84) that
pn(z)=
1
n+ 1
n+1∑
k=1
(
n+ 1
k
)
Bn+1−k z
k (n ∈ N). (85)
By analogy with the Bohr-Mollerup (Theorem 1.1), which guarantees the unique-
ness of the Gamma function 0, one can give for the double Gamma function and (more
generally) for the multiple Gamma functions of order n (n ∈ N) a definition of Artin
by means of the following theorem (see Vignéras [1199, p. 239]):
Theorem 1.5 For all n ∈ N, there exists a unique meromorphic function Gn(z)
satisfying each of the following properties:
(a) Gn(z+ 1)= Gn−1(z)Gn(z) (z ∈ C);
(b) Gn(1)= 1;
(c) For x = 1, Gn(x) are infinitely differentiable and
dn+1
dxn+1
{logGn(x)}= 0;
(d) G0(x)= x.
It may be remarked in passing that
G1(z)= 01(z)= 0(z)
and the case n= 2 of (80) can readily be seen to produce the double Gamma func-
tion (5).
We observe that the function {0n(z)}−1 defined in (80) is an entire function of order
n, all of whose zeros are the nonpositive integers 0, −1, −2, . . . , just as the Gamma
function {0(z)}−1, and the multiplicity of the zeros of {0n(z)}−1 at z=−k (k ∈ N0) is
equal to the number of solutions of the equation:
L(m)= k+ 1
(
m ∈ N0n−1×N
)
.
58 Zeta and q-Zeta Functions and Associated Series and Integrals
The number of the solutions for m ∈ N0n is the coefficient of xk+1 in the Maclaurin
series expansion of
(1− x)−n, that is,
(
n+ k
k+ 1
)
.
Therefore, the number of the solutions for m ∈ N0n−1×N is equal to(
n+ k
k+ 1
)
−
(
n+ k− 1
k+ 1
)
=
(
n+ k− 1
n− 1
)
.
We, thus, conclude that {0n(z)}−1 is an entire function with zeros at z=−k (k ∈ N0),
whose multiplicities are(
n+ k− 1
n− 1
)
(n ∈ N; k ∈ N0). (86)
The Triple Gamma Function 03
When n= 3 in (80), we readily obtain an explicit form of the triple Gamma func-
tion 03:
03(1+ z)=G3(1+ z)
=exp
[
−
1
6
(
γ +
π2
6
+
3
2
)
z3+
1
4
(
γ + log(2π)+
1
2
)
z2+�z
]
·
∏
m∈N02×N
{(
1+
z
L(m)
)−1
exp
[
z
L(m)
−
1
2
(
z
L(m)
)2
+
1
3
(
z
L(m)
)3]}
,
(87)
where
�=
1
12
(
3
2
− γ − 3log(2π)+
π2
12
)
+
1
2
∞∑
n=1
(−1)n
ζ(n+ 2)
(n+ 3)(n+ 4)
and
L(m)= m1+m2+m3 with m= (m1,m2,m3) ∈ N02×N,
ζ(s) being the Riemann zeta function (see Section 2.3). Now the infinite sum in �
can be evaluated explicitly by using a known formula (see Choi and Srivastava [288,
p. 116, Eq. (2.63)]):
∞∑
k=3
(−1)k
ζ(k)
(k+ 1)(k+ 2)
=
1
2
+
γ
6
−
π2
72
− 2logA. (88)
Introduction and Preliminaries 59
We, thus, find that
�=
3
8
−
1
4
log(2π)− logA (89)
in terms of the Glaisher-Kinkelin constant A defined by (2).
If we set n= 3 in (86), we observe that {03(z)}−1 is an entire function with zeros
at z=−k (k ∈ N0), whose multiplicity is
1
2
(
k2+ 3k+ 2
)
(k ∈ N0). (90)
Furthermore, in view of (90), (87) can be written in the following equivalent form
analogous to (3):
03(1+ z)= G3(1+ z)
= eR(z)
∞∏
k=1
{(1+
z
k
)− 12 k(k+1)
· exp
[
1
2
(k+ 1)z−
1
4
(
1+
1
k
)
z2+
1
6k
(
1+
1
k
)
z3
]}
, (91)
where, for convenience,
R(z) :=−
1
6
(
γ +
π2
6
+
3
2
)
z3+
1
4
(
γ + log(2π)+
1
2
)
z2+
(
3
8
−
1
4
log(2π)− logA
)
z.
It follows that 03 satisfies several basic properties and characteristics, which are
summarized here in
Theorem 1.6 The triple Gamma function 03 is the unique meromorphic function sat-
isfying each of the following properties:
(a) 03(1)= 1;
(b) 03(z+ 1)= G(z)03(z) (z ∈ C);
(c) For x = 1, 03(x) is infinitely differentiable and
d4
dx4
{log03(x)}= 0.
Just as in (8), we now proceed to express the value of 03
(
1
2
)
in terms of the mathe-
matical constants π, e, A and B. We begin by recalling the following known results:
1
2
log(2π)= lim
n→∞
[
n∑
k=1
logk−
(
n+
1
2
)
logn+ n
]
(92)
60 Zeta and q-Zeta Functions and Associated Series and Integrals
and
log
(
1+
1
n
)
=
1
n
−
1
2n2
+
1
3n3
+O
(
n−4
)
(n→∞). (93)
By taking logarithms on both sides of (91) and setting z= 12 in the resulting equa-
tion, if we make use of 1.1(3), we obtain
log03
(
1+
1
2
)
=
3
16
−
1
16
log(2π)−
1
2
logA
+ lim
n→∞
[
−
n∑
k=1
k(k+ 1)
2
log
(
1+
1
2k
)
+
n2
8
+
5
16
n−
1
24
logn
]
.
(94)
We, first, consider the following sum:
Un : =
n∑
k=1
k(k+ 1)
2
log
(
1+
1
2k
)
=
1
2
n∑
k=1
k(k+ 1) log(2k+ 1)
−
log2
2
n∑
k=1
k(k+ 1)−
1
2
n∑
k=1
k2 logk−
1
2
n∑
k=1
k logk
=
1
8
(
n∑
k=1
(2k+ 1)2 log(2k+ 1)−
n∑
k=1
log(2k+ 1)
)
−
log2
2
n∑
k=1
k(k+ 1)−
1
2
n∑
k=1
k2 logk−
1
2
n∑
k=1
k logk
and so
Un =
1
8
(
2n+1∑
k=1
k2 logk− 4
n∑
k=1
k2 logk− 4log2
n∑
k=1
k2
−
2n+1∑
k=1
logk+
n∑
k=1
logk+ n log2
)
−
log2
2
n∑
k=1
k(k+ 1)−
1
2
n∑
k=1
k2 logk−
1
2
n∑
k=1
k logk,
Introduction and Preliminaries 61
which immediately leads us to
Un =
1
8
2n+1∑
k=1
k2 logk−
1
8
2n+1∑
k=1
logk−
n∑
k=1
k2 logk−
1
2
n∑
k=1
k logk
+
1
8
n∑
k=1
logk−
(
n3
3
+
3
4
n2+
7
24
n
)
log2. (95)
Upon substituting from (95) into (94), if we apply (2), (69) and (92), we obtain
log03
(
1+
1
2
)
=
3
16
−
1
16
log(2π)+
7
8
logB
+ lim
n→∞
[
−
(
n3
3
+
3
4
n2+
7
24
n−
1
16
)
log
(
1+
1
2n
)
+
n2
6
+
n
3
−
35
288
+
1
16
log2
]
=
3
16
−
1
16
logπ +
7
8
logB
+ lim
n→∞
[
−
n2
6
−
n
3
−
19
288
+O
(
n−1
)
+
n2
6
+
n
3
−
35
288
]
=−
1
16
logπ +
7
8
logB,
where we have also used (93) for the second equality. Thus, we find that
03
(
1+
1
2
)
= π−
1
16 ·B
7
8 , (96)
which, in view of (8) and the assertion (b) of Theorem 1.5, yields
03
(
1
2
)
= 2−
1
24 ·π
3
16 ·e−
1
8 ·A
3
2 ·B
7
8 . (97)
A Multiplication Formula for the 0n
Nishizawa [867] obtained a multiplication formula for the n-ple Gamma function 0n,
by using his product formula for the multiple Gamma function 0n and other asymp-
totic formulas.
Here, by employing the same method used by Choi and Quine [278], Choi and
Srivastava [300] showed how the following multiplication formula for the multiple
Gamma function 0n can be obtained rather easily and nicely:
0n(z)=
fn(z,p,r)
pζn(0,z)
p−1∏
q1,··· ,qr=0
0n
(
z+ q1+ q2+ ·· ·+ qr
p
)
(p, r, n ∈ N), (98)
62 Zeta and q-Zeta Functions and Associated Series and Integrals
where
fn(z,p,r) :=
n∏
m=1
R
(−1)m
{
( zm−1)−
∑p−1
q1,... ,qr=0
(
z+q1+q2+···+qr
p
)
m−1
}
n−m+1
and Rm are given as in 2.1(28). Thus, in view of 2.1(16), ζn(0,z) can be expressed in
terms of the generalized Bernoulli polynomials of degree n.
The special case of (98), when n= 2 and r = 2, reduces to 1.4(21) by letting z= pa.
Indeed, starting with
p−1∑
q1,...,qr=0
ζn
(
s, a+
q1+ q2+ ·· ·+ qr
p
)
=
p−1∑
q1,... ,qr=0
∞∑
k1, ... ,kn
(
a+
q1+ q2+ ·· ·+ qr
p
+ k1+ k2+ ·· ·+ kn
)−s
= ps
∞∑
k1, ... ,kn
p−1∑
q1,... ,qr=0
(pa+ q1+ q2+ ·· ·+ qr + pk1+ pk2+ ·· ·+ pkn)
−s
= ps
∞∑
k1, ... ,kn
(pa+ k1+ k2+ ·· ·+ kn)
−s
= ps ζn (s, pa),
which yields
p−1∑
q1,... ,qr=0
ζn
(
s, a+
q1+ q2+ ·· ·+ qr
p
)
= ps ζn (s, pa). (99)
Differentiating each side of (99) with respect to s and setting s= 0 in the resulting
equation, and if we take the exponential in the last identity and then consider 2.1(27)
with pa= z, we obtain our desired multiplication formula (98).
A special case of (98), when n= 1, yields a multiplication formula for the Gamma
function 0 as follows:
p−1∏
q1,... ,qr=0
0
(
z+ q1+ q2+ ·· ·+ qr
p
)
= (2π)
1
2 (p
r
−1) p
1
2−z0(z) (p, r ∈ N).
(100)
Introduction and Preliminaries 63
1.5 The Gaussian Hypergeometric Function
and its Generalization
The Gauss Hypergeometric Equation
The second-order linear ordinary differential equation:
z(1− z)
d2w
dz2
+ [c− (a+ b+ 1)z]
dw
dz
− abw= 0 (1)
or, equivalently,
{δ(δ+ c− 1)− z(δ+ a)(δ+ b)}w= 0
(
δ := z
d
dz
)
, (2)
in which a, b and c are real or complex parameters, is called the Gauss hypergeometric
equation. Its only singularities are at
z= 0, 1, ∞;
each singularity is easily seen to be of regular kind.
The hypergeometric equation (1) or (2) is the most celebrated equation of the Fuch-
sian class, which consists of differential equations, whose only singularities (including
the point at infinity) are regular singular points. Its importance stems, in part, from a
well-known theorem that every homogeneous linear differential equation of the sec-
ond order, whose singularities (including the point at infinity) are regular and at most
three in number, can be transformed into the hypergeometric equation.
Power-series solutions of the hypergeometric equation (1) valid in the neighbor-
hoods of the regular singular points z= 0, 1 or∞ can be developed by direct appli-
cation of the classical method of Frobenius. Thus, if c is not an integer, the general
solution of (1) valid in a neighborhood of the origin (z= 0) is found to be
w= C1 2F1 (a, b; c; z)+C2 z
1−c
2F1 (a− c+ 1, b− c+ 1; 2− c; z) (c 6∈ Z),
(3)
where C1 and C2 are arbitrary constants, and (for convenience)
2F1 (a, b; c; z) := 1+
ab
1 ·c
z+
a(a+ 1)b(b+ 1)
1 ·2 ·c(c+ 1)
z2+ ·· ·
=
∞∑
n=0
(a)n (b)n
(c)n
zn
n!
(
c 6∈ Z−0
)
,
(4)
in terms of the Pochhammer symbol (λ)n defined by 1.1(5).
64 Zeta and q-Zeta Functions and Associated Series and Integrals
Gauss’s Hypergeometric Series
The infinite series in (4) obviously reduces to the elementary geometric series:
∞∑
n=0
zn = 1+ z+ z2+ ·· ·+ zn+ ·· · (5)
in its special cases, when
(i) a= c and b= 1 or (ii) a= 1 and b= c. (6)
Hence, it is called the hypergeometric series or, more precisely, Gauss’s hypergeomet-
ric series after the famous German mathematician, Carl Friedrich Gauss (1777–1855),
who in the year 1812 introduced this series into analysis and gave the F–notation for
it. (See also Equation 1.5(51) in which the general notation pFq is introduced, p and q
being any nonnegative integers.)
By d’Alembert’s ratio test, it is easily seen that the hypergeometric series in (4)
converges absolutely within the unit circle, that is, when |z|< 1, provided that the
denominator parameter c is neither zero nor a negative integer. Notice, however, that,
if either or both of the numerator parameters a and b in (4) is zero or a negative integer,
the hypergeometric series terminates, and the question of convergence does not enter
into the discussion.
Further tests show that the hypergeometric series in (4), when |z| = 1 (that is, on
the unit circle), is
(i) absolutely convergent, if <(c− a− b) > 0;
(ii) conditionally convergent, if −1< <(c− a− b)5 0 (z 6= 1);
(iii) divergent, if <(c− a− b)5−1.
As a matter of fact, in Case (i), we are led to the well-known Gauss’s summation
theorem:
2F1 (a, b; c; 1)=
0(c)0(c− a− b)
0(c− a)0(c− b)(
<(c− a− b) > 0; c 6∈ Z−0
)
.
(7)
An obvious special case of (7) occurs when the numerator parameter a or b is a
nonpositive integer −n. We, thus, have the Chu-Vandermonde summation formula:
2F1 (−n, b; c; 1)=
(c− b)n
(c)n
(
n ∈ N0; c 6∈ Z−0
)
, (8)
which is, in fact, equivalent to Vandermonde’s convolutiontheorem:
n∑
k=0
(
λ
k
)(
µ
n− k
)
=
(
λ+µ
n
)
=
n∑
k=0
(
λ
n− k
)(
µ
k
)
(n ∈ N0; λ, µ ∈ C).
(9)
Introduction and Preliminaries 65
For a number of summation theorems for the hypergeometric series (4), when z
takes on other special values, see Bailey [87, pp. 9–11], Erdélyi et al. [421, pp. 104–
105], Slater [1040, p. 243] and Luke [779, pp. 271–273].
The Hypergeometric Series and Its Analytic Continuation
We have seen that the hypergeometric series in (4) converges absolutely, when |z|< 1
and, thus, defines a function:
2F1 (a, b; c; z),
which is analytic, when |z|< 1, provided that c is neither zero nor a negative integer.
This function is correspondingly called the hypergeometric function or Gauss’s hyper-
geometric function. Indeed, it is the only solution of the Fuchsian equation (1) that is
analytic at the point z= 0 and assumes the value 1 at that point.
The hypergeometric function 2F1 (a, b; c; z) can, in fact, be continued analytically
outside the unit circle in a number of ways. If, for convenience, we use this same
notation for the analytically continued function, one way is to employ Euler’s integral
representation:
2F1 (a, b; c; z)=
0(c)
0(a)0(c− a)
1∫
0
ta−1 (1− t)c−a−1 (1− zt)−b dt
(<(c) > <(a) > 0; |arg(1− z)|5 π − � (0< � < π))
(10)
or, equivalently,
2F1 (a, b; c; z)=
0(c)
0(b)0(c− b)
1∫
0
tb−1 (1− t)c−b−1 (1− zt)−a dt
(<(c) > <(b) > 0; |arg(1− z)|5 π − � (0< � < π)),
(11)
since, by the series definition (4),
2F1 (a, b; c; z)≡ 2F1 (b, a; c; z). (12)
An analytic continuation without such parametric constraints as in (10) and (11)
can be provided by employing the following Mellin-Barnes contour integral repre-
sentation of the hypergeometric function:
0(a)0(b)
0(c)
2F1 (a, b; c; z)
=
1
2π i
i∞∫
−i∞
0(a+ ζ )0(b+ ζ )0(−ζ )
0(c+ ζ )
(−z)ζ dζ
(
|arg(−z)|5 π − � (0< � < π); a, b 6∈ Z−0
)
,
(13)
66 Zeta and q-Zeta Functions and Associated Series and Integrals
where the path of integration is the imaginary axis (in the complex ζ -plane), which is
indented, if necessary, to ensure that the poles of 0(a+ ζ )0(b+ ζ ), viz
ζ =−a− n and ζ =−b− n (n ∈ N0), (14)
lie to the left of the path and the poles of 0(−ζ ), viz
ζ = 0, 1, 2, . . . , (15)
lie to the right of the path.
The integrals in (10), (11) and (13) define analytic functions of z, which are single-
valued in the domain |arg(−z)|< π, that is, in the whole z-plane, with the exception
of the points on the negative real axis. Since the Gaussian function 2F1 (a, b; c; z)
is defined by the hypergeometric series (4) everywhere within the unit circle |z| = 1
(including, for example, at points on the negative real axis from z= 0 to z= 1− �,
� being an arbitrarily small positive number), we can use either of these integrals to
provide the analytic continuation of the hypergeometric function to the whole complex
z-plane cut along the real axis from z= 1 to z=∞. This analytic continuation of
2F1 (a, b; c; z) is conveniently denoted by the same symbol 2F1 (a, b; c; z), which will
henceforth represent a branch (the principal branch) of the analytic function generated
by the Gaussian hypergeometric series in (4).
By evaluating the contour integral in (13) using the sum of the residues of the
integrand at the poles given by (14), it can be shown that
1
0(c)
2F1 (a, b; c; z)
=
0(b− a)
0(b)0(c− a)
(−z)−a 2F1
(
a, 1− c+ a; 1− b+ a;
1
z
)
+
0(a− b)
0(a)0(c− b)
(−z)−b 2F1
(
b, 1− c+ b; 1− a+ b;
1
z
)
(|arg(−z)|5 π − � (0< � < π); a− b 6∈ Z).
(16)
This is an important result, for it may be used to provide the analytic continuation
of 2F1 (a, b; c; z) into the domain |z|> 1, that is, outside the circle of convergence of
the hypergeometric series (4). It also shows that the function so defined possesses a
branch point at infinity and that, asymptotically,
2F1 (a, b; c; z)∼A(−z)−a+B (−z)−b
(|z| →∞; |arg(−z)|5 π − � (0< � < π)),
(17)
where, for convenience,
A :=
0(c)0(b− a)
0(b)0(c− a)
and B :=
0(c)0(a− b)
0(a)0(c− b)
. (18)
Introduction and Preliminaries 67
Linear, Quadratic and Cubic Transformations
Pfaff-Kummer transformations:
2F1 (a, b; c; z)= (1− z)
−a
2F1
(
a, c− b; c;
z
z− 1
)
(19)(
c 6∈ Z−0 ; |arg(1− z)|5 π − � (0< � < π)
)
;
2F1 (a, b; c; z)= (1− z)
−b
2F1
(
b, c− a; c;
z
z− 1
)
(20)
(c 6∈ Z−0 ; |arg(1− z)|5 π − � (0< � < π)).
Euler’s transformation:
2F1 (a, b; c; z)= (1− z)
c−a−b
2F1 (c− a, c− b; c; z)
(c 6∈ Z−0 ; |arg(1− z)|5 π − � (0< � < π)).
(21)
The (Pfaff-Kummer) transformations (19) and (20) can be proven, for example, by
setting t = 1− s in Euler’s integral representations (10) and (11), respectively, and
then appealing to the principle of analytic continuation. The Euler transformation (21)
follows, if we perform the transformations (19) and (20) successively.
For an extensive list of quadratic and cubic transformations of the Gaussian hyper-
geometric function, see Erdélyi et al. [421, pp. 110–114] and Luke [779, Vol. I, p. 92
et seq.].
Hypergeometric Representations of Elementary Functions
(1− z)−a =
∞∑
n=0
(a)n
zn
n!
= 2F1 (a, b; b; z); (22)
log(1+ z)= z 2F1 (1, 1; 2; −z); (23)
log
(
1+ z
1− z
)
= 2z 2F1
(
1
2
, 1;
3
2
; z2
)
; (24)
sin −1z= z 2F1
(
1
2
,
1
2
;
3
2
; z2
)
; (25)
tan−1 z= z 2F1
(
1
2
, 1;
3
2
; −z2
)
; (26)
sinh −1 z= log
(
z+
√
1+ z2
)
= z 2F1
(
1
2
,
1
2
;
3
2
; −z2
)
; (27)
68 Zeta and q-Zeta Functions and Associated Series and Integrals
{
1
2
(
1+
√
1− z
)}1−2a
= 2F1
(
a, a−
1
2
; 2a; z
)
(28)
=
√
1− z 2F1
(
a, a+
1
2
; 2a; z
)
;
(
1+
√
z
)−2a
+
(
1−
√
z
)−2a
= 2 2F1
(
a, a+
1
2
;
1
2
; z
)
. (29)
Hypergeometric Representations of Other Functions
Legendre functions of the first and second kind:
Pµν (z)=
1
0(1−µ)
(
z+ 1
z− 1
)1
2µ
2F1
(
−ν, ν+ 1; 1−µ;
1− z
2
)
; (30)
Qµν (z)=
√
π eiµπ
2ν+1
0(µ+ ν+ 1)
0
(
ν+ 32
) (z2− 1)12µ
zµ+ν+1
(31)
·2F1
[
1
2
(µ+ ν+ 1),
1
2
(µ+ ν)+ 1; ν+
3
2
; z−2
]
.
The incomplete Beta function:
Bz(α,β)= α
−1 zα 2F1 (α, 1−β; α+ 1; z) . (32)
Complete Elliptic integrals of the first and second kind:
K(k)=
π/2∫
0
dθ
√
1− k2 sin 2 θ
=
1
2
π 2F1
(
1
2
,
1
2
; 1; k2
)
; (33)
E(k)=
π/2∫
0
√
1− k2 sin 2 θ dθ =
1
2
π 2F1
(
−
1
2
,
1
2
; 1; k2
)
. (34)
Jacobi polynomials:
P(α,β)n (z)= (−1)
n P(β,α)n (−z)
=
(
α+ n
n
)
2F1
[
−n, α+β + n+ 1;
α+ 1;
1− z
2
]
. (35)
Introduction and Preliminaries 69
Gegenbauer (or ultraspherical) polynomials:
Cνn(z)=
(
n+ ν− 12
n
)−1(
n+ 2ν− 1
n
)
P
(
ν− 12 ,ν−
1
2
)
n (z)
=
(
n+ 2ν− 1
n
)
2F1
−n, 2ν+ n;
ν+
1
2
;
1− z
2
 . (36)
Legendre (or spherical) polynomials:
Pn(z)= P
(0,0)
n (z)= C
1
2
n (z)= 2F1
(
−n, n+ 1; 1;
1− z
2
)
. (37)
Tchebycheff polynomials of the first and second kind:
Tn(z)=
(
n− 12
n
)−1
P
(
−
1
2 ,−
1
2
)
n (z)=
1
2
nC0n(z)= 2F1
(
−n, n;
1
2
;
1− z
2
)
, (38)
where
C0n(z) := lim
ν→0
{
ν−1 Cνn(z)
}
; (39)
Un(z)=
1
2
(
n+ 12
n+ 1
)−1
P
(
1
2 ,
1
2
)
n (z)= C
1
n(z)
= (n+ 1)2F1
(
−n, n+ 2;
3
2
;
1− z
2
)
. (40)
The Confluent Hypergeometric Function
If, in Gauss’s hypergeometric equation (1), we replace z by z/b, the resulting equation
will have three regular singularities at
z= 0, b, ∞.
By letting |b| →∞, this transformed equation reduces to the form:
z
d2w
dz2
+ (c− z)
dw
dz
− aw= 0 (41)
or, equivalently,
{δ(δ+ c− 1)− z(δ+ a)}w= 0
(
δ := z
d
dz
)
. (42)
70 Zeta and q-Zeta Functions and Associated Series and Integrals
Equation (41) or (42) has a regular singularity at z= 0 and an irregular singularity
at z=∞, which is formed by the confluence of two regular singularities at b and∞
of Gauss’s equation (1) with z replaced by z/b. Consequently, (41) or (42) is called
the confluent hypergeometric equation or Kummer’s differential equation after Ernst
Eduard Kummer (1810–1893), who presented a detailed study of its solutions in 1836
(cf. Kummer [707]).
The simplest solution of (1) or (2) is Kummer’s function F(a, c, z), which, in the
notations ofthis section, is
1F1 (a; c; z) := 1+
a
1 ·c
z+
a(a+ 1)
1 ·2 ·c(c+ 1)
z2+ ·· ·
(43)
=
∞∑
n=0
(a)n
(c)n
zn
n!
(c 6∈ Z−0 ; |z|<∞).
Moreover, since
lim
|λ|→∞
{
(λ)n
( z
λ
)n}
= zn = lim
|µ|→∞
{
(µz)n
(µ)n
}
(n ∈ N0; |z|<∞),
(44)
we readily have
1F1 (a; c; z)= lim
|b|→∞
2F1
(
a, b; c;
z
b
)
. (45)
In view of the principle of confluence involved in (45), the solution (43) is also called
the confluent hypergeometric function.
Other popular notations for the series solution (43) are (i) M(a, c, z), used by Airey
and Webb [15], and (ii) 8(a; c; z) or 8(a, c, z), introduced by Humbert [573, 574].
Important Properties of Kummer’s Confluent Hypergeometric Function
The principle of confluence exhibited by (45) is useful to deduce properties of the con-
fluent hypergeometric 1F1 function from those of Gauss’s hypergeometric 2F1 func-
tion. Thus, from (10) and (20), we obtain
1F1 (a; c; z)=
0(c)
0(a)0(c− a)
1∫
0
ta−1 (1− t)c−a−1 ezt dt
(<(c) > <(a) > 0);
(46)
1F1 (a; c; z)= e
z
1F1 (c− a; c; −z), (47)
Introduction and Preliminaries 71
which is known as Kummer’s first transformation. We also have Kummer’s second
transformation:
e−z 1F1 (a; 2a; 2z)= 0F1
(
−; a+
1
2
;
1
4
z2
)
(2a 6= −1,−3,−5, . . .), (48)
where
0F1 (−; c; z)= lim
|a|→∞
1F1 (a; c; z/a)
=
∞∑
n=0
zn
n!(c)n
(c 6∈ Z−0 ; |z|<∞). (49)
It follows immediately from the definition (43) that
1F1 (a; a; z)= 0F0 (−; −; z)= e
z. (50)
The Generalized (Gauss and Kummer) Hypergeometric Function
A natural generalization of the hypergeometric functions 2F1, 1F1, et cetera (consid-
ered in this section) is accomplished by the introduction of an arbitrary number of
numerator and denominator parameters. The resulting series:
pFq
[
α1, . . . , αp;
β1, . . . , βq;
z
]
=
∞∑
n=0
(α1)n · · ·(αp)n
(β1)n · · ·(βq)n
zn
n!
= pFq(α1, . . . , αp; β1, . . . , βq; z),
(51)
where (λ)n is the Pochhammer symbol defined by 1.1(5), is known as the generalized
Gauss (and Kummer) series, or simply, the generalized hypergeometric series. Here,
p and q are positive integers or zero (interpreting an empty product as 1), and we
assume (for simplicity) that the variable z, the numerator parameters α1, · · ·, αp and
the denominator parameters β1, . . ., βq take on complex values, provided that no zeros
appear in the denominator of (51), that is, that
(βj 6∈ Z−0 ; j= 1, . . .,q). (52)
Thus, if a numerator parameter is a negative integer or zero, the pFq series termi-
nates in view of the identity 1.1(47), and we are led to a (generalized hypergeometric)
72 Zeta and q-Zeta Functions and Associated Series and Integrals
polynomial of the type:
p+1Fq
[
−n, α1, . . . , αp;
β1, . . . , βq;
z
]
=
n∑
k=0
(−n)k(α1)k . . . (αp)k
(β1)k . . . (βq)k
zk
k!
=
(α1)n . . . (αp)n
(β1)n . . . (βq)n
(−z)n
·q+1Fp
[
−n, 1−β1− n, . . . , 1−βq− n;
1−α1− n, . . . , 1−αp− n;
(−1)p+q
z
]
(n ∈ N0),
(53)
where we have reversed the order of the terms of the polynomial by using 1.1(25) and
1.1(26).
Assuming that none of the numerator parameters is zero or a negative integer
(otherwise the question of convergence will not arise) and with the usual restriction
given by (52), the pFq series in (51)
(i) converges for |z|<∞, if p≤ q,
(ii) converges for |z|< 1, if p= q+ 1, and
(iii) diverges for all z, z 6= 0, if p> q+ 1.
Furthermore, if we set
ω :=
q∑
j=1
βj−
p∑
j=1
αj, (54)
then it is known that the pFq series in (51), with p= q+ 1, is
(i) absolutely convergent for |z| = 1, if <(ω) > 0,
(ii) conditionally convergent for |z| = 1 (z 6= 1), if −1< <(ω)5 0, and
(iii) divergent for |z| = 1, if <(ω)5−1.
Analytic Continuation of the Generalized Hypergeometric Function
The generalized hypergeometric pFq function is defined by the (absolutely) convergent
series (51), whenever
(i) p 5 q and |z|<∞
or
(ii) p= q+ 1 and |z|< 1.
As in the above-detailed case of the 2F1 function, the generalized hypergeometric
function pFq (with p= q+ 1) can be continued analytically to the domain |arg(1− z)
|< π, that is, to the whole complex z-plane cut along the real axis from z= 1 to
Introduction and Preliminaries 73
z=∞, by using (51) in conjunction with the Mellin–Barnes contour integral repre-
sentation:
0(α1) . . .0(αp)
0(β1) . . .0(βq)
pFq
[
α1, . . . , αp;
β1, . . . , βq;
z
]
=
1
2π i
i∞∫
−i∞
0(α1+ ζ ) . . .0(αp+ ζ )0(−ζ )
0(β1+ ζ ) . . .0(βq+ ζ )
(−z)ζ dζ
(αj 6∈ Z−0 ; j= 1, . . . , p; |arg(−z)|< π),
(55)
where the path of integration is the imaginary axis (in the complex ζ -plane), which is
indented, if necessary, to separate the poles of 0(αj+ ζ ) (j= 1, · · · , p) from those of
0(−ζ ).
Functions Expressible in Terms of the pFq Function
The following list is in addition to the important functions (considered above in this
section) that can be expressed as a hypergeometric 2F1 function.
Elementary functions:
ez = 0F0 [−;−; z]; (56)
(1− z)−α = 1F0 [α;−; z]; (57)
cos z= 0F1
[
−;
1
2
; −
1
4
z2
]
; (58)
sin z= z 0F1
[
−;
3
2
; −
1
4
z2
]
. (59)
Bessel functions:
Jν(z)=
(
1
2 z
)ν
0(ν+ 1)
0F1
[
−;ν+ 1; −
1
4
z2
]
=
(
1
2 z
)ν
0(ν+ 1)
exp(±iz)1F1
[
ν+
1
2
; 2ν+ 1; ∓2iz
]
; (60)
Iν(z)=
(
1
2 z
)ν
0(ν+ 1)
0F1
[
−;ν+ 1;
1
4
z2
]
=
(
1
2 z
)ν
0(ν+ 1)
exp(±z)1F1
[
ν+
1
2
; 2ν+ 1; ∓2z
]
. (61)
74 Zeta and q-Zeta Functions and Associated Series and Integrals
Lommel functions:
sµ,ν(z)=
zµ+1
(µ− ν+ 1)(µ+ ν+ 1)
·1F2
 1;1
2
(µ− ν+ 3),
1
2
(µ+ ν+ 3);
−
1
4
z2
 ; (62)
Sµ,ν(z)=sµ,ν(z)+ 2
µ−10
(
µ− ν+ 1
2
)
0
(
µ+ ν+ 1
2
)
(63)
·
{
sin
[
1
2
(µ− ν)π
]
Jν(z)− cos
[
1
2
(µ− ν)π
]
Yν(z)
}
,
where Jν(z) is given by (60), and
Yν(z)= csc(νπ) [cos (νπ)Jν(z)− J−ν(z)] . (64)
Struve functions:
Hν(z)=
(
1
2 z
)ν+1
0
(
3
2
)
0
(
ν+ 32
) 1F2
 1;3
2
, ν+
3
2
;
−
1
4
z2

=
21−ν
√
π 0
(
ν+ 12
) sν,ν(z) (65)
= Yν(z)+
21−ν
√
π 0
(
ν+ 12
) Sν,ν(z);
Lν(z)=
(
1
2 z
)ν+1
0
(
3
2
)
0
(
ν+ 32
) 1F2
 1;3
2
, ν+
3
2
;
1
4
z2

= exp
[
−
1
2
(ν+ 1)iπ
]
Hν(iz). (66)
Whittaker function of the first kind:
Mκ,µ(z)= z
µ+ 12 exp
(
−
1
2
z
)
1F1
(
µ− κ +
1
2
; 2µ+ 1; z
)
= zµ+
1
2 exp
(
1
2
z
)
1F1
(
µ+ κ +
1
2
; 2µ+ 1; −z
)
. (67)
Incomplete Gamma function:
γ (α,z)= α−1 zα 1F1 (α; α+ 1; −z) . (68)
Introduction and Preliminaries 75
Error function:
Erf(z)=
√
π
2
erf(z)=
1
2
γ
(
1
2
,z2
)
= z 1F1
(
1
2
;
3
2
; −z2
)
. (69)
Hermite polynomials:
Hn(z)=
[n/2]∑
k=0
(−1)k n!(2z)n−2k
k!(n− 2k)!
= (2z)n 2F0
(
−
1
2
n,
1
2
−
1
2
n; −; −z−2
)
(n ∈ N0). (70)
Laguerre functions and polynomials:
L(α)ν (z)=
0(ν+α+ 1)
0(ν+ 1)0(α+ 1)
1F1 (−ν; α+ 1; z) ; (71)
L(α)n (z)=
n∑
k=0
(
n+α
n− k
)
(−z)k
k!
=
(
n+α
n
)
1F1 (−n; α+ 1; z)
=
(−z)n
n!
2F0
(
−n,−α− n; −; −
1
z
)
(n ∈ N0). (72)
Poisson–Charlier polynomials:
cn(x;α)=
n∑
k=0
(−1)k
(
n
k
)(
x
k
)
k!α−k (α > 0; x ∈ N0)
= n!(−α)−n L(x−n)n (α)
= (−α)−n n!
(
x
n
)
1F1 (−n; x− n+ 1; α). (73)
In an alternative notation, we have (see Szegö [1141, p. 35])
pn(x)= α
−
1
2 n
(x− n+ 1)n
√
n!
1F1 (−n; x− n+ 1; α)
= α−
1
2 n
√
n!
(
x
n
)
1F1 (−n; x− n+ 1; α). (74)
76 Zeta and q-Zeta Functions and Associated Series and Integrals
1.6 Stirling Numbers of the First and Second Kind
Stirling Numbers of the First Kind
The Stirling numbers s(n,k) of the first kind are defined by the generating functions:
z(z− 1) · · ·(z− n+ 1)=
n∑
k=0
s(n,k)zk (1)
and
{log(1+ z)}k = k!
∞∑
n=k
s(n,k)
zn
n!
(|z|< 1). (2)
We have the following recurrence relations satisfied by s(n,k):
s(n+ 1,k)= s(n,k− 1)− ns(n,k) (n = k = 1); (3)
(
k
j
)
s(n,k)=
n−j∑
l=k−j
(
n
l
)
s(n− l, j)s(l,k− j) (n = k = j). (4)
From the definition (1) of s(n,k), the Pochhammer symbol in 1.1(5) can be written
in the form:
(z)n = z(z+ 1) · · ·(z+ n− 1)=
n∑
k=0
(−1)n+k s(n,k)zk, (5)
where (−1)n+k s(n,k) denotes the number of permutations of n symbols, which has
exactly k cycles.
It is not difficult to see also that
s(n,0)=
{
1 (n= 0)
0 (n ∈ N),
s(n,n)= 1,
s(n,1)= (−1)n+1(n− 1)!, s(n,n− 1)=−
(
n
2
) (6)
and
n∑
k=1
s(n,k)= 0 (n ∈ N\{1});
n∑
k=0
(−1)n+k s(n,k)= n!;
n∑
j=k
s(n+ 1, j+ 1)n j−k = s(n,k).
(7)
Introduction and Preliminaries 77
Yet another recursion formula for s(n,k) is given by (see Shen [1024]):
(k− 1)s(n,k)=−
k−1∑
m=1
s(n,k−m)H(m)n−1, (8)
where H(s)n is the generalized harmonic numbers of order s, defined by
H(s)n :=
n∑
k=1
1
ks
(n ∈ N; s ∈ C) (9)
and H(1)n := Hn (n ∈ N) is the harmonic numbers.
Here, for (8), we assume H(0)0 := 1 and H
(m)
0 := 0 (m ∈ N).
It readily follows from the recursion formula (8) that
s(n,2)= (−1)n (n− 1)!Hn−1;
s(n,3)= (−1)n+1
(n− 1)!
2
[
(Hn−1)
2
−H(2)n−1
]
;
s(n,4)= (−1)n
(n− 1)!
6
[
(Hn−1)
3
− 3Hn−1 H
(2)
n−1+ 2H
(3)
n−1
]
; (10)
s(n,5)= (−1)n+1
(n− 1)!
24
·
[
(Hn−1)
4
+ 8Hn−1 H
(3)
n−1− 6 (Hn−1)
2 H(2)n−1+ 3
(
H(2)n−1
)2
− 6H(4)n−1
]
.
In view of (5), by logarithmically differentiating 1.1(19) and then using 1.2(7), we
obtain
d
dz
{(z)n} =
n∑
k=1
(−1)n+k k s(n,k)zk−1
= (z)n
(
n∑
k=1
1
z+ k− 1
)
, (11)
which, upon employing Leibniz’s rule for differentiation, yields a more general
formula:
dj+1
dzj+1
{(z)n} =
n∑
k=j+1
(−1)n+k+j+1 (−k)j+1 s(n,k)z
k−j−1
=
j∑
l=0
(−1)l
j!
(j− l)!
(
n∑
k=1
1
(z+ k− 1)l+1
)(
dj−l
dzj−l
{(z)n}
)
(j ∈ N0; n ∈ N).
(12)
78 Zeta and q-Zeta Functions and Associated Series and Integrals
For j= 1 and j= 2, (12) immediately yields
n∑
k=2
(−1)n+k k(k− 1)s(n,k)zk−2
= (z)n
( n∑
k=1
1
z+ k− 1
)2
−
n∑
k=1
1
(z+ k− 1)2
 (13)
and
n∑
k=3
(−1)n+k k(k− 1)(k− 2)s(n,k)zk−3
= (z)n
( n∑
k=1
1
z+ k− 1
)3
− 3
(
n∑
k=1
1
z+ k− 1
)(
n∑
k=1
1
(z+ k− 1)2
)
+
n∑
k=1
1
(z+ k− 1)3
]
.
(14)
Stirling Numbers of the Second Kind
The Stirling numbers S(n,k) of the second kind are defined by the generating func-
tions:
zn =
n∑
k=0
S(n,k)z(z− 1) · · ·(z− k+ 1), (15)
(ez− 1)k = k!
∞∑
n=k
S(n,k)
zn
n!
, (16)
and
(1− z)−1 (1− 2z)−1 · · ·(1− kz)−1 =
∞∑
n=k
S(n,k)zn−k (|z|< k−1), (17)
where S(n,k) denotes the number of ways of partitioning a set of n elements into k
nonempty subsets.
It is not difficult to see also that
S(n,0)=
{
1 (n= 0)
0 (n ∈ N),
S(n,1)= S(n,n)= 1, and S(n,n− 1)=
(
n
2
)
. (18)
Introduction and Preliminaries 79
The recurrence relations for S(n,k) are given by
S(n+ 1,k)= k S(n,k)+ S(n,k− 1) (n = k = 1) (19)
and
(
k
j
)
S(n,k)=
n−j∑
i=k−j
(
n
i
)
S(n− i, j)S(i,k− j) (n = k = j). (20)
The numbers S(n,k) can be expressed in an explicit form:
S(n,k)=
1
k!
k∑
j=0
(−1)k−j
(
k
j
)
jn. (21)
Some additional properties of S(n,k) are recalled here as follows:
n∑
k=0
(−1)n−k k!S(n,k)= 1; (22)
n∑
j=k
S(j− 1,k− 1)kn−j = S(n,k); (23)
S(n,k)=
n−k∑
j=0
(−1) j
(
n− 1+ j
n− k+ j
)(
2n− k
n− k− j
)
S(n− k+ j, j); (24)
n∑
j=k
S(j,k)S(n, j) = δkn, (25)
where δmn denotes the Kronecker delta defined by
δmn =
{
0 (m 6= n),
1 (m= n).
and δm =
{
0 (m 6= 0),
1 (m= 0).
(26)
Relationships Among Stirling Numbers of the First and Second Kind
and Bernoulli Numbers
Akiyama and Tanigawa [17] presented, to evaluate multiple zeta values at nonpositive
integers, the following identities:
n∑
`=k
1
`
S(n,`)s(`,k)=
1
n
(
n
k
)
Bn−k+ δn−k−1 (n, k ∈ N; n = k), (27)
80 Zeta and q-Zeta Functions and Associated Series and Integrals
where Bk is the Bernoulli number given in 1.6.
n∑
`=0
(
n
`
)
Bn−` S(`,k)=
n
k
S(n− 1,k− 1) (n, k ∈ N). (28)
n∑
`=k−1
S(n,`)s(`+ 1,k)
`+ 1
=
Bn+1−k
n+ 1
(
n+ 1
k
)
(n, k ∈ N; n = k− 1), (29)
which, upon setting k = 1 and using (21), yields
Bn =
n∑
`=0
(−1)` `!
`+ 1
S(n,`)
=
n∑
`=0
1
`+ 1
∑̀
j=0
(−1) j
(
`
j
)
jn (n ∈ N0) , (30)
or, equivalently,
n∑
`=0
s(n,`)B` =
(−1)n n!
n+ 1
(n ∈ N0) . (31)
We also recall some known formulas (see, e.g., [982]):
max{k, j}+1∑
`=0
s(`, j)S(k,`)= δjk. (32)
max{k, j}+1∑
`=0
s(k,`)S(`, j)= δjk. (33)
s(n, i)=
n∑
k=i
k∑
j=0
s(n,k)s(k, j)S(j, i). (34)
S(n, i)=
n∑
k=i
k∑
j=0
S(n,k)S(k, j)s(j, i). (35)
S(n,m)=
n−m∑
k=0
(−1)k
(
k+ n− 1
k+ n−m
)(
2n−m
n− k−m
)
s(k−m+ n,k). (36)
s(n,m)=
n−m∑
k=0
(−1)k
(
k+ n− 1
k+ n−m
)(
2n−m
n− k−m
)
S(k−m+ n,k). (37)
Introduction and Preliminaries 81
1.7 Bernoulli, Euler and Genocchi Polynomials
and Numbers
Bernoulli Polynomials and Numbers
The Bernoulli polynomials Bn(x) are defined by the generating function:
zexz
ez− 1
=
∞∑
n=0
Bn(x)
zn
n!
(|z|< 2π). (1)
The numbers Bn := Bn(0) are called the Bernoulli numbers generated by
z
ez− 1
=
∞∑
n=0
Bn
zn
n!
(|z|< 2π). (2)
It easily follows from (1) and (2) that
Bn(x)=
n∑
k=0
(
n
k
)
Bk x
n−k. (3)
The Bernoulli polynomials Bn(x) satisfy the difference equation:
Bn(x+ 1)−Bn(x)= nx
n−1 (n ∈ N0), (4)
which yields
Bn(0)= Bn(1) (n ∈ N \ {1}). (5)
Setting x= 1 in (3), in view of (5), we have
Bn =
n∑
k=0
(
n
k
)
Bk, (6)
which gives a recursion formula for computing Bernoulli numbers. The first few of
the Bernoulli numbers are already listed with the Euler-Maclaurin summation formula
1.4(68), and (for the sake of completeness) we have the following list:
B0 = 1, B1 =−
1
2
, B2 =
1
6
, B4 =−
1
30
, B6 =
1
42
, B8 =−
1
30
, B10 =
5
66
,
B12 =−
691
2730
, B14 =
7
6
, B16 =−
3617
510
, B18 =
43867
798
, B20 =−
174611
330
,
B22 =
854513
138
, B24 =−
236364091
2730
, B26 =
8553103
6
, . . . ,B2n+1 = 0 (n ∈ N).
(7)
82 Zeta and q-Zeta Functions and Associated Series and Integrals
The first few of the Bernoulli polynomials are given below:
B0(x)= 1, B1(x)= x−
1
2
, B2(x)= x
2
− x+
1
6
,
B3(x)= x
3
−
3
2
x2+
1
2
x, B4(x)= x
4
− 2x3+ x2−
1
30
,
B5(x)= x
5
−
5
2
x4+
5
3
x3−
1
6
x,
B6(x)= x
6
− 3x5+
5
2
x4−
1
2
x2+
1
42
,
B7(x)= x
7
−
7
2
x6+
7
2
x5−
7
6
x3+
1
6
x, . . . .
(8)
It is not difficult to derive the following identities for the Bernoulli polynomials:
B′n(x)= nBn−1(x) (n ∈ N); (9)
Bn(1− x)= (−1)
n Bn(x) (n ∈ N0); (10)
(−1)n Bn(−x)= Bn(x)+ nx
n−1 (n ∈ N0). (11)
Multiplication formula:
Bn(mx)= m
n−1
m−1∑
k=0
Bn
(
x+
k
m
)
(n ∈ N0, m ∈ N). (12)
Addition formula:
Bn(x+ y)=
n∑
k=0
(
n
k
)
Bk(x)y
n−k (n ∈ N0). (13)
Integral formulas:
y∫
x
Bn(t)dt =
Bn+1(y)−Bn+1(x)
n+ 1
; (14)
x+1∫
x
Bn(t)dt = x
n
; (15)
1∫
0
Bn(t)Bm(t)dt = (−1)
n−1 m!n!
(m+ n)!
Bm+n (m, n ∈ N). (16)
Introduction and Preliminaries 83
It follows from (14) and (15) that the finite sum of powers is expressed as Bernoulli
polynomials and numbers:
m∑
k=1
kn =
Bn+1(m+ 1)−Bn+1
n+ 1
(m, n ∈ N). (17)
By writing 2n for n in (3), we can deduce that
B2n(x)+ nx
2n−1
=
n∑
k=0
(
2n
2k
)
B2k x
2n−2k,
which, upon integrating from 0 to 12 , yields
n∑
k=0
22k B2k
(2k)!(2n− 2k+ 1)!
=
1
(2n)!
(n ∈ N0), (18)
where we have applied (14).
It is readily shown that
B2n
(
1
2
)
=
(
21−2n− 1
)
B2n and B2n+1
(
1
2
)
= 0 (n ∈ N). (19)
The Generalized Bernoulli Polynomials and Numbers
The generalized Bernoulli polynomials B(α)n (x) of degree n in x are defined by the
generating function:(
z
ez− 1
)α
exz =
∞∑
n=0
B(α)n (x)
zn
n!
(
|z|< 2π; 1α := 1
)
(20)
for arbitrary (real or complex) parameter α. Clearly, we have
B(α)n (x)= (−1)
n B(α)n (α− x), (21)
so that
B(α)n (α)= (−1)
n B(α)n (0)=: (−1)
n B(α)n , (22)
in terms of the generalized Bernoulli numbers B(α)n defined by the generating function:(
z
ez− 1
)α
=
∞∑
n=0
B(α)n
zn
n!
(|z|< 2π; 1α := 1). (23)
84 Zeta and q-Zeta Functions and Associated Series and Integrals
It is easily observed that
B(1)n (x)= Bn(x) and B
(1)
n = Bn (n ∈ N0). (24)
From the generating function (20), it is fairly straightforward to deduce the addition
theorem:
B(α+β)n (x+ y)=
n∑
k=0
(
n
k
)
B(α)k (x)B
(β)
n−k(y), (25)
which, for x= β = 0, corresponds to the elegant representation:
B(α)n (x)=
n∑
k=0
(
n
k
)
B(α)k x
n−k (26)
for the generalized Bernoulli polynomials as a finite sum of the generalized Bernoulli
numbers.
Srivastava et al. [1101, p. 442, Eqs. (4.4) and (4.5)] gave two new classes of addi-
tion theorems for the generalized Bernoulli polynomials:B(α+λγ )n (x+ γ y)=
n∑
k=0
γ + n
γ + k
(
n
k
)
B(α−λk)k (x− ky)B
(λk+λγ )
n−k (ky+ γ y) (<(γ ) > 0);
(27)
B(α+β+n+1)n (x+ y+ n)=
n∑
k=0
(
n
k
)
B(α+k+1)k (x+ k)B
(β+n−k+1)
n−k (y+ n− k). (28)
Srivastava and Todorov [1110, p. 510, Eq. (3)] proved the following explicit for-
mula for the generalized Bernoulli polynomials:
B(α)n (x)=
n∑
k=0
(
n
k
)(
α+ k− 1
k
)
k!
(2k)!
k∑
j=0
(−1) j
(
k
j
)
j2k (x+ j)n−k
·2F1[k− n, k−α; 2k+ 1; j/(x+ j)], (29)
in terms of the Gaussian hypergeometric function (see Section 1.5). They also applied
the representation (29) to derive certain interesting special cases considered earlier by
Gould [499] and Todorov [1153]. Indeed, by the Chu-Vandermonde theorem 1.5(9),
we have
2F1(−N, b; c; 1)=
(
c− b+N− 1
N
)(
c+N− 1
N
)−1
(N ∈ N0),
Introduction and Preliminaries 85
which, for N = n− k, b= k−α and c= 2k+ 1, readily yields
2F1(k− n, k−α; 2k+ 1; 1)=
(
α+ n
n− k
)
(n− k)!(2k)!
(n+ k)!
(0≤ k ≤ n). (30)
In view of (30), the special case of the Srivastava-Todorov formula (29), when x= 0,
gives the following representation for the generalized Bernoulli numbers:
B(α)n =
n∑
k=0
(
α+ n
n− k
)(
α+ k− 1
k
)
n!
(n+ k)!
k∑
j=0
(−1) j
(
k
j
)
jn+k (31)
or, equivalently,
B(α)n =
n∑
k=0
(−1)k
(
α+ n
n− k
)(
α+ k− 1
k
)
n!
(n+ k)!
∆k 0n+k, (32)
where, for convenience,
∆k ar =∆k xr
∣∣∣∣
x=a
=
k∑
j=0
(−1)k−j
(
k
j
)
(a+ j)r, (33)
∆ being the difference operator defined by (cf. Comtet [337, p. 13 et seq.])
∆ f (x)= f (x+ 1)− f (x), (34)
so that, in general,
∆k f (x)=
k∑
j=0
(−1)k−j
(
k
j
)
f (x+ j). (35)
Alternatively, since (Comtet [337, p. 204, Theorem A]; see also Eq. 1.5(20))
S(n,k)=
1
k!
∆k 0n, (36)
where S(n,k) denotes the Stirling number of the second kind defined by 1.6(14), that
is, by
zn =
n∑
k=0
(
z
k
)
k!S(n,k), (37)
86 Zeta and q-Zeta Functions and Associated Series and Integrals
the representation (31) or (32) can be written also as (Todorov [1153, p. 665, Eq. (3)])
B(α)n =
n∑
k=0
(−1)k
(
α+ n
n− k
)(
α+ k− 1
k
)(
n+ k
k
)−1
S(n+ k,k). (38)
Formula (38) provides an interesting generalization of the following known result
for the Bernoulli numbers Bn:
Bn =
n∑
k=0
(−1)k
(
n+ 1
k+ 1
)(
n+ k
k
)−1
∆k 0n+k
k!
, (39)
which was considered, for example, by Gould [499, p. 49, Eq. (17)].
Euler Polynomials and Numbers
The Euler polynomials En(x) and the Euler numbers En are defined by the following
generating functions:
2exz
ez+ 1
=
∞∑
n=0
En(x)
zn
n!
(|z|< π) (40)
and
2ez
e2z+ 1
= sechz=
∞∑
n=0
En
zn
n!
(
|z|<
π
2
)
, (41)
respectively.
The following formulas are readily derivable from (40) and (41):
En(x+ 1)+En(x)= 2x
n (n ∈ N0); (42)
E′n(x)= nEn−1(x) (n ∈ N); (43)
En(1− x)= (−1)
n En(x) (n ∈ N0); (44)
(−1)n+1 En(−x)= En(x)− 2x
n (n ∈ N0); (45)
En(x+ y)=
n∑
k=0
(
n
k
)
Ek(x)y
n−k (n ∈ N0); (46)
En(x)=
n∑
k=0
(
n
k
)
Ek
2k
(
x−
1
2
)n−k
(n ∈ N0), (47)
which, upon taking x= 12 , yields
En = 2
n En
(
1
2
)
(n ∈ N0); (48)
Introduction and Preliminaries 87
n∑
k=0
(
2n
2k
)
E2k = 0 (n ∈ N). (49)
Multiplication formulas:
En(mx)= m
n
m−1∑
k=0
(−1)k En
(
x+
k
m
)
(n ∈ N0; m= 1, 3, 5, . . .); (50)
En(mx)=−
2
n+ 1
mn
m−1∑
k=0
(−1)k Bn+1
(
x+
k
m
)
(n ∈ N0; m= 2, 4, 6, . . .).
(51)
Integral formulas:
y∫
x
En(t)dt =
En+1(y)−En+1(x)
n+ 1
(m, n ∈ N0); (52)
1∫
0
En(t)Em(t)dt = (−1)
n 4
(
2m+n+2− 1
) m!n!
(m+ n+ 2)!
Bm+n+2 (m, n ∈ N0).
(53)
An alternating finite sum of powers can be expressed as the Euler polynomials:
m∑
k=1
(−1)m−k kn =
1
2
[
En(m+ 1)+ (−1)
m En(0)
]
(m, n ∈ N). (54)
Fourier Series Expansions of Bernoulli and Euler Polynomials
By employing suitable contour integrations in complex function theory, we can obtain
the following Fourier series expansions of Bernoulli and Euler polynomials:
B2n(x)=
(−1)n−1 2(2n)!
(2π)2n
∞∑
k=1
cos 2kπx
k2n
(n ∈ N; 0 5 x 5 1); (55)
B2n−1(x)=
(−1)n 2(2n− 1)!
(2π)2n−1
∞∑
k=1
sin 2kπx
k2n−1
(n= 1 and 0< x< 1; n ∈ N \ {1} and 0 5 x 5 1); (56)
E2n(x)=
(−1)n 4(2n)!
π2n+1
∞∑
k=0
sin (2k+ 1)πx
(2k+ 1)2n+1
(n= 0 and 0< x< 1; n ∈ N and 0 5 x 5 1);
(57)
88 Zeta and q-Zeta Functions and Associated Series and Integrals
E2n−1(x)=
(−1)n 4(2n− 1)!
π2n
∞∑
k=0
cos (2k+ 1)πx
(2k+ 1)2n
(n ∈ N; 0 5 x 5 1), (58)
which, for x= 12 , yields
E2n+1 = 0 (n ∈ N0), (59)
where use is made of the relationship (48).
The first few of the Euler numbers En are given below:
E0 = 1, E2 =−1, E4 = 5, E6 =−61, E8 = 1385, E10 =−50521, . . . . (60)
Relations Between Bernoulli and Euler Polynomials
The following relationships between the Bernoulli and Euler polynomials follow
easily from the definitions (1) and (40):
En(x)=
2n+1
n+ 1
{
Bn+1
(
x+ 1
2
)
−Bn+1
( x
2
)}
=
2
n+ 1
{
Bn+1(x)− 2
n+1 Bn+1
( x
2
)}
(n ∈ N0), (61)
which, in view of (10), can also be written in the form:
En(1− x)= (−1)
n 2
n+ 1
[
2n+1 Bn+1
(
x+ 1
2
)
−Bn+1(x)
]
(n ∈ N0). (62)
Two additional formulas involving these polynomials are given below:
En−2(x)= 2
(
n
2
)−1 n−2∑
k=0
(
n
k
) (
2n−k− 1
)
Bn−k Bk(x) (n ∈ N \ {1}); (63)
Bn(x)= 2
−n
n∑
k=0
(
n
k
)
Bn−k Ek(2x) (n ∈ N0). (64)
The Generalized Euler Polynomials and Numbers
The generalized Euler polynomials E(α)n (x) and the generalized Euler numbers E
(α)
n
are defined by the generating functions:
(
2
ez+ 1
)α
exp(xz)=
∞∑
n=0
E(α)n (x)
zn
n!
(|z|< π; 1α := 1) (65)
Introduction and Preliminaries 89
and (
2ez
e2z+ 1
)α
=
∞∑
n=0
E(α)n
zn
n!
(
|z|<
π
2
; 1α := 1
)
(66)
for arbitrary (real or complex) parameter α. Clearly, we have
E(1)n (x)= En(x) and E
(1)
n = En (n ∈ N0). (67)
It is easy to find from (65) and (66) that
2n E(α)n
(α
2
)
= E(α)n (n ∈ N0), (68)
which, for α = 1, reduces to (48).
Srivastava et al. [1101, p. 443, Eq. (4.12)] proved the following interesting addition
theorem for the generalized Euler polynomials:
E(α+λγ )n (x+ γ y)=
n∑
k=0
γ + n
γ + k
(
n
k
)
E(α−λk)k (x− ky)E
(λk+λγ )
n−k (ky+ γ y) (<(γ ) > 0).
(69)
From the generating functions (20) and (65), it is easily seen that
B(0)n (x)= E
(0)
n (x)= x
n (n ∈ N0). (70)
Recently, by making use of some fairly standard techniques based on series rear-
rangement, Srivastava and Pintér [1105] derived each of the following elegant theo-
rems (cf. [1105, p. 379, Theorem 1; p. 380, Theorem 2]). The following relationship
(cf. [1105, p. 379, Theorem 1]):
B(α)n (x+ y)=
n∑
k=0
(
n
k
)[
B(α)k (y)+
k
2
B(α−1)k−1 (y)
]
En−k(x)(α ∈ C; n ∈ N0) (71)
holds true between the generalized Bernoulli polynomials and the classical Euler
polynomials.
The following relationship (cf. [1105, p. 380, Theorem 2]):
E(α)n (x+ y)=
n∑
k=0
2
k+ 1
(
n
k
)[
E(α−1)k+1 (y)−E
(α)
k+1(y)
]
Bn−k(x)(α ∈ C; n ∈ N0),
(72)
holds true between the generalized Euler polynomials and the classical Bernoulli
polynomials.
90 Zeta and q-Zeta Functions and Associated Series and Integrals
Upon setting α = 1 in (71), if we let y→ 0 and make use of (70), we can deduce
the aforementioned main relationship in Cheon’s work (cf. [254, p. 368, Theorem 3]):
Bn(x)=
n∑
k=0
(k 6=1)
(
n
k
)
BkEn−k(x) (n ∈ N0), (73)
just as it was accomplished by Srivastava and Pintér [1105, p. 379].
Genocchi Polynomials and Numbers
The Genocchi polynomials G(α)n (x) of (real or complex) order α are usually defined
by means of the following generating function:(
2z
ez+ 1
)α
· exz =
∞∑
n=0
G(α)n (x)
zn
n!
(
|z|< π; 1α := 1
)
, (74)
so that, obviously, the classical Genocchi polynomials Gn(x), given by
Gn (x) := G
(1)
n (x) (n ∈ N0) , (75)
are defined by the following generating function:
2zexz
ez+ 1
=
∞∑
n=0
Gn (x)
zn
n!
(|z|< π). (76)
For the classical Genocchi numbers Gn, we have (see also Problem 54)
Gn := Gn (0)= G
(1)
n (0) . (77)
Finally, in light of the definition (74), we find for the Genocchi numbers G(α)n of
(real or complex) order α that
(
2z
ez+ 1
)α
=
∞∑
n=0
G(α)n
zn
n!
(
|z|< π; 1α := 1
)
, (78)
so that, just as we observedabove in Equation (77),
Gn = G
(1)
n (n ∈ N0).
Various properties and characteristics of the above-defined Genocchi polynomials
and numbers, analogous to those that hold true for the Bernoulli and Euler polynomials
and numbers, can be found in the widely-scattered literature, which is cited by (for
example) Luo and Srivastava [791] (see also Section 1.8 below).
Introduction and Preliminaries 91
1.8 Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi
Polynomials and Numbers
Apostol-Bernoulli Polynomials and Numbers
Some interesting analogues of the classical Bernoulli polynomials and numbers
were investigated by Apostol [58, p. 165, Eq. (3.1)] and (more recently) by
Srivastava [1087, pp. 83–84]. We begin by recalling, here, Apostol’s definitions as
follows:
The Apostol-Bernoulli polynomialsBn(x;λ) are defined by means of the generating
function (Apostol [58]; see also Srivastava [1087]):
zexz
λez− 1
=
∞∑
n=0
Bn(x;λ)
zn
n!
(1)
(|z|< 2π, when λ= 1; |z|< |logλ| , when λ 6= 1)
with, of course,
Bn(x)= Bn(x;1) and Bn (λ) := Bn (0;λ) (2)
where Bn (λ) denotes the so-called Apostol-Bernoulli numbers.
Under the assumption λ 6= 1, Apostol [58] gave the main properties of Bn (a,λ),
including, for example, the summation formula:
Bn (a,λ)=
n∑
k=0
(
n
k
)
Bk (λ) an−k (n ∈ N0), (3)
from which it is seen that Bn (a,λ) are polynomials in a, satisfying the difference
equation:
λBn (a+ 1,λ)−Bn (a,λ)= nan−1 (n ∈ N), (4)
which yields the following special cases:
λB1(1,λ)= 1+B1(λ) (5)
and
λBn(1,λ)= Bn(λ) (n ∈ N\{1}). (6)
92 Zeta and q-Zeta Functions and Associated Series and Integrals
Equations (5), (6) and (3), with a= 1, together, aid us in computing Bn(λ) recur-
sively. Thus, we have
B0(λ)= 0, B1(λ)=
1
λ− 1
, B2(λ)=−
2λ
(λ− 1)2
, B3(λ)=
3λ(λ+ 1)
(λ− 1)3
,
B4(λ)=−
4λ
(
λ2+ 4λ+ 1
)
(λ− 1)4
, B5(λ)=
5λ
(
λ3+ 11λ2+ 11λ+ 1
)
(λ− 1)5
,
B6(λ)=−
6λ
(
λ4+ 26λ3+ 66λ2+ 26λ+ 1
)
(λ− 1)6
(7)
and (in general)
Bn(λ)=
nλ
(λ− 1)n
n−1∑
k=1
(−1)k k!λk−1 (λ− 1)n−1−k S(n− 1,k), (8)
in terms of the Stirling numbers of the second kind (see Section 1.6).
We recall here several further properties of the functions Bn(a,λ) as follows:
∂p
∂ap
{Bn(a,λ)} =
n!
(n− p)!
Bn−p(a,λ) (p= 0, 1, . . . , n); (9)
Bn(a+ b,λ)=
n∑
k=0
(
n
k
)
Bk(a,λ)bn−k; (10)
b∫
a
Bn(t,λ)dt =
Bn+1(b,λ)−Bn+1(a,λ)
n+ 1
; (11)
m−1∑
k=0
kn =
λ− 1
n+ 1
m∑
k=1
Bn+1(k,λ)+
Bn+1(m,λ)−Bn+1(λ)
n+ 1
, (12)
which obviously generalizes the familiar result 1.7(17).
Motivated by the success of the generalizations in 1.7(20) and 1.7(66) of the clas-
sical Bernoulli polynomials and the classical Euler polynomials involving a real or
complex parameter α, Luo and Srivastava [788, 789] introduced and investigated the
so-called Apostol-Bernoulli polynomials B(α)n (x;λ) of order α and the Apostol-Euler
polynomials E (α)n (x;λ) of order α, which are defined as follows:
The Apostol-Bernoulli polynomials B(α)n (x;λ) of (real or complex) order α are
defined by means of the following generating function:(
z
λez− 1
)α
· exz =
∞∑
n=0
B(α)n (x;λ)
zn
n!
(13)(
|z|< 2π, when λ= 1; |z|< |logλ| , when λ 6= 1; 1α := 1
)
,
Introduction and Preliminaries 93
with, of course,
B(α)n (x)= B(α)n (x;1) and B(α)n (λ) := B(α)n (0;λ), (14)
where B(α)n (λ) denotes the so-called Apostol-Bernoulli numbers of order α.
The Apostol-Euler polynomials E (α)n (x;λ) of (real or complex) order α are defined
by means of the following generating function (cf. Luo [781]):
(
2
λez+ 1
)α
· exz =
∞∑
n=0
E (α)n (x;λ)
zn
n!
(
|z|< |log(−λ)| ; 1α := 1
)
, (15)
with, of course,
E(α)n (x)= E (α)n (x;1) and E (α)n (λ) := E (α)n (0;λ), (16)
where E (α)n (λ) denotes the so-called Apostol-Euler numbers of order α.
Luo and Srivastava [788, 789] presented a variety of properties and relations
between other mathematical functions. Among them, we choose to record, here, the
following results:
The Apostol-Euler polynomials E (α)n (x;λ) of order α is represented by
E (α)n (x;λ)= e−x logλ
∞∑
k=0
E(α)n+k (x)
(logλ)k
k!
(n ∈ N0) (17)
in series of the familiar Euler polynomials E(α)n (x) of order α.
The Aposto-Bernoulli polynomials B(`)n (x;λ) of order ` is represented by
B(`)n (x;λ)= e−x logλ
∞∑
k=0
(
n+ k− `
k
)(
n+ k
k
)−1
B(`)n+k (x)
(logλ)k
k!
(18)
(n, ` ∈ N0)
in series of the familiar Bernoulli polynomials B(`)n (x) of order `.
An explicit representation of B(`)n (λ) is given:
B(`)n (λ)= `!
(
n
`
) n−∑̀
k=0
(
`+ k− 1
k
)
k!(−λ)k
(λ− 1)k+`
S (n− `,k) , (19)(
n, ` ∈ N0; λ ∈ C \ {1}
)
.
94 Zeta and q-Zeta Functions and Associated Series and Integrals
An explicit formula for the Apostol-Bernoulli polynomials B(α)n (x;λ) involving the
Stirling numbers of the second kind is also given as follows:
B(`)n (x;λ)= `!
n∑
k=`
(
n
k
)(
k
`
)
xn−k
k−∑̀
j=0
(
`+ j− 1
j
)
j!(−λ)j
(λ− 1)j+`
S (k− `, j) (20)
(
n, ` ∈ N0; λ ∈ C \ {1}
)
.
The following relationship:
B(α)n (x;λ)=
n∑
k=0
(
n
k
)
B(α−1)n−k (λ)Bk (x;λ) (n ∈ N0) (21)
holds true between the Apostol-Bernoulli polynomials B(α)n (x;λ) of order α and the
Apostol-Bernoulli numbers B(α−1)n (λ) of order α− 1.
Let n ∈ N0. Suppose also that α and λ are suitable (real or complex) parameters.
Then,
B(α)n (x;λ)=
n∑
k=0
(
n
k
)
B(α)k (λ)x
n−k and B(0)n (x;λ)= xn, (22)
E (α)n (x;λ)=
n∑
k=0
(
n
k
)E (α)k (λ)
2k
(
x−
α
2
)n−k
, (23)
λB(α)n (x+ 1;λ)−B(α)n (x;λ)= nB
(α−1)
n−1 (x;λ), (24)
∂
∂x
{
B(α)n (x;λ)
}
= nB(α)n−1 (x;λ), (25)
b∫
a
B(α)n (x;λ),dx=
B(α)n+1 (b;λ)−B
(α)
n+1 (a;λ)
n+ 1
, (26)
B(α+β)n (x+ y;λ)=
n∑
k=0
(
n
k
)
B(α)k (x;λ)B
(β)
n−k (y;λ), (27)
E (α+β)n (x+ y;λ)=
n∑
k=0
(
n
k
)
E (α)k (x;λ)E
(β)
n−k(y;λ), (28)
B(α)n (α− x;λ)=
(−1)n
λα
B(α)n
(
x;λ−1
)
, (29)
E (α)n (α− x;λ)=
(−1)n
λα
E (α)n (x;λ−1), (30)
B(α)n (α+ x;λ)=
(−1)n
λα
B(α)n
(
−x;λ−1
)
, (31)
nxB(α)n−1 (x;λ)= (n−α)B
(α)
n (x;λ)+αλB(α+1)n (x+ 1;λ) (32)
Introduction and Preliminaries 95
and
B(α+1)n (x;λ)=
(
1−
n
α
)
B(α)n (x;λ)+ n
( x
α
− 1
)
B(α)n−1 (x;λ) . (33)
From the generating functions (13) and (15), it follows also that (see [788] and
[781])
λB(α)n (x+ 1;λ)−B(α)n (x;λ)= nB
(α−1)
n−1 (x;λ) (34)
and
λE (α)n (x+ 1;λ)+ E (α)n (x;λ)= 2E (α−1)n (x;λ), (35)
respectively. Now, since
B(0)n (x;λ)= E (0)n (x;λ)= xn (n ∈ N0), (36)
upon setting β = 0 in the addition theorems (27) and (28), if we interchange x and y,
we obtain
B(α)n (x+ y;λ)=
n∑
k=0
(
n
k
)
B(α)k (y;λ)x
n−k (37)
and
E (α)n (x+ y;λ)=
n∑
k=0
(
n
k
)
E (α)k (y;λ)x
n−k, (38)
respectively.
Next, by combining (34) and (37) (with x= 1 and y 7−→ x), we find that
B(α−1)n (x;λ)=
1
n+ 1
[
λ
n+1∑
k=0
(
n+ 1
k
)
B(α)k (x;λ)−B
(α)
n+1(x;λ)
]
(n ∈ N0), (39)
which, in the special case when α = 1, yields the following expansion:
xn =
1
n+ 1
[
λ
n+1∑
k=0
(
n+ 1
k
)
Bk(x;λ)−Bn+1(x;λ)
]
(n ∈ N0), (40)
in series of the Apostol-Bernoulli polynomials {Bn(x;λ)}∞n=0.
In the special case of (40), when λ= 1, we obtain the following familiar expansion
(cf., e.g., [795, p. 26]):
xn =
1
n+ 1
n∑
k=0
(
n+ 1
k
)
Bk(x) (n ∈ N0) (41)
96 Zeta and q-Zeta Functions and Associated Series and Integrals
in series of the classical Bernoulli polynomials {Bn(x)}∞n=0.
In precisely the same manner, the addition theorem (38) in conjunction with (35)
would lead us to the following companions of (39) and (40):
E (α−1)n (x;λ)=
1
2
[
λ
n∑
k=0
(
n
k
)
E (α)k (x;λ)+ E
(α)
n (x;λ)
]
(n ∈ N0) (42)
and
xn =
1
2
[
λ
n∑
k=0
(
n
k
)
Ek(x;λ)+ En(x;λ)
]
(n ∈ N0). (43)
In view of (36), this last expansion (43) in series of the Apostol-Euler polynomials
{En(x;λ)}∞n=0 is indeed an immediate consequence of (42), when α = 1.
By using (13) (with α = 1) and (15) (with α = 1), we have
∞∑
n=0
Bn(x;λ2)
tn
n!
=
text
λ2et− 1
=
t/2
λet/2− 1
·
2ext
λet/2+ 1
=
∞∑
n=0
2−nBn(λ)
tn
n!
·
∞∑
n=0
2−nEn(2x;λ)
tn
n!
=
∞∑
n=0
[
2−n
n∑
k=0
(
n
k
)
Bn−k(λ)Ek(2x;λ)
]
tn
n!
,
which yields the following relationship between the Apostol-Bernoulli and Apostol-
Euler polynomials:
Bn(x;λ2)=2−n
n∑
k=0
(
n
k
)
Bn−k(λ)Ek(2x;λ) (44)
or, equivalently,
2nBn
( x
2
;λ2
)
=
n∑
k=0
(
n
k
)
Bk(λ)En−k(x;λ). (45)
By applying similar arguments, it is not difficult to get the following explicit repre-
sentation for the Apostol-Euler polynomials En(x;λ) in terms of the Apostol-Bernoulli
polynomials Bn(x;λ):
En−1(x;λ)=
2n
n
[
λBn
(
x+ 1
2
;λ2
)
−Bn
( x
2
;λ2
)]
(46)
Introduction and Preliminaries 97
or, equivalently,
En−1(x;λ)=
2
n
[
Bn(x;λ)− 2nBn
( x
2
;λ2
)]
. (47)
In addition, from the relationships (46) (with x= 0) and (47) (with x= 0), we find
that
λBn
(
1
2
;λ2
)
= 2−nBn(λ)+ n ·2−n−1En−1(0;λ). (48)
Thus, by substituting for En−1(0;λ) from (47) (with x= 0) into (48), we obtain the
above-asserted relationship:
Bn(λ) := Bn(0;λ)=
(−1)n
λ
Bn
(
1;
1
λ
)
= 2n
[
1
2
Bn(λ2)+
λ
2
Bn
(
1
2
;λ2
)]
(n ∈ N0),
(49)
that is,
Bn(λ)= 2n−1
[
Bn(λ2)+ λBn
(
1
2
;λ2
)]
(n ∈ N0). (50)
The following relationship:
B(α)n (x+ y;λ)=
n∑
k=0
(
n
k
)[
B(α)k (y;λ)+
k
2
B(α−1)k−1 (y;λ)
]
En−k(x;λ) (51)
(α,λ ∈ C; n ∈ N0)
holds true between the generalized Apostol-Bernoulli polynomials and the Apostol-
Euler polynomials.
Luo [781] obtained the following general recursion formulas for the generalized
Apostol-Euler polynomials E (α)n (x;λ) and the generalized Apostol-Euler numbers
E (α)n (λ) (see [781, Equations (20) and (29)]):
E (α)n (x;λ)= 2α
n∑
k=0
(
n
k
)
xn−k
k∑
j=0
(
α+ j− 1
j
)
j!(−λ) j
(λ+ 1)j+α
S(k, j)
(52)
(α,λ ∈ C; n ∈ N0)
and
E (α)n (λ)= (−1)n
n∑
k=0
(
n
k
)
2k+ααn−k
k∑
j=0
(
α+ j− 1
j
)
j!(−λ) j
(λ+ 1)j+α
S(k, j)
(53)
(α,λ ∈ C; n ∈ N0).
98 Zeta and q-Zeta Functions and Associated Series and Integrals
Luo and Srivastava [789] gave an addition formula for each of the generalized
Apostol-Bernoulli and the generalized Apostol-Euler polynomials:
B(α)n (x+ y;λ)=
n∑
k=0
(
x
k
)
k!
n−k∑
j=0
(
n
j
)
B(α)j (y;λ)S(n− j,k) (54)
and
E (α)n (x+ y;λ)=
n∑
k=0
(
x
k
)
k!
n−k∑
j=0
(
n
j
)
E (α)j (y;λ)S(n− j,k) (55)
(α,λ ∈ C; n ∈ N0)
hold true between the generalized Apostol-Bernoulli polynomials, the generalized
Apostol-Euler polynomials and the Stirling numbers of the second kind. By setting
λ= 1 in (54) and (55), it is easy to deduce the following interesting identities:
B(α)n (x+ y)=
n∑
k=0
(
x
k
)
k!
n−k∑
j=0
(
n
j
)
B(α)j (y)S(n− j,k) (α ∈ C; n ∈ N0) (56)
and
E(α)n (x+ y)=
n∑
k=0
(
x
k
)
k!
n−k∑
j=0
(
n
j
)
E(α)j (y)S(n− j,k) (α ∈ C; n ∈ N0) (57)
for the generalized Bernoulli polynomials and the generalized Euler polynomials of
order α.
Apostol-Genocchi Polynomials and Numbers
Since the publication of the works by Luo and Srivastava (see [780], [781], [788] and
[789]), many further investigations of the above-mentioned Apostol type polynomials
have appeared in the literature. Boyadzhiev [162] gave some properties and repre-
sentations of the Apostol-Bernoulli polynomials and the Eulerian polynomials. Garg
et al. [467] studied the Apostol-Bernoulli polynomials of order α and obtained some
new relations and formulas involving the Apostol type polynomials and the Hurwitz
(or generalized) zeta function ζ(s,a) defined by 2.2(1) below. Luo (see [782] and
[783]) obtained the Fourier expansions and integral representations for the Apostol-
Bernoulli and the Apostol-Euler polynomials and gave the multiplication formulas for
the Apostol-Bernoulli and the Apostol-Euler polynomials of order α. Prévost [915]
investigated the Apostol-Bernoulli and the Apostol-Euler polynomials by using the
Introduction and Preliminaries 99
Padé approximation methods. Wang et al. (see [1207] and [1208]) further developed
some results of Luo and Srivastava [789] and obtained some formulas involving power
sums of the Apostol type polynomials. Zhang and Yang [1258] gave several identi-
ties for the generalized Apostol-Bernoulli polynomials. Conversely, Cenkci and Can
[227] gave a q-analogue of the Apostol-Bernoulli polynomials Bn(x;λ). Choi et al.
[267] gave the q-extensions of the Apostol-Bernoulli polynomials of order α and the
Apostol-Euler polynomials of order α (see also [268]). Hwang et al. [579] and Kim
et al. [663] also gave q-extensions of Apostol’s type Euler polynomials.
On the subject of the Genocchi polynomials Gn(x) and their various extensions,
a remarkably large number of investigations have appeared in the literature (see,
e.g., [229], [268], [566], [567], [568], [604], [655], [656], [657], [720], [721], [722],
[773], [784], [785], [786], [886] and [1257]; see also the references cited in each of
these works). Moreover, Luo (see [784] and [786]) introduced and investigated the
Apostol-Genocchi polynomials of (real or complex) order α, which are defined as
follows.
The Apostol-Genocchi polynomials
G(α)n (x;λ) (λ ∈ C)
of (real or complex) order α are defined by means of the following generating function:(
2z
λez+ 1
)α
· exz =
∞∑
n=0
G(α)n (x;λ)
zn
n!
(
|z|< |log(−λ)| ; 1α := 1
)
(58)
with, of course,
G(α)n (x)= G(α)n (x;1) , G(α)n (λ) := G(α)n (0;λ), (59)
Gn (x;λ) := G(1)n (x;λ) and Gn (λ) := G(1)n (λ), (60)
where Gn (λ), G(α)n (λ) and Gn (x;λ) denote the so-called Apostol-Genocchi numbers,
the Apostol-Genocchi numbers of order α and the Apostol-Genocchi polynomials,
respectively.
Important Remarks and Observations
The constraints on |z|, which we have used in the definitions (1), (13), (15) and (58)
above, are meant to ensure that the generating functions in (1), (13), (15) and (58)
are analytic throughout the prescribed open disks in the complex z-plane (centred
at the origin z= 0) to have the corresponding convergent Taylor-Maclaurin series
expansions (about the origin z= 0) occurring on their right-hand sides (each with
a positive radius of convergence). Moreover, throughout this investigation, logz is
tacitly assumed to denote the principal branch of the many-valued function logz
100 Zeta and q-Zeta Functions and Associated Series and Integrals
with the imaginary part I
(
logz
)
constrained by−π < I
(
logz
)
5 π . More importantly,
throughout this presentation, wherever | logλ| and | log(−λ)| appear as the radii of the
open disks in the complex z-plane (centred at the origin z= 0) in which the defining
generating functions are analytic, it is tacitly assumed that the obviously exceptional
cases when λ= 1 and λ=−1, respectively, are to be treated separately. Naturally,
therefore, the corresponding constraints on |z| in the earlier investigations (see, e.g.,
[781], [788], [789] and [1087]) and elsewhere in the literature dealing with one or
the other of these Apostol type polynomials (see also [58]) should also be modified
accordingly.
Generalizations and Unified Presentations of the Apostol Type
Polynomials
The mutual relationships among the families of the generalized Apostol-Bernoulli
polynomials, the generalized Apostol-Euler polynomials and the generalized Apostol-
Genocchi polynomials, which are asserted by Problems 70, 71 and 72 below, can be
appropriately applied with a view to translating various formulas involving one family
of these generalized polynomials into the corresponding results involving each of the
other two families of these generalized polynomials. Nevertheless, we find it useful to
investigate properties and results involving these three families of generalized Apos-
tol type polynomials in a unified manner. In fact, the following interesting unification
(and generalization) of the generating functions of the three families of Apostol type
polynomials was recently investigated, rather systematically, by Ozden et al. (cf. [886,
p. 2779, Equation (1.1)]):
21−κ zκ exz
βbez− ab
=
∞∑
n=0
Yn,β(x;κ,a,b)
zn
n!
(61)
(
|z|< 2π when β = a; |z|<
∣∣∣∣b log(βa
)∣∣∣∣ when β 6= a;
1α := 1; κ,β ∈ C; a,b ∈ C \ {0}
)
,
where we have not only suitably relaxed the constraints on the parameters κ , a and
b, but we have also strictly followed the above remarks and observations regarding
the open disk in the complex z-plane (centred at the origin z= 0) within which thegenerating function in (61) is analytic to have the corresponding convergent Taylor-
Maclaurin series expansion (about the origin z= 0) occurring on the right-hand side
(with a positive radius of convergence).
Here, in conclusion of our present section, we first define the following unification
(and generalization) of the generating functions of the above-mentioned three families
of the generalized Apostol type polynomials.
Introduction and Preliminaries 101
Definition 1.1 The generalized Apostol type polynomials
F (α)n (x;λ;µ;ν) (α,λ,µ,ν ∈ C)
of (real or complex) order α are defined by means of the following generating
function:(
2µ zν
λez+ 1
)α
· exz =
∞∑
n=0
F (α)n (x;λ;µ;ν)
zn
n!
(
|z|< |log(−λ)| ; 1α := 1
)
, (62)
so that, by comparing Definition 1 with the corresponding definitions given above, we
have
B(α)n (x;λ)= (−1)α F (α)n (x;−λ;0;1) , (63)
E (α)n (x;λ)= F (α)n (x;λ;1;0) (64)
and
G(α)n (x;λ)= F (α)n (x;λ;1;1) . (65)
Furthermore, if we compare the generating functions (61) and (62), we have
Yn,β(x;κ,a,b)=−
1
ab
F (1)n
(
x;−
(
β
a
)b
;1− κ;κ
)
. (66)
We, thus, see from the relationships (63), (64), (65) and (66), that the generating
function of F (α)n (x;λ;µ;ν) in (62) includes, as its special cases, not only the gener-
ating function of the polynomials Yn,β(x;κ,a,b) in (61) and the generating functions
of all three of the generalized Apostol type polynomials B(α)n (x;λ), E (α)n (x;λ) and
G(α)n (x;λ), but also the generating functions of their special cases B(α)n (x), E(α)n (x)
and G(α)n (x).
The various interesting properties and results involving the new unified family of
generalized Apostol type polynomials F (α)n (x;λ;µ;ν), given by Definition 1 above,
can also be derived in a manner analogous to that of our investigation in this presen-
tation.
The following natural generalization and unification of the Apostol-Bernoulli
polynomials B(α)n (x;λ) of order α, as well as the generalized Bernoulli numbers
Bn (a,b) studied by Guo and Qi [519] and the generalized Bernoulli polynomials
Bn (x;a,b) studied by Luo et al. [787], was introduced and investigated recently by
Srivastava et al. [1095].
102 Zeta and q-Zeta Functions and Associated Series and Integrals
Definition 1.2 (cf. [1095, p. 254, Equation (20)]). The generalized Apostol-Bernoulli
type polynomials B(α)n (x;λ;a,b,c) of order α ∈ C are defined by the following gen-
erating function:(
z
λbz− az
)α
· cxz =
∞∑
n=0
B(α)n (x;λ;a,b,c)
zn
n!
(67)|z|<
∣∣∣∣∣∣ logλlog( ba)
∣∣∣∣∣∣ ; a ∈ C \ {0}; b,c ∈ R+; a 6= b; 1α := 1
 .
In a sequel to the work by Srivastava et al. [1095], a similar generalization of each
of the families of Euler and Genocchi polynomials were introduced and investigated
(see, for details, [1096, Section 4]).
Definition 1.3 (cf. [1096, Section 2]). The generalized Apostol-Euler type polynomi-
als E(α)n (x;λ;a,b,c) of order α ∈ C are defined by the following generating function:(
2
λbz+ az
)α
· cxz =
∞∑
n=0
E(α)n (x;λ;a,b,c)
zn
n!
(68)|z|<
∣∣∣∣∣∣ log(−λ)log( ba)
∣∣∣∣∣∣ ; a ∈ C \ {0}; b,c ∈ R+; a 6= b; 1α := 1
 .
Definition 1.4 (cf. [1096, Section 4]). The generalized Apostol-Genocchi type polyno-
mials G(α)n (x;λ;a,b,c) of order α ∈ C are defined by the following generating func-
tion: (
2z
λbz+ az
)α
· cxz =
∞∑
n=0
G(α)n (x;λ;a,b,c)
zn
n!
(69)|z|<
∣∣∣∣∣∣ log(−λ)log( ba)
∣∣∣∣∣∣ ; a ∈ C \ {0}; b,c ∈ R+; a 6= b; 1α := 1
 .
Remark 1 In their special case when
a= 1 and b= c= e,
the generalized Apostol-Bernoulli type polynomials B(α)n (x;λ;a,b,c) defined by
(67), the generalized Apostol-Euler type polynomials E(α)n (x;λ;a,b,c) defined by
(68) and the generalized Apostol-Genocchi type polynomials G(α)n (x;λ;a,b,c)
defined by (69) would reduce at once to the Apostol-Bernoulli polynomials B(α)n (x;λ),
the Apostol-Euler polynomials E (α)n (x;λ) and the Apostol-Genocchi polynomials
G(α)n (x;λ), respectively.
Introduction and Preliminaries 103
Since the parameter λ ∈ C, by comparing Definitions 2, 3 and 4 above, we can
easily deduce the following potentially useful lemma (see also Lemmas 1, 2 and 3).
Lemma 1.7 The families of the generalized Apostol-Bernoulli type polynomials
B(l)n (x;λ;a,b,c) (l ∈ N0)
and the generalized Apostol-Euler type polynomials
E(l)n (x;λ;a,b,c) (l ∈ N0)
are related by
B(l)n (x;λ;a,b,c)=
(
−
1
2
)l n!
(n− l)!
E
(l)
n−l (x;−λ;a,b,c) (n, l ∈ N0; n = l)
(70)
or, equivalently, by
E(l)n (x;λ;a,b,c)= (−2)
l n!
(n+ l)!
B
(l)
n+l (x;−λ;a,b,c) (n, l ∈ N0). (71)
Furthermore, the families of the generalized Apostol-Bernoulli type polynomials
B(l)n (x;λ;a,b,c) (l ∈ N0)
and the generalized Apostol-Euler type polynomials
E(l)n (x;λ;a,b,c) (l ∈ N0)
are related to the generalized Apostol-Genocchi type polynomials
G(l)n (x;λ;a,b,c) (l ∈ N0)
as follows:
G(α)n (x;λ;a,b,c)= (−2)
α B(α)n (x;−λ;a,b,c)
(
α ∈ C; 1α := 1
)
(72)
and
G(l)n (x;λ;a,b,c)= (−1)
l (−n)l E
(l)
n−l (x;λ;a,b,c)=
n!
(n− l)!
E
(l)
n−l (x;λ;a,b,c)
(
n, l ∈ N0; n = l; λ ∈ C
)
.
104 Zeta and q-Zeta Functions and Associated Series and Integrals
The inter-relationships asserted by the above Lemma do aid in translating the var-
ious properties and results involving any one of these three families of generalized
Apostol type polynomials in terms of the corresponding properties and results involv-
ing the other two families. Nonetheless, it would occasionally seem more appropri-
ately convenient to investigate these three families in a unified manner by means of
Definition 5 below.
Definition 1.5 A unification of the generalized Apostol-Bernoulli type polynomials
B(α)n (x;λ;a,b,c) ,
the generalized Apostol-Euler type polynomials
E(α)n (x;λ;a,b,c)
and the generalized Apostol-Genocchi type polynomials
G(α)n (x;λ;a,b,c)
of order α ∈ C is defined by the following generating function:
(
2µ zν
λbz+ az
)α
· cxz =
∞∑
n=0
Z(α)n (x;λ;a,b,c;µ;ν)
zn
n!
(73)|z|<
∣∣∣∣∣∣ log(−λ)log( ba)
∣∣∣∣∣∣ ; a ∈ C \ {0}; b,c ∈ R+; a 6= b; α,λ,µ,ν ∈ C; 1α := 1
 ,
so that, by comparing Definition 5 with Definitions 1 through 4, we have
F (α)n (x;λ;µ;ν)= Z(α)n (x;λ;1,e,e;µ;ν), (74)
B(α)n (x;λ;a,b,c)= (−1)
α Z(α)n (x;−λ;a,b,c;0;1) , (75)
E(α)n (x;λ;a,b,c)= Z(α)n (x;λ;a,b,c;1;0) (76)
and
G(α)n (x;λ;a,b,c)= Z(α)n (x;−λ;a,b,c;1;1) . (77)
Thus, clearly, Definitions 1 and 5 above provide us with remarkably powerful and
extensive generalizations of the various families of the Apostol type polynomials and
Apostol type numbers. Properties and results involving these generalizations deserve
to be investigated further (see also [791], [886], [1095] and [1096]; see also [1091]
and Problem 73 onwards of this chapter).
Introduction and Preliminaries 105
1.9 Inequalities for the Gamma Function and the Double
Gamma Function
The Gamma Function and Its Relatives
Recently, many research articles were published providing inequalities for the Gamma
function and its relatives. We refer to Gautschi’s survey paper [476] and the compre-
hensive bibliography complied by Sándor [1003].
We begin by proving that log0 is convex on (0,∞) (cf. Theorem 1.1):
0
(
x
p
+
y
q
)
5 [0(x)]1/p [0(y)]1/q
(
1< p<∞;
1
p
+
1
q
= 1
)
. (1)
Indeed, if 1< p<∞ and 1/p+ 1/q= 1, then we have
0
(
x
p
+
y
q
)
=
∞∫
0
t
x
p+
y
q−1 e−t dt
=
∞∫
0
t
x−1
p t
y−1
q e−(1/p+1/q)t dt
5
 ∞∫
0
(
t
x−1
p e−t/p
)p
dt
1/p  ∞∫
0
(
t
y−1
q e−t/q
)q
dt
1/q
=
 ∞∫
0
tx−1 e−t dt
1/p  ∞∫
0
ty−1 e−t dt
1/q
= [0(x)]1/p [0(y)]1/q,
where we made use of the well-known Hölder’s inequality:
For 1 5 p 5∞, the spaces Lp(µ) are Banach spaces and, if f ∈ Lp(µ) and
g ∈ Lq(µ) (with 1/p+ 1/q= 1), then fg ∈ L1(µ) and∫
|f (µ)g(µ)|dµ5
(∫
| f (µ)|p dµ
)1/p (∫
|g(µ)|q dµ
)1/q
. (2)
We recall, here (see, e.g., [115, 406, 932, 939, 1184, 1226]), that a function f is said
to be strictly completely monotonic on an interval I ⊂ R, if
(−1)n f (n)(x) > 0 (x ∈ I; n ∈ N0). (3)
If (−1)n f (n)(x)= 0 for all x ∈ I and n ∈ N0, then f is called completely monotonic on
I. We recall also (see, e.g.,[76, 935, 939, 943, 944, 948, 950]) that a positive function
106 Zeta and q-Zeta Functions and Associated Series and Integrals
f is said to be logarithmically completely monotonic on an interval I ⊂ R, if
(−1)n {ln f (x)}(n) = 0 (x ∈ I; n ∈ N). (4)
For convenience, C[I] and L[I] denote, respectively, the sets of completely mono-
tonic functions and the logarithmically completely monotonic functions on an interval
I ⊂ R. It is known (see [115, 926, 935, 939, 943, 944]) that L[I]⊂ C[I].
Alzer [22] remarked that completely monotonic functions play a dominant rôle
in areas, such as numerical analysis, probability theory and physics. The concept of
complete and logarithmically complete monotonicities has also played an important
rôle to prove some inequalities involving the gamma function. Here, we investigate
some known inequalities involving the gamma function, by, mainly, focusing on C[I]
and L[I].
Wendel [1223] proved the following double inequality:(
x
x+ a
)1−a
5
0(x+ a)
xa0(x)
5 1 (0< a< 1; x> 0), (5)
which can be rewritten as follows:
x1−a 5
0(x+ 1)
0(x+ a)
5 (x+ a)1−a (0< a< 1; x> 0), (6)
to establish the well-known asymptotic relation (see 1.1(57)):
lim
x→∞
0(x+ a)
xa0(x)
= 1 (a, x ∈ R), (7)
by using the Hölder’s inequality (2).
Komatu [689] proved the inequality
1
x+
√
x2+ 2
< ex
2
∞∫
x
e−t
2
dt 5
1
x+
√
x2+ 1
(0 5 x<∞). (8)
Pollak [905] has improved the upper bound in (8), by showing that
ex
2
∞∫
x
e−t
2
dt 5
1
x+
√
x2+ 4/π
. (9)
Gautschi [470] proved more general inequalities than those in (8) and (9):
1
2
[(
xp+ 2
)1/p
− x
]
< ex
p
∞∫
x
e−t
p
dt 5 cp
[(
xp+
1
cp
)
− x
]
(0 5 x<∞),
(10)
Introduction and Preliminaries 107
where
cp :=
{
0
(
1+
1
p
)}p/(p−1)
(p ∈ N \ {1}).
For p= 2, the right-hand inequality of (10) reduces to (9), whereas the left-hand
inequality reduces to the corresponding inequality in (8). The integral in (10) for p= 3
occurs in heat transfer problems [1242], for p= 4 in the study of electrical discharge
through gases [1009]. An application of (10) for general p is given in [840].
Gautschi [470] also derived inequalities for the following Gamma-function ratio:
0(n+ 1)
0(n+ s)
,
which are given by
n1−s 5
0(n+ 1)
0(n+ s)
5 exp[(1− s)ψ(n+ 1)] (0 5 s 5 1; n ∈ N) (11)
and
n1−s 5
0(n+ 1)
0(n+ s)
5 (n+ 1)1−s (0 5 s 5 1; n ∈ N), (12)
where ψ denotes the Psi-function defined in 1.3(1). The inequalities (11) have
attracted remarkable interest, and several intriguing papers on the subject were sub-
sequently published by, for example, Erber [420], Kec̆kić and Vasić [642], Laforgia
[724] and Zimering [1266], providing new bounds for the following Gamma-function
ratio:
0(n+ 1)
0(n+ s)
.
However, it is observed that the upper bound in (12) is not better and the range in (12)
is not broader than the corresponding ones in (6).
Kershaw [645] gave proofs of the following closer bounds than (11):
exp
[
(1− s)ψ
(
x+ s1/2
)]
<
0(x+ 1)
0(x+ s)
< exp
[
(1− s)ψ
(
x+
s+ 1
2
)]
(13)
and
(
x+
s
2
)1−s
<
0(x+ 1)
0(x+ s)
<
[
x−
1
2
+
(
s+
1
4
)1/2]1−s
, (14)
each being valid for x> 0 and 0< s< 1.
108 Zeta and q-Zeta Functions and Associated Series and Integrals
Bustoz and Ismail [196] established a remarkably more general result. They
showed that the two functions
f1(x)=
0(x+ s)
0(x+ 1)
exp
[
(1− s)ψ
(
x+
s+ 1
2
)]
(0< s< 1) (15)
and
f2(x)=
0(x+ 1)
0(x+ s)
(
x+
s
2
)s−1
(0< s< 1) (16)
are strictly completely monotonic on (0,∞). Since
lim
x→∞
f1(x)= lim
x→∞
f2(x)= 1,
the inequalities (13) and (14) are immediate consequences of the fact that f1 and
f2 are strictly decreasing on (0,∞). Alzer [22] refined one of the Kec̆kić-Vasić’s
inequalities in [642] and also gave the following interesting result:
For every s ∈ (0, 1), the function
x 7−→ fα(x,s)=
0(x+ s)
0(x+ 1)
(x+ 1)x+1/2
(x+ s)x+s−1/2
· exp
(
s− 1+
1
12
[
ψ ′(x+ 1+α)−ψ ′(x+ s+α)
])
(α > 0)
(17)
is strictly completely monotonic on (0,∞), if and only if α = 12 . Furthermore, for
every s ∈ (0, 1), the function
x 7−→
1
fβ(x,s)
(β = 0)
is strictly completely monotonic on (0,∞), if and only if β = 0.
Gurland [527] obtained the following inequality
[0(λ+α)]2
0(λ)0(λ+ 2α)
<
λ
α2+ λ
(α, λ ∈ R; α+ λ > 0; λ > 0; α 6= 0; α 6= 1), (18)
by making a novel use of the Cramér-Rao lower bound for the variance of an unbiased
estimator (see Cramér [343]). Indeed, we consider the density function given by
f (x)=
1
θλ0(λ)
exp
(
−
x
θ
)
xλ−1 (x> 0; λ > 0; θ > 0). (19)
From the fact that the expression:
0(λ)
0(α+ λ)
xα
Introduction and Preliminaries 109
is an unbiased estimator of θα , we have the following Cramér-Rao bound for the
variance:
V
[
0(λ)
0(α+ λ)
xα
]
=
1
E
[
∂
∂θα
{log f (x)}
]2 ,
which yields the inequality (18).
A special case of (18) with λ= n/2 and α = 1/2 is seen to be reduced to Gurland’s
formula [526]:
4n+ 3
(2n+ 1)2
[
(2n)!!
(2n− 1)!!
]2
< π <
4
4n+ 1
[
(2n)!!
(2n− 1)!!
]2
(n ∈ N), (20)
where
(2n)!! := 2 ·4 · · ·(2n− 2)(2n) and (2n− 1)!! := 1 ·3 ·5· · ·(2n− 3)(2n− 1).
The inequality (20) is a sharper inequality for approximating π than that given by
Wallis:
2
2n+ 1
[
(2n)!!
(2n− 1)!!
]2
< π <
1
n
[
(2n)!!
(2n− 1)!!
]2
(n ∈ N). (21)
Gokhale [492] obtained a similar inequality to (18) as follows:
0(λ)0(λ+ 2α)
[0(λ+α)]2
> 1+
α2 (λ− 2)
(λ+α− 1)2
(α, λ ∈ R; λ+ 2α > 0,
λ > 2, α 6= 0,α 6= −1).
(22)
Gurland’s inequality (18) can be written in the form:
0(λ)0(λ+ 2α)
[0(λ+α)]2
> 1+
α2
λ
. (23)
We note that the inequality (22) is seen to be stronger than that in (23) for a certain
range of α. Indeed, when
α2 (λ− 2)
(λ+α− 1)2
>
α2
λ
, that is,
− (λ− 1)−
√
λ(λ− 2) < α <−(λ− 1)+
√
λ(λ− 2).
110 Zeta and q-Zeta Functions and Associated Series and Integrals
By employing the multivariate generalization of (19), that is, the Wishart distribu-
tion, Olkin [878] obtained
p−1∏
j=0
[0(λ+α− j/2)]2
0(λ− j/2)0(λ+ 2α− j/2)
≤
λ2
(
p2− 1
)2
+ λp4
λ2
(
p2− 1
)2
+ λp4+α2(
α, λ ∈ R; α+ λ >
p− 1
2
; λ > 0; p> 0
)
.
(24)
The special case of (24) with p= 1 reduces to (18). Selliah [1018], by employing the
multiparameter version of the Cramér-Rao lower bound, using the information matrix
for the same problem, obtained the following inequality:
p−1∏
j=0
[0(λ+α− j/2)]2
0(λ− j/2)0(λ+ 2α− j/2)
≤
λ
λ+ pα2(
α, λ ∈ R; α+ λ >
p− 1
2
; λ > 0; p> 0
)
,
(25)
which is seen to be sharper than that given in (24).
For x ∈ (−α,∞), define the function zs,t(x) by
zs,t(x) :=

[
0(x+ t)
0(x+ s)
]1/(t−s)
− x (s 6= t)
eψ(x+s)− x (s= t),
(26)
where s, t ∈ R+0 , R
+
0 being the set of nonnegative real numbers, and α =min{s, t}.
A monotonicity and convexity of zs,t(x) was proved (see [241, 406, 929, 945]) so
that the function zs,t(x) is either convex and decreasing for |t− s|< 1 or concave and
increasing for |t− s|> 1. From this fact, the best bounds in the Kershaw’s double
inequality (14) could be deduced. Qi [930, 931] further generalized this result.
For x ∈ (−ρ,∞), define the function Ha,b,c(x) by
Ha,b,c(x) := (x+ c)
b−a 0(x+ a)
0(x+ b)
(a, b, c ∈ R; ρ =min{a, b, c}). (27)
Very recently, Qi [932, Theorem 1] proved the following results:
Ha,b,c(x) ∈ L[(−ρ,∞)]
(
(a, b, c) ∈ D1(a, b, c)
)
(28)
and [
Ha,b,c(x)
]−1
∈ L[(−ρ,∞)]
(
(a, b, c) ∈ D2(a, b, c)
)
, (29)
Introduction and Preliminaries 111
where, for convenience,
D1(a, b, c) :=
{
(a, b, c) | a+ b = 1, c 5 b< c+
1
2
}
∪
{
(a, b, c) | a> b = c+
1
2
}
∪
{
(a, b, c) | 2a+ 1 5 a+ b 5 1, a< c
}
∪
{
(a, b, c) | b− 1 5 a< b 5 c
}
\ {(a, b, c) | a= c+ 1, b= c}
and
D2(a, b, c) :=
{
(a, b, c) | a+ b = 1, c 5 a< c+
1
2
}
∪
{
(a, b, c) | b> a = c+
1
2
}
∪
{
(a, b, c) | b< a 5 c
}
∪
{
(a, b, c) | b+ 1 5 a, c 5 a 5 c+ 1
}
∪
{
(a, b, c) | b+ c+ 1 5 a+ b 5 1
}
\ {(a, b, c) | a= c+ 1, b= c}
\ {(a, b, c) | b= c+ 1, a= c} .
Qi [932, Theorem 2] made use of (28), (29) and 1.1(37) to prove the following inequal-
ities:
(x+ c)a−b <
0(x+ a)
0(x+ b)
(x ∈ (−ρ,∞); (a, b, c) ∈ D1(a, b,c)) (30)
and
0(x+ a)
0(x+ b)
5
0(δ+ a)
0(δ+ b)
(
x+ c
δ+ c
)a−b
(x ∈ [δ,∞); (a, b, c) ∈ D1(a, b, c)) , (31)
where a, b, c ∈ R, ρ =min{a, b, c}, and δ is a constant greater than −ρ. If (a, b, c) ∈
D2(a, b, c), then inequalities in (30) and (31) are reversed, respectively, in (−ρ,∞)
and [δ,∞).
Qi [932] then observed the following facts: setting a= 1 and 0< b< 1 in (30)
reveals that
(x+ b)1−b <
0(x+ 1)
0(x+ b)
(0< b< 1; x ∈ (−b,∞)) (32)
holds true. It is obvious that the inequality in (32) not only refines the lower bound,
but also extends the range of the left-hand side of the inequality in (14).
Taking a= 1, 0< b< 1 and δ = 1 in (31) shows that
0(x+ 1)
0(x+ b)
5
1
0(1+ b)
(
x+ b
1+ b
)1−b
(0< b< 1; x ∈ [1,∞)) (33)
holds true. A usual argument shows that, if
112 Zeta and q-Zeta Functions and Associated Series and Integrals
x =
(
1
2 −
√
b+ 14
)
(1+ b) 1−b
√
0(1+ b)+ 1
(1+ b) 1−b
√
0(1+ b)− 1
:= λ(b),
then the inequality (33) would be better than the right-hand side of (14). It is easy to
find that
lim
b→0+
λ(b) and lim
b→1−
λ(b)=
e+
(
1−
√
5
)
eγ
2eγ − e
∼= 0.6123686 · · ·< 1,
where γ denotes the Euler-Mascheroni constant defined by 1.1(3). This implies that
the inequality in (33) refines the right-hand side of (14), if b is close enough to 1 and
that the upper bound in (33) is better than the one in (14), if x is sufficiently large.
The Double Gamma Function
In a striking contrast to an abundant literature on the inequalities for the Gamma
function and its relatives, there has been a few known results for the inequalities for the
double Gamma function (see, e.g., [446], [105] and [244]). Presumably, Batir [105]
initiated the study of the following class of inequalities for the double Gamma function
02(x+ 1)= 1/G(x+ 1):
(0(x))
x
2 xx (2π)
x
2 e−
x
2−
x2
2 < G(x+ 1) <
(
0(x)
0(x/2)
)x
(8π)
x
2 e−
x
2−
x2
2
(
x ∈ R+
)
;
(34)
(2π)
x
2 e
x
2−
x2
2 +
x2
2 ψ(x/2) < G(x+ 1) < (2π)
x
2 e−
x2
2 +
x2
2 ψ(x)
(
x ∈ R+
)
; (35)
(2π)
x
2 (0(x+ 1))x exp
(
−
x
2
−
x2
2
−
x2
2
ψ(α(x))
)
(36)
< G(x+ 1) < (2π)
x
2 (0(x+ 1))x exp
(
−
x
2
−
x2
2
−
x2
2
ψ(β(x))
) (
x ∈ R+
)
,
where R+ denotes the set of positive real numbers,
α(x)=
x
3
and β(x)=
x2
2
1
(x+ 1) log(x+ 1)− x
.
Problems
1. Show that each form of the Gamma function, defined by 1.1(1), 1.1(2) and 1.1(7), is ana-
lytic in its given domain.
(Whittaker and Watson [1225, Chapter 12]; Rainville [959, pp. 15–18])
Introduction and Preliminaries 113
2. Use the Wielandt’s theorem [Theorem 1.3] to derive each of the following results:
l the Gauss product 1.1(4) from the Euler integral 1.1(1),
l Gauss’s multiplication formula 1.1(31),
l the representation of the Beta function by Gamma functions 1.1(42),
l Stirling’s formula 1.1(33).
(Remmert [973])
3. Prove the following series representations for the Euler-Mascheroni constant γ :
γ = 1+
∞∑
n=3
(−1)n
n
(
log(n− 1)
log2
)
and
γ =
∞∑
n=1
(−1)n
n
(
logn
log2
)
,
where the bracket denotes the greatest integer function.
(Gerst [480]; Sandham [1001])
4. Deduce the explicit Weierstrass canonical product form 1.3(3) for the double Gamma
function 02, by using the three properties associated with Barnes’s definition of 02.
(Barnes [94, pp. 265–269]; Whittaker and Watson [1225, p. 264])
5. Prove the following generalization of Gauss’s multiplication formula 1.1(51) for the
Gamma function:
m−1∏
`=0
0
(
kz+
k`
m
)
= (2π)
1
2 (m−k)
(
k
m
)mkz+ 12 (mk−m−k)
·
k−1∏
n=0
0
(
mz+
mn
k
)
(k, m ∈ N).
(Choi and Quine [278, p. 131]; Magnus et al. [795, p. 3])
6. For the Euler-Mascheroni constant γ , the sequence {sn}∞n=0, defined by
sn :=
(
1+
1
2
+ ·· ·+
1
n− 1
)
− logn (n ∈ N \ {1}),
satisfies the asymptotic property:
sn = γ +O
(
n−1
)
(n→∞).
Show that γ − sn (n ∈ N \ {1}) can be expressed, as follows, as an infinite sum with rational
terms:
γ − sn =
1
n
∞∑
m=0
tm+2(m+n
m
) (n ∈ N \ {1}),
114 Zeta and q-Zeta Functions and Associated Series and Integrals
where
tm+2 = −
1
(m+ 1)!
1∫
0
(0− x)(1− x) · · · (m− x)dx (m ∈ N0) .
(Elsner [416, p. 1537]; Jolley [613, pp. 14–15]).
7. Prove the von Staudt-Clausen theorem (Theorem 6.1):
B2n = In−
∑
p−1|2n
1
p
(n ∈ N),
where Bn is the Bernoulli number, In is an integer and the sum is taken over all primes p
such that p− 1 divides 2n.
(Carathéodory [210, pp. 281–284]; Apostol [65, pp. 274–275])
8. Prove the following rapidly converging series expansion for ψ(z):
ψ(z)=
1
2
log
(
(z+ n)2+ (z+ n)+
1
3
)
−
n∑
k=0
1
z+ k
−
1
9
∞∑
k=n+1
(k− n)Ak,
where
Ak :=
1∫
0
(z+ k+ t)−2
(
(z+ k+ t)4−
1
3
(z+ k+ t)2+
1
9
)−1
dt.
(Shafer and Lossers [1020])
9. Let Sn denote the area of the sphere of radius 1 in Rn. Prove that
Sn =
(
∞∫
−∞
e−t
2
dt
)n
∞∫
0
e−r2 rn−1 dr
=
2
(√
π
)n
0
( n
2
) (n ∈ N).
(Campbell [208, pp. 126–128])
10. Let
In :=
∫
· · ·
∫
f
(
n∑
k=1
tk
)
tα1−11 · · · t
αn−1
n dt1 · · ·dtn(
αk > 0; tk = 0; k = 1, · · · , n;
n∑
k=1
tk 5 1
)
,
where f is a continuous function. Prove that
In =
0(α1) · · ·0(αn)
0(α1+ ·· ·+αn)
1∫
0
f (τ )τα1+···+αn−1 dτ.
(Whittaker and Watson [1225, p. 258]; Campbell [208, pp. 128–133])
Introduction and Preliminaries 115
11. Prove that
π∫
0
(sin θ)λ−1 e−µθ dθ =
π0(λ)e−
1
2πµ
2λ−10
(
λ+iµ+1
2
)
0
(
λ−iµ+1
2
) (<(λ) > 0; µ ∈ C).
(Nielsen [862, p. 159])
12. Show that
∞∫
−∞
e−2π izt
cosh (π t)
dt =
1
cosh (πz)
(
−
1
2
< =(z) <
1
2
)
(cf. Equation 1.1(46))
13. Making use of the Pochhammer symbol defined by 1.1(5), show that the Vandermonde
convolution theorem 1.4(9) can be rewritten in the form:
n∑
k=0
(α)k (β)n−k
k!(n− k)!
=
(α+β)n
n!
=
n∑
k=0
(α)n−k (β)k
(n− k)!k!
(α, β ∈ C; n ∈ N0) .
14. Show that Euler’s transformation 1.5(21) is equivalent to the Pfaff-Saalschütz theorem:
3F2
[
a, b,−n;
c, a+ b− c− n+ 1;
1
]
=
(c− a)n (c− b)n
(c)n (c− a− b)n
(
c 6∈ Z−0 ; n ∈ N0
)
.
15. Deduce Gauss’s summation theorem 1.5(7) as a limit case of the Pfaff-Saalschütz theorem
(see Problem 13 above).
[Hint: Let n→∞, and apply the asymptotic expansion 1.1(37).]
16. For the Bernoulli polynomials Bn(x) generated by 1.7(1), show that
xn =
n∑
k=0
1
n+ 1
(
n+ 1
k
)
Bk(x) (n ∈ N0) .
Hence (or otherwise), deduce the following representation for the Laguerre polynomials
L(α)n (x) defined by 1.5(72):
L(α)n (x)=
n∑
k=0
(−1)k
k!
(
n+α
n− k
)
Bk(x) ·2F2(−n+ k, 1; α+ k+ 1, 2 ; 1).
(cf. Magnus et al. [795, p. 26]; see also Popov [907])
17. Derive the following asymptotic expansion for the Psi (or Digamma) function:
ψ(z)= logz−
1
2z
−
1
12z2
+
1
120z4
−
1
252z6
+O
(
z−8
)
(|z| →∞; |arg(z)|5 π − � (0< � < π)).
116 Zeta and q-Zeta Functions and Associated Series and Integrals
18. For a, b ∈ N, let (a,b) and [a,b] denote the greatest common divisor and least common
multiple, respectively, of a and b, and {x} = x− [x] denote the fractional part of x. Then,
show that, if <(s) > 12 ,
1∫
0
ζ(1− s, {ax})ζ(1− s, {bx})dx=
2 {0(s)}2 ζ(2s)
(2π)2s
(
(a,b)
[a,b]
)s
,
where ζ(s,a) is the Hurwitz (or generalized) Zeta function defined by 2.2(1).
Show also that
1∫
0
Bn({ax})Bn({bx})dx= (−1)
n−1 B2n
(2n)!
(
(a,b)
[a,b]
)n
(n ∈ N).
(cf. Mordell [844, p. 372]; see also Mikolás [827])
19. Prove that
1/
√
8∫
0
t2 log t(
t2+ 1
) 1
2
dt =
1
96
[
2π2+ 30(log2)2− 39 log2− 9
]
.
(Knuth [679, p. 138])
20. Prove that
∞∫
0
cosh x · logx
cosh (2x)− cos (2πa)
dx
=
π
2 sin (πa)
log
0
(
1
2 a+
1
2
)
0
(
1
2 a
)
+ 1
2
log
(
2π cot
aπ
2
) (0< a< 1).
(Williams and Zhang [1232, p. 44])
21. Prove the following inequality:
cosec2 x−
1
2m+ 1
<
m∑
k=−m
1
(x− kπ)2
< cosec2 x (m ∈ N; 0< |x|5
1
2
π; x ∈ R).
Also, apply this inequality to show that
∞∫
0
(
sin x
x
)2
dx=
π
2
.
(Neville [858, pp. 629–630])
22. Prove that
∞∫
0
ex logx
e2x− 2ex cos (2πa)+ 1
dx=
π
2
1
sin (2πa)
log
(
(2π)1−2a
0(1− a)
0(a)
)
(0< a< 1).
(Zhang and Williams [1252, p. 377])
Introduction and Preliminaries 117
23. Prove that
π2∫
π
4
log(log(tanx)) dx=
π
2
log
0
(
3
4
)
0
(
1
4
)√2π
 .
(cf. Vardi [1189, p. 308]; see also Gradshteyn and Ryzhik [505, p. 532])
24. Let
P=
∞∏
n=1
(n− a1) (n− a2) · · · (n− ak)
(n− b1) (n− b2) · · · (n− bk)
 k∑
j=1
aj =
k∑
j=1
bj
 ,
where no aj or bj is a positive integer. Show that
P=
k∏
j=1
0
(
1− bj
)
0
(
1− aj
) .
(cf. Rainville [959, pp. 13–15]; see also Melzak [821, p. 101])
25. Let Mn and mn denote the maximum and minimum of the Bernoulli polynomial Bn(x) for
0 5 x 5 1. Show that
M4n = B4n
(
1
2
)
=
(
1− 21−4n
)
|B4n| (n ∈ N);
M4n+2 = B4n+2 (0)= B4n+2 (n ∈ N0) ;
m4n = B4n (0)=−|B4n| (n ∈ N);
m4n+2 = B4n+2
(
1
2
)
=−
(
1− 2−1−4n
)
B4n+2 (n ∈ N0) .
(Lehmer [740, pp. 534])
26. Prove the following integrals:
1∫
0
logu
u
[log(1+ u)]2 du=A−
π4
288
and
π∫
0
u
[
log
(
2 cos
u
2
)]2
du=A+
31π4
480
,
where
A=
∞∑
n=1
(−1)n−1
(n+ 1)2
(
n∑
k=1
1
k2
)
.
(Rutledge and Douglass [991, p. 30])
118 Zeta and q-Zeta Functions and Associated Series and Integrals
27. Suppose that
(a) f (z) is a meromorphic function of a complex variable z in C;
(b) f (z) is analytic and vanishes nowhere on C \Z−0 ;
(c) f (z) satisfies the functional equation
nz−
1
2
n−1∏
k=0
f
(
z+ k
n
)
= (2π)
n−1
2 f (z)
(
z ∈ C \Z−0
)
,
where n is an arbitrarily fixed positive integer greater than 1 and nz−
1
2 denotes the principal
value exp
[(
z− 12
)
Logn
]
. Then, show that the function f must be of the form:
f (z)= exp
[
a
(
z−
1
2
)
+
2mπ i
n− 1
]
0(z),
where a is an arbitrary complex constant and m is an arbitrary integer.
(cf. Haruki [540, p. 174]; see also Theorem 1.1 (Bohr-Mollerup))
28. Suppose that a function f : R+→ R satisfies the functional equation:
f (x+ 1)= x f (x)
(
x ∈ R+
)
and, moreover, that
lim
x→∞
( e
x
)x √ x
2π
f (x)= 1.
Show that f (x)= 0(x) for all x ∈ R+.
(Kuczma [705, p. 129])
29. Suppose that a function f : R+→ R satisfies the following properties:
(a) f (x+ 1)= x f (x);
(b)
( e
x
)x f (x) is decreasing for x ∈ R+;
(c) f (1)= 1.
Show that f (x)= 0(x) for x ∈ R+.
(Anastassiadis [37, p. 117])
30. Suppose that a function f : R+→ R+ satisfies the following properties:
(a) f (x+ 1)= xx+y f (x) (y> 0);
(b) f (x) is decreasing for x ∈ R+;
(c) f (1)= 1y .
Show that f (x)= B(x,y) for x ∈ R+.
(Anastassiadis [38, pp. 25–26])
31. Prove that, for p≥ 5 and p ∈ N,
B(p)p ≡−
1
2
p2 (p− 1)!
(
modp5
)
,
where B(p)p denotes the generalized Bernoulli numbers defined by 1.6(22).
(Carlitz [216, p. 112])
Introduction and Preliminaries 119
32. Prove that, for m ∈ N and i=
√
−1,
2F1(1, 1 ; 2m ; i)
= (2m− 1) im 2m−1
 log2
2
−
iπ
4
−
m−1∑
j=1
(−1) j
i j
2 j
(
1
2j
+
1+ i
2j− 1
)
and
2F1(1, 1 ; 2m+ 1 ; i)
= m(1+ i) im 2m
 log2− 1
2
− i
2−π
4
−
m−1∑
j=1
(−1) j
i j
2 j
(
1
2j
+
1− i
4j+ 2
) .
(Butzer and Hauss [198, p. 355])
33. Prove that
∞∑
n=0
1
(2n+ 1)2
(2n
n
) = 8
3
G−
π
3
log
(
2+
√
3
)
.
(Borwein and Borwein [148, p. 386])
34. Prove that
1∫
−1
(2− 2x)α (2+ 2x)β
(
1− x2
)− 12
dx=
0
(
α+ 12
)
0
(
β + 12
)
π
0(α+β + 1)
[
0
(
1
2
)]2 22α+2β .
(Askey [74, p. 357])
35. Prove that
log0(z+ 1)=
1
2
(
z+
1
2
) [
log
(
z+
1
2
+
1
2
√
3
)
+ log
(
z+
1
2
−
1
2
√
3
)]
− z−
1
2
+
1
2
log(2π)+
∞∫
0
e−zt φ(t)dt,
where, for convenience,
φ(t) :=
1
t
(
1
2
−
1
t
+
1
et − 1
)
−
1
2t
[
1− e−
1
2 t cosh
(
t
2
√
3
)]
−
d
dt
1− e
−
1
2 t cosh
(
t
2
√
3
)
t
 ;
moreover,
φ(t)=
e−
1
2 t
t
[
1
2
cosech
(
1
2
t
)
+
1
2
√
3
sinh
(
t
2
√
3
)
−
1
t
cosh
(
t
2
√
3
)]
.
(cf. Watson [1212, p. 5]; see also Eq. (29))
120 Zeta and q-Zeta Functions and Associated Series and Integrals
36. Prove that, for fixed p ∈ R+,
γ = lim
n→∞
∞∑
k=0
(
nk
k!
)p
(Hk − logn)
∞∑
k=0
(
nk
k!
)p ,
where γ is the Euler-Mascheroni constant and Hk denotes the harmonic numbers defined
by 3.2(36).
(Brent and McMillan [174, pp. 310])
37. Let u be the row vector {uk = 1/k : k ∈ N} and let M be the matrix with entries {mij =
1/(i− j+ 1) if j 5 i, mij = 0 if j> i : i, j ∈ N}. Let v be the column vector {vn = 1/(n+ 1) :
n ∈ N}. Show that the product u
(
M−1v
)
exists (as a convergent series), and is equal to
Euler-Mascheroni constant γ defined by 1.1(3).
(Kenter [644, p. 452])
38. Let �(z) be the function of the complex variable z defined by
�(z) :=
0(z+ a1) 0(z+ a2)
0(z+ b1) 0(z+ b2)
in which a1, a2, b1, b2 are to be regarded as any constants (real or complex). Show that,
for |arg z|< π and large |z|,
�(z)∼
(
1
z
)p (
1+
c1
z+ 1
+
c2
(z+ 1)(z+ 2)
+
c3
(z+ 1)(z+ 2)(z+ 3)
+ ·· ·
)
,
where the cn are constants and p= b1+ b2− a1− a2.
(Engen [1183, pp. 124–125])
39. Prove the following Lipschitz summation formula:
∞∑
m=−∞
(2m+ τ)−λ =
e−π iλ/2πλ
0(λ)
∞∑
n=1
nλ−1 eπ iτ n (<(τ ) > 0; λ > 1).
(Knopp [677, p. 65])
40. Prove the following series-integral representation of the Catalan constant G:
G=
1
2
∞∑
n=0
(−1)n

π
2∫
0
cos nt dt

2
.
(Catalan [224, p. 51])
41. Prove that
1
2
ψ ′(x)ψ ′′′(x) <
(
ψ ′′(x)
)2
(x> 0),
where ψ(x) denotes the Psi (or Digamma) function.
(cf. English and Rousseau [419, p. 432]; see also Alzer and Wells [33, p. 1459])
Introduction and Preliminaries 121
42. A function f is said to be completely monotonic on an interval I, if f ∈ C∞(I) and
(−1)k f (k)(x)= 0 (∗)
for all x ∈ I and for all k ∈ N0. If the inequality (∗) is strict for all x ∈ I and for all k ∈ N0,
then f is said to be strictly completely monotonic on I. Consider the function
Fn(x;c) :=
(
ψ (n)(x)
)2
− cψ (n−1)(x)ψ (n+1)(x)(
x ∈ R+; c ∈ R; n ∈ N \ {1}
)
.
Let n ∈ N \ {1}, and let α, β ∈ R. Show that the function.
x 7→ Fn(x;α)
is strictly completely monotonic on (0,∞), if and only if
α 5
n− 1
n
,
and that the function:
x 7→ −Fn(x;β)
is strictly completely monotonic on (0,∞), if and only if
β =
n
n+ 1
.
(Alzer and Wells [33, p. 1460])
43. Let the function g(x) be defined by
g(x) := x2ψ ′(1+ x)− xψ(1+ x)+ log0(x+ 1) (x>−1).
Show that g(x) strictly decreases from∞ to 0 on (−1,0] and strictly increases from 0 to
∞ on [0,∞).
(Elbert and Laforgia [404, Theorem 2])
44. Prove that
xx−1 ey
yy−1 ex
5
0(x)
0(y)
5
xx−
1
2 ey
yy−
1
2 ex
(
x = y> 1
)
.
(Kec̆kić and Vasić [642, p. 107])
45. Prove that
(a) For n, an odd positive integer,
G=−T
(
1
4
)
=
2n+ 1
n+ 1
n∑
j=1
(−1) j T
(
2j− 1
8n+ 4
)
;
(b) For n, an even positive integer,
G=−T
(
1
4
)
=
2n+ 1
n
n∑
j=1
(−1) j+1 T
(
2j− 1
8n+ 4
)
,
122 Zeta and q-Zeta Functions and Associated Series and Integrals
where, for convenience, T is defined by
T(r) :=
rπ∫
0
log(tanθ)dθ
(
0 5 r 5
1
2
)
.
(Bradley [165, pp. 164–165])
46. For the Stirling numbers s(n,k) defined by 1.6(1), show that
(a)
∞∑
n=0
(−1)n
(n+ k)(a)n
s(n+ k,k)=−
(1− a)k
k!
ψ (k)(a− k);
(b)
∞∑
n=0
(−1)n
(n+ k− 1)(a)n
s(n+ k,k)=
(−a)k
a(k− 1)!
ψ (k−1)(a− k).
Also deduce the special cases of each of these summation formulas, when
a= k+ 1 and a= k+ 2.
(Cf. Jordan [614, p. 343]; see also Hansen [531, p. 348])
47. Verify that the special case of Problem 46(a) above, when a= k+ 1 is precisely the rela-
tionship 3.5(16).
(Cf. Problem 5 (Chapter 3))
48. Show that
∞∫
0
cos x
xp
dx=
π
20(p) cos (pπ/2)
(0< p< 1)
and
∞∫
0
sin x
xp
dx=
π
20(p) sin (pπ/2)
(0< p< 1).
(Andrews [49, p. 83 and p. 85])
49. For p ∈ N, prove the following integral:
1∫
0
· · ·
1∫
0
 p∏
j=1
uj
x−1 
p∏
j=1
(1− uj)

y−1
|1(u)|2z du1 · · ·dup
=
p∏
ν=1
0(1+ νz)0(x+ (ν− 1)z)0(y+ (ν− 1)z)
0(1+ z)0(x+ y+ (p+ ν− 2)z)
(
<(x) > 0; <(y) > 0; <(z) >−min
{
1
p
,
<(x)
p− 1
,
<(y)
p− 1
})
,
where
1(u)=1(u1, u2, . . . , up) :=
∏
i<j
(
uj− ui
)
.
(Selberg [1013]; see aso [1015, pp. 204–213])
Introduction and Preliminaries 123
50. Prove the result asserted by Equation 1.3(57) in two ways, as noted there.
(Choi and Cvijović [271, Theorem 1])
51. Show that the generalized Barnes G-function Gm defined by
Gm(1+ z) = exp
−
m∑
j=1
ζ( j−m)j
(−z) j−
γ +Hm
m+ 1
(−z)1+m
+
m∑
j=1
(
m
j
)
ζ ′(j−m)(−z) j

·
∞∏
k=1
{(
1+
z
k
)km
exp
((
m+1∑
r=1
1
r kr
(−z)r
)
km
)}
(m ∈ N0)
is an entire function of order m with the zeros-k with multiplicity km (k ∈ N). Moreover,
there is the zero 0 with multiplicity 1 for m= 0. The used infinite product is absolutely
convergent. One has Gm(1)= 1, and Gm satisfies the functional equation
Gm(z+ 1)=
m∏
r=0
{Gm−r(z)}
(−1)r (mr) (m ∈ N).
Here, as usual, ζ(s) is the Riemann zeta function, γ the Euler-Mascheroni constant, and
Hm the harmonic numbers.
(Schuster [1010, Theorem A])
52. Show that the following integrals:
Ik(q) :=
∞∫
0
t(
1+ t2
)k+1 (e2πqt − 1) dt (k ∈ N)
are given by
Ik(q)=−
1
4k
−
(2k
k
)
22k+2 q
+
1
k 22k
k∑
j=1
(−1) j+1
( j− 1)!
(
2k− j− 1
k− j
)
2j−1 q jψ (j)(q).
(See Boros et al. [141, Theorem 3.2])
53. For arbitrary n ∈ N, show that
1
2n+ 11−γ − 2
5 Hn− logn− γ <
1
2n+ 13
,
where γ is the Euler-Mascheroni constant and Hn the harmonic numbers. Here, the con-
stants 11−γ − 2 and
1
3 are the best possible.
(See Qi et al. [938, Theorem 2])
54. The Genocchi numbers Gn are defined by means of the following generating function:
2t
et + 1
=:
∞∑
n=0
Gn
tn
n!
= t+
∞∑
n=1
G2n
t2n
(2n)!
,
124 Zeta and q-Zeta Functions and Associated Series and Integrals
which are directly related to the Bernoulli numbers Bn by means of
Gn = 2
(
1− 22n
)
Bn.
Show that
G2n =−n−
1
2
n−1∑
k=1
(
2n
2k
)
G2k and G2n =−1−
n−1∑
k=1
(
2n
2k− 1
)
G2k
2k
.
(See Herrmann [555, Proposition 2.1])
55. Let α > 0 and 0< c 6= 1 be real numbers. Show that the following inequality:(
0(x+ y+ c)
0(x+ y)
)1/α
<
(
0(x+ c)
0(x)
)1/α
+
(
0(y+ c)
0(y)
)1/α
holds true for all positive real numbers x and y, if and only if α = max {1, c}. The reverse
inequality is valid for all x, y> 0, if and only if α 5 min {1, c}.
(See Alzer [27])
56. Hadamard’s Gamma function H(x) is defined by
H(x)=
1
0(1− x)
d
dx
{
log
(
0(1/2− x/2)
0(1− x/2)
)}
,
where 0(x) denotes the classical Euler’s Gamma function given in 1.1(1). H(x) is an entire
function and satisfies the following relationship:
H(n)= (n− 1)! (n ∈ N).
Show that H can be expressed as follows:
H(x)= 0(x)
[
1+
sin (πx)
2π
{
ψ
( x
2
)
−ψ
(
x+ 1
2
)}]
in terms of the sine, Gamma and Psi (or Digamma) functions. Furthermore, Hadamard’s
Gamma function H(x) satisfies the following remarkable functional equation, which is a
counterpart of 1.1(9):
H(x+ 1)= xH(x)+
1
0(1− x)
,
which is a counterpart of 1.1(9).
(See Alzer [28]; Luschny [792]; Newton [859])
57. Show that the following inequalities hold true:
ψm(x)ψn(x)=
[
ψ m+n
2
(x)
]2 (
x> 0; m, n,
m+ n
2
∈ N
)
and
(s+ 1)
ζ(s)
ζ(s+ 1)
= s
ζ(s+ 1)
ζ(s+ 2)
(s> 1) ,
where ψn(x) := ψ (n)(x) and ζ(s) is the Riemann Zeta function given in 2.3(1).
(See Laforgia and Natalini [725])
Introduction and Preliminaries 125
58. Show that
γ =
2n
e2n
∞∑
r=0
2r n
(r+ 1)!
r∑
s=0
1
s+ 1
− n ln2+O
(
1
2n e2n
)
,
where γ denotes the Euler-Mascheroni constant (see Section 1.2).
(See Bailey [84]; Mortici [846])
59. Let m ∈ N and z ∈ C. Suppose also that 1(m;λ) abbreviates the following array of m
parameters:
λ
m
,
λ+ 1
m
, . . . ,
λ+m− 1
m
(m ∈ N).
Show that
n∑
k=0
(2mk+ 1)!!
(n− k)!(mk+ 1)!
zk
k!
=
1
n!
n∑
k=0
(
2mk+ 1
mk
)(
n
k
)( z
2m
)k
=
1
n! m+1
Fm
−n,1
(
m; 32
)
;
1(m;2) ;
− 2m · z
 . (37)
(Sofo and Srivastava [1044]; see, for special cases,
Srivastava [1089] and Samoletov [1000])
60. Show that each of the following limit formulas holds true:
lim
z→−k
0(nz)
0(qz)
=
(−1)(n−q)k · q
n
·
(qk)!
(nk)!
(n,q ∈ N; k ∈ N0)
and
lim
z→−k
ψ(nz)
ψ(qz)
=
q
n
(n,q ∈ N; k ∈ N0).
(Prabhu and Srivastava [912])
61. For a suitably bounded sequence {Cn}n∈N0 of essentially arbitrary complex numbers, show
that
∞∑
n=0
Cn
α+Nn
ωn zα+Nn
n!
=
∞∑
n=0
γ (α+ n,z)
n!
[n/N]∑
k=0
(
n
Nk
)
(Nk) !
k!
Ck ω
k (N ∈ N) ,
126 Zeta and q-Zeta Functions and Associated Series and Integrals
provided that each member exists. Also, deduce the following hypergeometric form of this
expansion formula:
zα p+1Fq+1
 α/N,(αp) ;
(α+N)/N,
(
βq
)
;
ω
(
−
z
N
)N
= α
∞∑
n=0
γ (α+ n,z)
n!
p+NFq
1(N;−n) ,(αp) ;(
βq
)
;
ω
 (N ∈ N) .
(Lin et al. [764, p. 518]; see, for special cases, Gautschi et al. [477])
62. For a suitably bounded sequence {�(n)}n∈N0 of essentially arbitrary complex numbers,
show that
∞∑
k=0
n∑
k1,··· ,kr=0
�(k)
(−n)k1 . . . (−n)kr (γ + n)k+k1+···+kr
k1! · · ·kr!(γ )k+k1+···+kr
·
(β1)k1 . . . (βr)kr
(β1+ 1)k1 · · ·(βr + 1)kr
=
(n!)r (γ −β1− ·· ·−βr)n
(γ )n (β1+ 1)n . . . (βr + 1)n
∞∑
k=0
�(k)
(γ −β1− ·· ·−βr + n)k
(γ −β1− ·· ·−βr)k
,
provided that the series involved are absolutely convergent.
(Carlitz [219, p. 169]; see also Lin and Srivastava [766, p. 310]
and Problem 26 of Chapter 6)
63. Let the harmonic numbers Hn and the generalized harmonic numbers H
(s)
n be defined, as
usual, by
Hn := H
(1)
n and H
(s)
n :=
n∑
k=1
1
ks
(s ∈ C; n ∈ N := {1,2,3, . . .}),
(a) For αj,βj = 0 (j= 1,2,3,4) and m ∈ N0, show that
∞∑
n=1
tn
(
n+m− 1
n− 1
)
[ψ(β1+ 1+α1n)−ψ(β1+ 1)]
n4
4∏
i=1
(
αin+βi
βi
)
=−α1α2α3α4
1∫
0
1∫
0
1∫
0
1∫
0
(1− x)β1 ln(1− x) · (1− y)β2 (1− z)β3 (1−w)β4
xyzw
·
xα1 yα2 zα3 wα4
(1− txα1 yα2 zα3 wα4)m+1
dx dy dz dw.
Introduction and Preliminaries 127
(b) Let p and q be positive integers. In terms of the Bernoulli polynomials Bn (x) and the
generalized Clausen functions Cln (z) , show that
ψ(n)
(
p
q
)
ψ(n)
(
p
q
)
= n!q
n

(−1)
⌊n
2
⌋
(2π)n+1
2(n+ 1) !
q∑
s=1
Bn+1
(
s
q
)
cos
(
2πps
q
)
sin
(
2πps
q
)

±
q∑
s=1
Cln+1
(
2πs
q
)
sin
(
2πps
q
)
cos
(
2πps
q
)


({
n= 2m− 1
n= 2m
}
; m ∈ N; 1 5 p 5 q
)
,
where the Bernoulli polynomials Bn (x) are generated, as usual, by
text
et − 1
=
∞∑
n=0
Bn (x)
tn
n!
(|t|< 2π)
and the generalized Clausen functions Cln (θ) are given by
Cln (θ)=

∞∑
k=1
sin (kθ)
kn
(n even)
∞∑
k=1
cos (kθ)
kn
(n odd).
(c) Let
α1 = 0, α2 = 0, α3 = 0 and α4 = 0
be positive real numbers. Also, let p ∈ N0 and
j,k, l,m ∈ N0, 0 5 p 5 3k− 4 and |t|5 1.
Then show that
∞∑
n=1
nptn [ψ( j+ 1+α1n)−ψ( j+ 1)](
α1n+ j
j
)(
α2n+ k
k
)(
α3n+ l
l
)(
α4n+m
m
)
= −α1klm
1∫
0
1∫
0
1∫
0
1∫
0
(1− x) j
x
(1− y)k−1 (1− z)l−1 (1−w)m−1
· ln(1− x)Li−p−1
(
txα1 yα2 zα3 wα4
)
dx dy dz dw,
128 Zeta and q-Zeta Functions and Associated Series and Integrals
where the Polylogarithmic function Liq (β) is given by
Liq (z)=
∞∑
r=1
zr
rq
.
(See Sofo and Srivastava [1045])
64. Let [τ ] denote the greatest integer in τ ∈ R. Then, for an essentially arbitrary sequence
{�n}n∈N0 of complex numbers, show that the following general combinatorial series rela-
tionship holds true:
[n/m]∑
k=0
(
λ+ k
k
) [k/m]∑
j=0
(
k
mj
)
�j z
j
=
[n/m]∑
k=0
(
λ+ n+ 1
n−mk
)(
λ+mk
mk
)
�k z
k
(λ ∈ C; m ∈ N; n ∈ N0)
or, equivalently,
[n/m]∑
k=0
(
λ+ k
k
) [k/m]∑
j=0
(
k
mj
)
�j z
j
=
(
λ+ n+ 1
n
) [n/m]∑
k=0
λ+ 1
λ+mk+ 1
(
n
mk
)
�k z
k
(λ ∈ C; m ∈ N; n ∈ N0),
provided that both members of each of these assertions exist.
Also, deduce the following general Fox-Wright hypergeometric series relationship:
p+39
∗
q+1
 (−n,m), (α,m), (a1,A1), . . . , (ap,Ap);
(α+ 1,m), (b1,B1), . . . , (bq,Bq);
z

=
(
α+ n
n
)−1 [n/m]∑
k=0
(
α+ k− 1
k
)
p+19
∗
q
 (−k,m), (a1,A1), . . . , (ap,Ap);
(b1,B1), . . . , (bq,Bq);
z
 ,
provided that each member of this assertion exists. Here, the Fox-Wright function
p9q (p,q ∈ N0) or p9∗q (p,q ∈ N0), with p numerator parameters a1, · · · ,ap and q
denominator parameters b1, · · · ,bq, such that
aj ∈ C ( j= 1, . . . ,p) and bj ∈ C \Z−0 ( j= 1, . . . ,q),
is defined by
p9
∗
q
 (a1,A1) , . . . ,(ap,Ap) ;
(b1,B1) , . . . ,
(
bq,Bq
)
;
z
 := ∞∑
k=0
(a1)A1k . . .
(
ap
)
Apk
(b1)B1k . . .
(
bq
)
Bqk
zk
k!
=
0(b1) · · ·0
(
bq
)0(a1) · · ·0
(
ap
) p9q
 (a1,A1) , . . . ,(ap,Ap) ;
(b1,B1) , . . . ,
(
bq,Bq
)
;
z

Aj > 0 ( j= 1, . . . ,p) ; Bj > 0 ( j= 1, . . . ,q) ; 1+ q∑
j=1
Bj−
p∑
j=1
Aj = 0
 ,
Introduction and Preliminaries 129
where the equality in the convergence condition holds true for suitably bounded values of
|z|, given by
|z|<
 p∏
j=1
A
−Aj
j
 ·
 q∏
j=1
B
Bj
j
 .
Clearly, in terms of the generalized hypergeometric function pFq (p,q ∈ N0), we have the
following relationship (see, for details, Section 1.5):
p9
∗
q
 (a1,1) , . . . ,(ap,1) ;
(b1,1) , . . . ,
(
bq,1
)
;
z
= pFq
 a1, . . . ,ap;
b1, . . . ,bq;
z

=
0(b1) · · ·0
(
bq
)
0(a1) · · ·0
(
ap
) p9q
 (a1,1) , . . . ,(ap,1) ;
(b1,1) , . . . ,
(
bq,1
)
;
z
 .
(See R. Srivastava [1118])
65. For the Bernoulli numbers Bn, which are usually given by the recurrence relation 1.7(6),
that is, by
Bn = (−1)
n
n∑
k=0
(
n
k
)
Bk (n ∈ N0)
and
Bn =−
1
n+ 1
n−1∑
k=0
(
n+ 1
k
)
Bk (n ∈ N),
derive each of the following computationally more advantageous recursion formulas:
B2n =−
1
(n+ 1)(2n+ 1)
n−1∑
k=0
(
n+ 1
k
)
(n+ k+ 1)Bn+k (n ∈ N)
or, equivalently,
Bn =−
1
n+ 1
n−1∑
k=0
(
n+ 1
k
)
Bn+k
(
Bn := (n+ 1)Bn; n ∈ N
)
,
B2n =
1
2n+ 1
−
1
(n+ 1)(2n+ 1)
n−1∑
k=0
(
2n+ 2
2k
)
B2k (n ∈ N)
and
B2n =
1
2
−
1
2n+ 1
n−1∑
k=0
(
2n+ 1
2k
)
B2k (n ∈ N).
(Cf. Srivastava and Miller [1104]; see also [625])
130 Zeta and q-Zeta Functions and Associated Series and Integrals
66. In terms of the sequences
{
a( j)n
}
n∈N0
( j= 1, . . . ,r), let
fj(z)=
∞∑
n=0
a( j)n z
n ( j= 1, . . . ,r).
Denote also the familiar multinomial coefficient by(
n
n1, . . . ,nr
)
:=
n!
n1! . . . nr!
(n,nj ∈ N0; j= 1, . . . ,r; r ∈ N).
By applying the following known series relationship involving product of power series:
r∏
j=1
{ fj(z)} =
∞∑
n=0
bn z
n
bn := ∑
n1+···+nr=n
r∏
j=1
{
a( j)nj
nj!
} ,
or otherwise, derive several properties of the generalized Bernoulli polynomials B(α)n (x)
and the generalized Euler polynomials E(α)n (x) of order α as follows:
B(α1+···+αr)n (x1+ ·· ·+ xr)=
∑
n1+···+nr=n
(
n
n1, . . . ,nr
) r∏
j=1
{
B
(αj)
nj
(
xj
)}
,
E(α1+···+αr)n (x1+ ·· ·+ xr)=
∑
n1+···+nr=n
(
n
n1, . . . ,nr
) r∏
j=1
{
E
(αj)
nj
(
xj
)}
and
n∑
k=0
(
n
k
)
B(α)n−k(x)E
(α)
k (y)= 2
n B(α)n
(
x+ y
2
)
.
(Cf. Brychkov [193])
67. In terms of the sequence {�n}n∈N0 , let the function 8(z) be defined by
8(z)=
∞∑
k=0
�k z
k (|z|< R; R ∈ R+).
Suppose also that
ω = exp
(
2π i
n
)
and m ∈ {0,1, . . . ,n− 1} (n ∈ N).
Introduction and Preliminaries 131
Then show that
∞∑
k=0
�nk+m z
nk+m
=
1
n
n−1∑
k=0
ω(n−m)k 8
(
ωk z
)
=
1
n
n∑
k=1
ω(n−m)k 8
(
ωk z
)
(|z|< R)
and apply these identities to derive the corresponding results involving the Fox-Wright
function p9q (p,q ∈ N0) or p9∗q (p,q ∈ N0) (see Problem 64 above).
(See, for details, Srivastava [1081])
68. In connection with the so-called Littlewood’s teaser about the hitherto nonexistent formula
for the sum:
n∑
r=0
(
n
r
)3
,
show that
n∑
r=0
(
n
r
)k
∼
2kn
√
k
(
2
πn
) 1
2 (k−1)
(n→∞; k ∈ N)
and
n∑
r=0
(
n
r
)k
xn−r yr =
[n/2]∑
r=0
(
n
2r
)(
2r
r
)(
n+ r
r
)
xr yr (x+ y)n−2r.
(See Pólya and Szegö [906, p. 65, Theorem 40; p. 239, Entry 40] and MacMahon
[794, p. 122]; see also Nanjundiah [854] and Srivastava [1075])
69. For the Bernoulli polynomials Bn(x), show that
Bn(x)=
n∑
k=0
1
k+ 1
(
(µ+ 1)k− x
k
) k∑
j=0
(−1) j
(
k
j
)
(µk+ j)n
+ (µ+ x)
n∑
k=0
1
k+ 1
(
(µ+ 1)k− x
k
)
Gk(µ;x)
k∑
j=0
(−1) j
(
k
j
)
(µk+ j)n,
where, for convenience,
G0(µ;x)= 0 and Gk(µ;x)=
k∑
`=1
1
x−µk− `
(k = 1, . . . ,n; µ ∈ C).
(See Srivastava [1074, p. 81, Eq. (27)])
70. Show that the following relationship holds true:
G(l)n (x;λ)= {n}l E
(l)
n−l (x;λ)=
n!
(n− l)!
E (l)n−l (x;λ)
(
n, l ∈ N0; 0 5 l 5 n; λ ∈ C
)
132 Zeta and q-Zeta Functions and Associated Series and Integrals
or, equivalently,
E (l)n (x;λ)=
1
{n+ l}l
G(l)n+l (x;λ)=
n!
(n+ l)!
G(l)n+l (x;λ) (n, l ∈ N0; λ ∈ C)
between the Apostol-Genocchi polynomial of order l and the Apostol-Euler polynomial of
order n− l, which are defined, respectively, by 1.8(58) and 1.8(15) with α = l (l ∈ N0).
(See Luo and Srivastava [791, Lemma 2])
71. Derive the following relationship:
G(α)n (x;λ)= (−2)α B(α)n (x;−λ)
(
α,λ ∈ C; 1α := 1
)
or, equivalently,
B(α)n (x;λ)=
1
(−2)α
G(α)n (x;−λ)
(
α ∈ C; 1α := 1
)
between the Apostol-Genocchi polynomials G(α)n (x;λ) and the Apostol-Bernoulli polyno-
mials B(α)n (x;λ), which are defined by 1.8(58) and 1.8(13), respectively.
(See Luo and Srivastava [791, Lemma 3])
72. Verify that the following relationship holds true:
B(l)n (x;λ)=
n!
(n− l)!(−2)l
E (l)n−l (x;−λ)
(
n, l ∈ N0; 0 5 l 5 n; λ ∈ C
)
or, equivalently,
E (l)n (x;λ)=
n!(−2)l
(n+ l)!
B(l)n+l (x;−λ) (n, l ∈ N0; λ ∈ C)
between the Apostol-Bernoulli polynomial of order l and Apostol-Euler polynomial of
order l, which are defined, respectively, by 1.8(13) and 1.8(15), with α = l (l ∈ N0).
(See Luo and Srivastava [791, Lemma 4])
73. For the generalized Bernoulli polynomials B(α)n (x;λ;a,b,c) of order α ∈ C, defined by
1.8(66), derive each of the following identities:
B(α)n (x+ 1;λ;a,b,c)=
n∑
k=0
(
n
k
)
(lnc)n−k B(α)k (x;λ;a,b,c) ;
B(α)n (x+α;λ;a,b,c)=B
(α)
n
(
x;λ;
a
c
,
b
c
,c
)
;
Introduction and Preliminaries 133
B(α)n (α− x;λ;a,b,c)=B
(α)
n
(
−x;λ;
a
c
,
b
c
,c
)
;
αλ ln
(
b
a
) n∑
k=0
(
n
k
)
(lnb)k B(α+1)n−k (x;λ;a,b,c)= (α− n)B
(α)
n (x;λ;a,b,c)
+ n(x lnc−α lna)B(α)n−1 (x;λ;a,b,c)
or, equivalently,
α ln
(
b
a
) n∑
k=0
(
n
k
)
(lna)k B(α+1)n−k (x;λ;a,b,c)= (α− n)B
(α)
n (x;λ;a,b,c)
+ n(x lnc−α lnb)B(α)n−1 (x;λ;a,b,c) ;
B
(α+β)
k (x+ y;λ;a,b,c)=
k∑
r=0
(
k
r
)
B
(α)
k−r (x;λ;a,b,c)B
(β)
r (y;λ;a,b,c) ;
B
(α)
k (x+ y;λ;a,b,c)=
k∑
r=0
(
k
r
)
(y lnc)r B(α)k−r (x;λ;a,b,c) ;
∂ l
∂xl
{
B(α)n (x;λ;a,b,c)
}
=
n!
(n− l) !
(lnc)l B(α)n−l (x;λ;a,b,c) (l ∈ N0)
and
η∫
ξ
B(α)n (x;λ;a,b,c) dx
=
1
(n+ 1) lnc
[
B
(α)
n+1 (η;λ;a,b,c)−B
(α)
n+1 (ξ ;λ;a,b,c)
]
(η > ξ).
(See, for details, Srivastava et al. [1095])
74. For the generalized Bernoulli polynomials B(α)n (x;λ;a,b,c) of order α ∈ C, defined by
1.8(66), show that each of the following explicit series representations holds true:
B(l)n (x;λ;a,b,c)= l!
(
n
l
) n−l∑
r=0
(
n− l
r
)(
l+ r− 1
r
)
λr
(λ− 1)r+l
(x lnc− l lna)n−r−l
·
r∑
j=0
(−1) j
(
r
j
) [
j ln
(
b
a
)]r
2F1
l+ r− n,1;r+ 1;− j ln
(
b
a
)
x lnc− l lna

(
a,b,c ∈ R+ (a 6= b); l ∈ N0; λ ∈ C \ {1}
)
134 Zeta and q-Zeta Functions and Associated Series and Integrals
and
B(l)n (x;λ;a,b,c)= e
−x
(
lnc lnλ
lnb−lna
)
(lnb− lna)n−l
·
∞∑
k=0
n+k∑
r=0
(−l)r
(
lna
lnb− lna
)r(k
r
)
·
(
n+ k− r− l
k
)(
n+ k− r
k− r
)−1
(ln λ)k
k!
n+k−r∑
p=0
(
n+ k− r
p
)(
l+ p− 1
p
)
p!
(2p) !
·
p∑
j=0
(−1)j
(
p
j
)
j2p
(
x
lnc
lnb− lna
+ j
)n+k−r−p
· 2F1
p− n− k+ r,p− l;2p+ 1; j ln
(
b
a
)
x lnc+ j ln
(
b
a
)

(
a,b,c ∈ R+ (a 6= b); l ∈ N0
)
,
where 2F1 (a,b;c;z) denotes the Gauss hypergeometric function, defined by 1.5(4).
(Srivastava et al. [1095, p. 258, Theorem 6; p. 260, Theorem 7])
75. For the generalized Apostol type polynomials Yn,β(x;k,a,b), defined by 1.8(61), derive
each of the following properties:
d
dx
{
Yn,β(x;k,a,b)
}
= nYn−1,β(x;k,a,b),
anb(m−1)mv−k
m−1∑
j=0
(
β
a
)bjn
Yv,βm
(
x
m
+
nj
m
;k,am,b
)
= amb(n−1)nv−k
n−1∑
l=0
(
β
a
)blm
Yv,βn
(
x
n
+
ml
n
;k,an,b
)
,
y∫
0
Yn,β(x;k,a,b)dx=

Yn+1,β(y;k,a,b)−Yn+1,β(k,a,b)
n+ 1
(n ∈ N)
0 (n= 0)
and
Yn+k−1,β(x;k,a,b)=
(
−
1
2
)k−1
Nn−1
N∑
j=1
(
β j−1
a j−N
)b
·Yn+k−1,βN
(
x+ j− 1
N
;k,aN ,b
)
.
(See, for details, Ozden et al. [886])
Introduction and Preliminaries 135
76. For the generalized Euler polynomials E(α)n (x;λ;a,b,c) of order α ∈ C, defined by 1.8(68),
derive each of the following identities:
E(α)n (x+ 1;λ;a,b,c)=
n∑
k=0
(
n
k
)
(lnc)n−k E(α)k (x;λ;a,b,c) ,
E(α)n (x+α;λ;a,b,c)=E
(α)
n
(
x;λ;
a
c
,
b
c
,c
)
,
E(α)n (α− x;λ;a,b,c)= E
(α)
n
(
−x;λ;
a
c
,
b
c
,c
)
,
E(α)n (α− x;λ;a,b,c)= (−1)
n E(α)n
(
x;λ;
c
a
,
c
b
,c
)
,
α
2
ln
(
b
a
) n∑
k=0
(
n
k
)
(lna)k E(α+1)n−k (x;λ;a,b,c)= E
(α)
n+1 (x;λ;a,b,c)
− (x lnc−α lnb) E(α)n (x;λ;a,b,c) ,
αλ
2
ln
(
b
a
) n∑
k=0
(
n
k
)(
ln
b
c
)k
E
(α+1)
n−k (x+ 1;λ;a,b,c)= E
(α)
n+1 (x;λ;a,b,c)
+ (x lnc−α lna) E(α)n (x;λ;a,b,c) ,
αλ
2
ln
(
b
a
) n∑
k=0
(
n
k
)
(lnb)k E(α+1)n−k (x;λ;a,b,c)= E
(α)
n+1 (x;λ;a,b,c)
− (x lnc−α lna) E(α)n (x;λ;a,b,c) ,
E
(α+β)
n (x+ y;λ;a,b,c)=
n∑
k=0
(
n
k
)
E
(α)
n−k (x;λ;a,b,c) E
(β)
k (y;λ;a,b,c) ,
E(α)n (x+ y;λ;a,b,c)=
n∑
k=0
(
n
k
)
(y lnc)n−k E(α)k (x;λ;a,b,c) ,
∂ l
∂xl
{
E(α)n (x;λ;a,b,c)
}
=
n!
(n− l) !
(lnc)l E(α)n−l (x;λ;a,b,c)
and
η∫
ξ
E(α)n (x;λ;a,b,c) dx=
1
(n+ 1) lnc
[
E
(α)
n+1 (η;λ;a,b,c)−E
(α)
n+1 (ξ ;λ;a,b,c)
]
(η > ξ),
it being understood (wherever needed) that
a,b,c ∈ R+ (a 6= b), x ∈ R and l ∈ N0.
(See, for details, Srivastava et al. [1096])
136 Zeta and q-Zeta Functions and Associated Series and Integrals
77. For the generalized Euler polynomials E(α)n (x;λ;a,b,c) of order α ∈ C, defined by 1.8(67),
show that each of the following explicit series representations holds true:
E(α)n (x;λ;a,b,c)= 2
α
n∑
k=0
(
n
k
)(
α+ k− 1
k
)
λk
(λ+ 1)α+k
[
ln
(
b
a
)]k
(x lnc−α lna)n−k
·
k∑
j=0
(−1) j
(
k
j
)
jk 2F1
k− n,1;k+ 1;− j ln
(
b
a
)
x lnc−α lna

(
a,b,c ∈ R+ (a 6= b); α ∈ C; λ ∈ C \ {−1}
)
,
E(α)n (x;λ;a,b,c)= 2
α
n∑
k=0
(
n
k
)(
α+ k− 1
k
)
λk
(λ+ 1)α+k
[
ln
(
b
a
)]k k∑
j=0
(−1) j
(
k
j
)
jk
·
[
x lnc−α lna+ j ln
(
b
a
)]n−k
2F1
k− n,k;k+ 1;− j ln
(
b
a
)
x lnc−α lna+ j ln
(
b
a
)

(
a,b,c ∈ R+ (a 6= b); α ∈ C; λ ∈ C \ {−1}
)
and
E(α)n (x;λ;a,b,c)= e
−x
(
lnc lnλ
lnb−lna
) [
ln
(
b
a
)]k ∞∑
k=0
n∑
j=0
(−α) j
(
n
j
)(
lna
lnb− lna
)j
·
(lnλ)k
k!
n+k−j∑
m=0
(
n+ k− j
m
) (
α+m− 1
m
) m∑
`=0
(−1)`
(
m
`
) (
`
2
)m
·
(
x lnc
lnb− lna
+ `
)n+k−j−m
2F1
m− n− k+ j,m;m+ 1; ` ln
(
b
a
)
x lnc+ ` ln
(
b
a
)
 ,
(
a,b,c ∈ R+ (a 6= b); α ∈ C
)
,
where 2F1 (a,b;c;z) denotes the Gauss hypergeometric function, defined by 1.5(4).
(Srivastava et al. [1096, pp. 294–295, Theorem 6; p. 298, Theorem 7])
78. Show that the following relationships hold true:
G(α)n (x;λ;a,b,c)= (−2)
α B(α)n (x;−λ;a,b,c)
(
α ∈ C; 1α := 1
)
and
G(l)n (x;λ;a,b,c)= (−1)
l (−n)l E
(l)
n−l (x;λ;a,b,c)=
n!
(n− l)!
E
(l)
n−l (x;λ;a,b,c)(
n, l ∈ N0; n = l; λ ∈ C
)
between the generalized Bernoulli polynomials B(α)n (x;λ;a,b,c), the generalized
Euler polynomials E(α)n (x;λ;a,b,c) and the generalized Genocchi polynomials
G
(α)
n (x;λ;a,b,c) of order α ∈ C, defined by 1.8(66), 1.8(67) and 1.8(68), respectively.
(Srivastava et al. [1096, p. 300, Lemma 3])
Introduction and Preliminaries 137
79. For the generalized Genocchi polynomials G(α)n (x;λ;a,b,c) of order α ∈ C, defined by
1.8(69), show that each of the following explicit series representations holds true:
G(l)n (x;λ;a,b,c)=
2l · n!
(n− l)!
n−l∑
k=0
(
n− l
k
)(
k+ l− 1
k
)
λk
(λ+ 1)k+l
[
ln
(
b
a
)]k
· (x lnc− l lna)n−k−l
k∑
j=0
(−1) j
(
k
j
)
jk 2F1
k+ l− n,1;k+ 1;− j ln
(
b
a
)
x lnc− l lna

(
a,b,c ∈ R+ (a 6= b); l ∈ N0; λ ∈ C \ {1}
)
and
G(l)n (x;λ;a,b,c)=
2l · n!
(n− l)!
n−l∑
k=0
(
n− l
k
)(
k+ l− 1
k
)
λk
(λ+ 1)k+l
[
ln
(
b
a
)]k
·
k∑
j=0
(−1) j
(
k
j
)
jk
[
x lnc− l lna+ j ln
(
b
a
)]n−k−l
· 2F1
k+ l− n,k;k+ 1;− j ln
(
b
a
)
x lnc− l lna+ j ln
(
b
a
)

(
a,b,c ∈ R+ (a 6= b); l ∈ N0; λ ∈ C \ {1}
)
,
where 2F1 (a,b;c;z) denotes the Gauss hypergeometric function, defined by 1.5(4).
(Srivastava et al. [1096, p. 301, Theorem 9])
80. For the Apostol-Genocchi polynomials Gn (x;λ) (λ ∈ C), defined by 1.8(58) (with α = 1),
derive the following exponential Fourier series representations:
Gn(x;λ)=
2 · n!
λx
∞∑
k=−∞
e(2k−1)π ix
[(2k− 1)π i− logλ]n
=
(2 · n!)in
λx
(
∞∑
k=0
exp
[( nπ
2 − (2k+ 1)πx
)
i
]
[(2k+ 1)π i+ logλ]n
+
∞∑
k=0
exp
[(
−
nπ
2 + (2k+ 1)πx
)
i
]
[(2k+ 1)π i− logλ]n
)
(n ∈ N; 0 5 x 5 1; λ ∈ C \ {0,−1}).
(See Luo and Srivastava [791, p. 5726, Theorem 20]; see also [786])
81. Derive each of the following properties of the Apostol-Genocchi polynomials G(α)n (x;λ)
of order α ∈ C, defined by 1.8(58):
G(α)n (λ)= G(α)n (0;λ), G(0)n (x;λ)= xn,
G(0)n (λ)= δn,0 and G
(α)
0 (x;λ)= G
(α)
0 (λ)= δα,0 (n ∈ N0; α ∈ C),
138 Zeta and q-Zeta Functions and Associated Series and Integrals
where δn,k denotes the Kronecker symbol;
G(α)n (x;λ)=
n∑
k=0
(
n
k
)
G(α)k (λ) x
n−k
and
G(α)n (x;λ)=
n∑
k=0
(
n
k
)
G(α−1)n−k (λ)Gk(x;λ);
λG(α)n (x+ 1;λ)+G(α)n (x;λ)= 2n G
(α−1)
n−1 (x;λ) (n ∈ N);
∂
∂x
{
G(α)n (x;λ)
}
= nG(α)n−1(x;λ) (n ∈ N)
and
∂p
∂xp
{
G(α)n (x;λ)
}
=
n!
(n− p)!
G(α)n−p(x;λ) (n,p ∈ N0; 0 5 p 5 n);
b∫
a
G(α)n (x;λ)dx=
G(α)n+1(b;λ)−G
(α)
n+1(a;λ)
n+ 1
and
b∫
a
G(α)n (x;λ)dx=
n∑
k=0
1
n− k+ 1
(
n
k
)
G(α)k (λ) (b
n−k+1
− an−k+1);
G(α+β)n (x+ y;λ)=
n∑
k=0
(
n
k
)
G(α)k (x;λ)G
(β)
n−k(y;λ);
G(α)n (α− x;λ)=
(−1)n+α
λα
G(α)n (x;λ−1)
and
G(α)n (α+ x;λ)=
(−1)n+α
λα
G(α)n (−x;λ−1);
(n−α) G(α)n (x;λ)= nx G
(α)
n−1(x;λ)−
αλ
2
G(α+1)n (x+ 1;λ)
and
α
2
G(α+1)n (x;λ)= n(α− x) G
(α)
n−1(x;λ)+ (n−α) G
(α)
n (x;λ).
(See Luo and Srivastava [791, pp. 5706–5607])
82. For the Apostol-Genocchi polynomials G(α)n (x;λ) of order l (l ∈ N0), defined by 1.8(58)
with α = l (l ∈ N0), show that the following explicit series representations hold true:
G(l)n (x;λ)= 2l l!
(
n
l
) n−l∑
k=0
(
n− l
k
)(
l+ k− 1
k
)
λk
(λ+ 1)l+k
·
k∑
j=0
(−1) j
(
k
j
)
jk (x+ j)n−k−l 2F1
(
l+ k− n,k;k+ 1;
j
x+ j
)
(n, l ∈ N0; λ ∈ C \ {−1})
Introduction and Preliminaries 139
and
G(l)n (x;λ)= e−x logλ
∞∑
k=0
(
n+ k− l
k
)(
n+ k
k
)−1(n+ k
l
)
l!(logλ)k
k!
·
n+k−l∑
r=0
1
2r
(
n+ k− l
r
)(
l+ r− 1
r
)
·
r∑
j=0
(−1) j
(
r
j
)
jr (x+ j)n+k−r−l 2F1
(
r+ l− n− k,r;r+ 1;
j
x+ j
)
(n, l ∈ N0; λ ∈ C),
where 2F1 (a,b;c;z) denotes the Gauss hypergeometric function defined by 1.5(4).
(See Luo and Srivastava [791, p. 5708, Theorem 1])
83. For the Apostol-Genocchi polynomials G(α)n (x;λ) of order l (l ∈ N0) defined by 1.8(58),
derive the following relationship:
G(α)n (x+ y;λ)=
n∑
k=0
2
k+ 1
(
n
k
)[
(k+ 1)G(α−1)k (y;λ)−G
(α)
k+1(y;λ)
]
Bn−k(x;λ)(
α,λ ∈ C; n ∈ N0
)
with the Apostol-Bernoulli polynomials Bn−k(x;λ) defined by 1.8(1).
(See Luo and Srivastava [791, p. 5710, Theorem 2])
84. For the Apostol-Genocchi polynomials Gn (x;λ), defined by 1.8(58) with α = 1, show that
the following integral representation holds true:
Gn(z;e2π iξ )= 2ne−2π izξ
∞∫
0
(
M(n;z, t)cosh (2πξ t)+ iN(n;x, t)sinh (2πξ t)
cosh (2π t)− cos (2πz)
)
tn−1 dt
(
n ∈ N; 0 5 <(z)5 1; |ξ |<
1
2
(ξ ∈ R)
)
,
where
M(n;z, t)=
[
eπ t cos
(
πz−
nπ
2
)
− e−π t cos
(
πz+
nπ
2
)]
and
N(n;z, t)=
[
eπ t sin
(
πz−
nπ
2
)
+ e−π t sin
(
πz+
nπ
2
)]
.
(See Luo and Srivastava [791, p. 5724, Theorem 19])
85. Let the so-called λ-Stirling numbers S(n,k;λ) of the second kind be defined by means of
the following generating function:
(λez− 1)k
k!
=
∞∑
n=0
S(n,k;λ)
zn
n!
(k ∈ N0; λ ∈ C),
140 Zeta and q-Zeta Functions and Associated Series and Integrals
so that, obviously,
S(n,k) := S(n,k;1)
for the Stirling numbers S(n,k) of the second kind, defined by 1.6(15). Show that each of
the following results holds true:
λx xn =
∞∑
k=0
(
x
k
)
k! S(n,k;λ) (k ∈ N0; λ ∈ C),
S(n,k;λ)=
1
k!
k∑
j=0
(−1)k−j
(
k
j
)
λ jjn (n,k ∈ N0; λ ∈ C),
S(n,k;λ)=
1
k!
k∑
j=0
(−1) j
(
k
j
)
λk−j(k− j)n (n,k ∈ N0; λ ∈ C),
S(n,k;λ)= S(n− 1,k− 1;λ)+ k S(n− 1,k;λ) (n,k ∈ N)
and
S(n,k;λ)=
n−1∑
j=0
(
n− 1
j
)
λn−j−1 S( j,k− 1;λ) (n,k ∈ N).
(See Luo and Srivastava [791, p. 5716, Theorems 9 to 11])
86. For the λ-Stirling numbers S(n,k;λ) of the second kind, defined by means of the generating
function in Problem 84 above, show that each of the following explicit relationships holdstrue:
S(n,k;λ)= n!
∞∑
j=n
(
j
n
)
(logλ) j−n
j!
S( j,k) (n,k ∈ N0; λ ∈ C)
and
S(n,k;λ)=
k∑
j=0
λ j(λ− 1)k−j
(k− j)!
S(n, j) (n,k ∈ N0; λ ∈ C),
with the Stirling numbers S(n,k) of the second kind, defined by 1.6(15).
(See Luo and Srivastava [791, pp. 5716–5717, Theorem 12])
2 The Zeta and Related Functions
This chapter aims at providing a self-contained theory of the Zeta and related func-
tions, which will be required in each of the next chapters. We first introduce (and
investigate the various properties and relationships satisfied by) the multiple Hurwitz
Zeta function ζn(s,a) (n ∈ N) and consider its relatively more familiar special case
when n= 1, that is, the Hurwitz (or generalized) Zeta function ζ(s,a). We then deal
rather systematically with the Riemann Zeta function, which (for the main purpose of
this book) happens to be the most important member of the significantly large family
of Zeta functions considered in this chapter. Other functions (introduced in this chap-
ter) include the Polylogarithm functions, Legendre’s Chi function, Clausen’s integral
(or Clausen’s function), the Hurwitz–Lerch Zeta function and so on.
2.1 Multiple Hurwitz Zeta Functions
Barnes [97] introduced and studied the generalized multiple Hurwitz Zeta function
ζn (s,a |w1, . . . , wn) defined for <(s) > n by the n-ple series
ζn (s,a |w1, . . . , wn) :=
∞∑
m1, ...,mn=0
1
(a+�)s
(<(s) > n; n ∈ N), (1)
where �= m1 w1+ ·· ·+mn wn and the general conditions for a and the parameters
w1, . . . , wn are given in Barnes [97], who used it in the study of the multiple Gamma
functions (see Section 1.4). In this section, we consider only the simple case of (1),
when
wj = 1( j= 1, . . . , n; j, n ∈ N)
and
ζn (s,a) :=
∞∑
k1, ...,kn=0
(a+ k1+ ·· ·+ kn)
−s (<(s) > n; a 6∈ Z−0 ), (2)
which is also referred to as n-ple (or, simply, multiple) Hurwitz Zeta function. We shall
give a rather detailed investigation of the properties and characteristics of the function
ζn (s,a) in (2), including its analytic continuation.
Zeta and q-Zeta Functions and Associated Series and Integrals. DOI: 10.1016/B978-0-12-385218-2.00002-5
c© 2012 Elsevier Inc. All rights reserved.
http://dx.doi.org/10.1016/B978-0-12-385218-2.00002-5
142 Zeta and q-Zeta Functions and Associated Series and Integrals
The Analytic Continuation of ζn (s,a)
In order to give the analytic continuation of ζn (s,a), we first recall a criterion for
the convergence of an n-ple series attributed to Eisenstein in the work of Forsyth
[457], who gave the proof for the double series. Here, we shall give a proof of the
general case:
Theorem 2.1 The n-ple series:
∞∑
m1, ...,mn=−∞
′
(
m21+ ·· ·+m
2
n
)−µ
(3)
converges, if µ > 12 n, where the prime denotes the exclusion of the case when mj = 0
( j= 1, . . . , n).
Proof. Let the series be arranged in the partial series: for this purpose, choose integers
kj, such that
2kj 5 mj < 2
kj+1 (1 5 j 5 nj; kj ∈ N0).
Let
D := {(m1, . . . ,mn) |2
kj 5 mj < 2
kj+1; 1 5 j 5 n}
and let Ij := [2kj ,2kj+1) (1 5 j 5 n) be half-open intervals. Since 2k+1− 2k = 2k, the
number of integers mj in Ij is 2kj (1 5 j 5 n). So the number of n-tuples in D is
2k1 · · ·2kn = 2k1+···+kn = 2nx,
where
x :=
k1+ ·· ·+ kn
n
.
Note that, for (m1, . . . ,mn) ∈ D,
n∑
j=1
22kj 5 m21+ ·· ·+m
2
n <
n∑
j=1
22kj+2
and, by comparing the arithmetic and geometric means of 22kj (1 5 j 5 n), we
also have
22x 5
1
n
n∑
j=1
22kj 5 m21+ ·· ·+m
2
n.
The Zeta and Related Functions 143
Now, for any (m1, . . . ,mn) in D and any positive real µ, we have
1
(m21+ ·· ·+m
2
n)
µ
5
1
22xµ
.
Then, the sum of all terms as 1/(m21+ ·· ·+m
2
n)
µ in D is less than or equal to
2nx
22xµ
=
1
2x(2µ−n)
=
1
2
2µ−n
n k1
· · ·
1
2
2µ−n
n kn
.
Let k =max{kj |1 5 j 5 n}. We find that the sum of all the partial series is less than or
equal to
k∑
k1=0
· · ·
k∑
kn=0
1
2
2µ−n
n k1
· · ·
1
2
2µ−n
n kn
=
 k∑
k1=0
1
2
2µ−n
n k1
n
=
1− 2−(k+1) 2µ−nn
1− 1
2
2µ−n
n
n .
(4)
Taking the limit in (4) as k→∞, we observe that the n-ple series in (4) converges, if
2µ− n> 0, that is, if µ > 12 n and the sum is 1
1− 1
2
2µ−n
n
n = 22µ−n(
2
2µ−n
n − 1
)n .
This completes the proof of Theorem 2.1. �
Let s= σ + it (σ, t ∈ R). First, for convergence, we consider ζn(s,a) in (2) for the
case when a> 0:
ζn(s,a)=
∞∑
k1, ...,kn=0
(a+ k1+ ·· ·+ kn)
−s (<(s)= σ > n; a> 0). (5)
Theorem 2.2 The series for ζn(s,a) in (5) converges absolutely for σ > n. The con-
vergence is uniform in every half-plane σ ≥ n+ δ (δ > 0), so ζn(s,a) is an analytic
function of s in the half-plane σ > n.
Proof. Observe that, for σ > 0,
∞∑
k1, ...,kn=0
′ (k1+ ·· ·+ kn)
−σ
=
∞∑
k1, ...,kn=0
′ [(k1+ ·· ·+ kn)
2]−
1
2σ
5
∞∑
k1, ...,kn=0
′ (k21+ ·· ·+ k
2
n)
−
1
2σ ,
144 Zeta and q-Zeta Functions and Associated Series and Integrals
in which the prime denotes the exclusion of the case when kj = 0 (1 5 j 5 n) and that
the last series converges for σ > n by Eisentein’s theorem (Theorem 2.1). Thus, all
statements in Theorem 2.2 follow from the inequality:
∞∑
k1, ...,kn=0
|(a+ k1+ ·· ·+ kn)
−s
| =
∞∑
k1, ...,kn=0
(a+ k1+ ·· ·+ kn)
−σ
5
∞∑
k1, ...,kn=0
(a+ k1+ ·· ·+ kn)
−n−δ.
We now choose to recall a convergence theorem concerning term-by-term
integration of monotonic sequences of functions, which is due to Lévi (see Apostol
[64, p. 268, Theorem 10.25]). �
Theorem 2.3 Let L(I) denote the set of all Lebesgue-integrable functions on an inter-
val I. Also, let {gn} be a sequence of functions in L(I), such that
(a) each gn is nonnegative almost everywhere on I
(b) the series
∑
∞
n=1
∫
I gn converges.
Then, the series
∑
∞
n=1 gn converges almost everywhere on I to a sum function g in
L(I). Moreover,∫
I
g=
∫
I
∞∑
n=1
gn =
∞∑
n=1
∫
I
gn.
Next, we present an integral representation of ζn(s,a), which is given by
Theorem 2.4 If <(s)= σ > n, then
0(s)ζn(s,a)=
∞∫
0
xs−1e−ax
(1− e−x)n
dx (<(s) > n; n ∈ N). (6)
Proof. It follows from 1.1(1) that, for <(s)= σ > 0,
0(s)=
∞∫
0
xs−1e−xdx.
We first keep s real, s> n (n ∈ N), and then extend the result to complex s by analytic
continuation. In this Eulerian integral for 0(s), we set
x= (a+ k1+ ·· ·+ kn)t (kj ∈ N0; j= 1, . . . , n),
The Zeta and Related Functions 145
and we find that
0(s)= (a+ k1+ ·· ·+ kn)
s
∞∫
0
e−(a+k1+···+kn)t t s−1dt,
so that
(a+ k1+ ·· ·+ kn)
−s0(s)=
∞∫
0
e−(k1+···+kn)te−att s−1dt.
Summing over all kj ∈ N0 (1 5 j 5 n), we obtain
ζn(s,a)0(s)=
∞∑
k1, ...,kn=0
∞∫
0
e−(k1+···+kn)te−att s−1 dt,
the series on the right being convergent, if s> n.
Now, we wish to interchange the order of summation and integration. The simplest
way to justify this process is to regard the integral as a Lebesgue integral. Since the
integrand is nonnegative, Lévi’s convergence theorem (Theorem 2.3) implies that the
series:
∞∑
k1, ...,kn=0
∞∫
0
e−(a+k1+···+kn)te−att s−1dt
converges almost everywhere to a sum function, which itself is Lebesgue-integrable
on [0,∞) and that
ζn(s,a)0(s)=
∞∫
0
∞∑
k1, ...,kn=0
e−(k1+···+kn)te−att s−1dt.
But, if t > 0, we have 0< e−t < 1, and, hence,
∞∑
k=0
e−kt =
1
1− e−t
,
the series being a geometric series. We, therefore, have
∞∑
k1, ...,kn=0
e−(k1+···+kn)te−att s−1 =
e−att s−1
(1− e−t)n
146 Zeta and q-Zeta Functions and Associated Series and Integrals
almost everywhere on [0,∞); in fact, everywhere except at 0, so we get
ζn(s,a)0(s)=
∞∫
0
e−att s−1
(1− e−t)n
dt.
This proves (6) for real s and s> n (n ∈ N).
To extend this integral representation to all complex s= σ + it with σ > n, we note
that both functions on the left-hand side of (6) are analytic for σ > n. To show that the
right member is also analytic, we assume that n+ δ 5 σ 5 c, where c> n and δ > 0,
and write
∞∫
0
∣∣∣∣ e−att s−1(1− e−t)n
∣∣∣∣dt =
∞∫
0
e−attσ−1
(1− e−t)n
dt
=
 1∫
0
+
∞∫
1
 e−attσ−1
(1− e−t)n
dt.
If 0 5 t 5 1, we have tσ−n 5 tδ , and, if t = 1, we have tσ−n 5 tc−n. Also, since
et−1= t for t = 0, we have
1∫
0
e−attσ−1
(1− e−t)ndt 5
1∫
0
e(n−a)ttδ+n−1
(et− 1)n
dt
5

en−a
∫ 1
0 t
δ−1 dt = e
n−a
δ
(0< a 5 n)
∫ 1
0 t
δ−1dt = 1
δ
(a> n)
and
∞∫
1
e−attσ−1
(1− e−t)n
dt 5
∞∫
0
e−attc−1
(1− e−t)n
dt = 0(c)ζn(c,a).
This shows that the integral in (6) converges uniformly in every strip n+ δ 5 σ 5 c,
where δ > 0 and, therefore, represents an analytic function in every such strip; hence,
also in the half-plane σ > n. Thus, by the principle of analytic continuation, (6) holds
true for all s with <(s)= σ > n (n ∈ N). �
In order to extend ζn(s,a) to the half-plane on the left of the line σ = n, we derive
another representation in terms of a contour integral. The contour C is essentially a
Hankel’s loop (cf., e.g., Whittaker and Watson [1225, p. 245]), which starts from∞
The Zeta and Related Functions 147
along the upper side of the positive real axis, encircles the origin once in the posi-
tive (counter-clockwise) direction and then returns to ∞ along the lower side of the
positive real axis.
The loop C is composed of three parts C1, C2 and C3, where C2 is a positively-
oriented circle of radius c< 2π about the origin and C1 and C3 are the upper and
lower edges of a cut in the complex z-plane along the positive real axis, traversed
as described above. Thus, we can use the parameterizations −z= re−π i on C1 and
−z= reπ i on C3, where r varies from c to∞.
Theorem 2.5 If a> 0, then the function defined by the following contour integral:
In(s,a)=−
1
2π i
(0+)∫
∞
(−z)s−1e−az
(1− e−z)n
dz (7)
is an entire function of s. Moreover,
ζn(s,a)= 0(1− s) In(s,a) (<(s)= σ > n). (8)
Proof. Here (−z)s means rse−π is on C1 and rseπ is on C3. We consider an arbitrary
compact disk |s|5 M and prove that the integrals along C1 and C3 converge uniformly
on every such disk. Since the integrand in (7) is an entire function of s, this will prove
that In(s,a) is also an entire function of s for a> 0. Along C1, we have, for r = 1,
|(−z)s−1| = rσ−1
∣∣∣e−π i(σ−1+it)∣∣∣= rσ−1 eπ t 5 rM−1eπM,
since |s|5 M. Similarly, along C3, we have, for r = 1,∣∣∣(−z)s−1∣∣∣= rσ−1|eπ i(σ−1+it)| = rσ−1e−π t 5 rM−1eπM.
Hence, on either C1 or C3, we have, for r = 1,∣∣∣∣ (−z)s−1e−az(1− e−z)n
∣∣∣∣5 rM−1eπMe−ar(1− e−r)n 5 eπM(1− e−1)n ·rM−1·e−ar.
But the integral:
∞∫
c
rM−1 e−ar dr
converges, if c> 0; this shows that the integrals along C1 and C3 converge uniformly
on every compact disk |s|5 M, and, hence, In(s,a) is an entire function of s. To prove
(8), we write
−2π i In(s,a)=
∫
C1
+
∫
C2
+
∫
C3
(−z)s−1g(−z)dz,
148 Zeta and q-Zeta Functions and Associated Series and Integrals
where
g(−z)=
e−az
(1− e−z)n
.
On C1 and C3, we have g(−z)= g(−r), and, on C2, we write −z= ceiθ , where θ
varies from 2π to 0. This gives us
−2π i In(s,a)=
c∫
∞
rs−1 e−π i(s−1) g(−r)dr− i
0∫
2π
cs−1e(s−1)iθceiθg(ceiθ )dθ
+
∞∫
c
rs−1eπ i(s−1)g(−r)dr
=−2isin(πs)
∞∫
c
rs−1g(−r)dr− ics
0∫
2π
eisθ g(ceiθ )dθ.
(9)
Dividing both sides of (9) by −2i, we obtain
π In(s,a)= sin(πs)I1(s,c)+ I2(s,c),
where
I1(s,c)=
∞∫
c
rs−1g(−r)dr and I2(s,c)=
cs
2
0∫
2π
eisθg(ceiθ )dθ.
Now, let c→ 0. We find that, in view of (6),
lim
c→0
I1(s,c)=
∞∫
0
rs−1e−ar
(1− e−r)n
dr = 0(s)ζn(s,a) (σ =<(s) > n).
We next show that
lim
c→0
I2(s,c)= 0.
To do this, we note that g(−z) is analytic in |z|< 2π, except for a pole of order
n at z=0. Therefore, zng(−z) is analytic everywhere inside |z|< 2π and, hence, is
bounded there, that is,
|g(−z)|5
A
|z|n
(|z| = c< 2π),
The Zeta and Related Functions 149
and A is a positive constant. We, thus, have
|I2(s,c)|5
cσ
2
2π∫
0
e−tθ ·
A
cn
dθ 5 πAe2π |t| cσ−n.
If σ > n and c→ 0, we find that I2(s,c)→ 0; hence
π In(s,a)= sin(πs)0(s)ζn(s,a), (10)
which is, in terms of 1.1(12), seen to be equivalent to (8). �
In the equation (8), valid for σ > n, the function In(s,a) is an entire function of s,
and 0(1− s) is analytic for every complex s for s ∈ C \N. We, therefore, can use this
equation to define ζn(s,a) for σ 5 n, that is, outside σ > n as desired.
Definition 2.1 If <(s)= σ 5 n, we define ζn(s,a) by
ζn(s,a) := 0(1− s) In(s,a), (11)
where In(s,a) is given in (7).
This equation (11) provides the analytic continuation of ζn(s,a) to the whole com-
plex s-plane.
Theorem 2.6 The function ζn(s,a), defined by (11), is analytic for all s except for
simple poles at s= k (1 5 k 5 n), with their respective residues given by
Res
s=k
ζn(s,a)=
1
(n− k)!(k− 1)!
lim
z→0
dn−k
dzn−k
zne−az
(1− e−z)n
(k = 1, . . . , n; n ∈ N).
(12)
In particular, when s= n, its residue is 1/(n− 1)!.
Proof. Since In(s,a) is an entire function of s for a> 0, the only possible singulari-
ties of ζn(s,a) are the poles of 0(1− s). Since 1/0(1− s) has simple zeros at s ∈ N,
0(1−s) has simple poles at s ∈ N. But Theorem 2.2 shows that ζn(s,a) is analytic in
<(s)= σ > n, and so s= 1, . . . ,n are the only poles of ζn(s,a).
We next prove the assertion (12). If s is any integer, say s= k, the integrand in the
contour integral for In(s,a) in (7) takes the same values on both C1 and C3, and, hence,
the integrals along C1 and C3 cancel, giving us
In(k,a)=−
1
2π i
∫
C2
(−z)k−1e−az(
1− e−z
)n dz
=−Res
z=0
f (z),
150 Zeta and q-Zeta Functions and Associated Series and Integrals
where, for convenience,
f (z) :=
(−z)k−1e−az
(1− e−z)n
.
We observe that, for 1 5 k 5 n, the function f (z) has a pole of order n+ 1− k at z=0.
We, therefore, have
In(k,a)=
(−1)k
(n− k)!
lim
z→0
dn−k
dzn−k
zne−az
(1− e−z)n
. (13)
To find the residue of ζn(s,a) at s= k (1 5 k 5 n), by using 1.1(12), we compute the
limit:
lim
s→k
(s− k)ζn(s,a)= lim
s→k
(s− k)0(1− s)In(s,a)
=
π In(k,a)
0(k)
· lim
s→k
s− k
sin(πs)
=
In(k,a)
(−1)k(k− 1)!
,
which, by virtue of (13), immediately yields (12). �
Relationship between ζn (s,x) and B
(α)
n (x)
By using the multiple Hurwitz Zeta function, Choi [262] derived the following explicit
formula for B(α)n (x) different from 1.7(28):
B(n)n+k(x)= n
(
n+ k
n
) n−1∑
j=0
(−1)j
Bk+j+1(x)
k+ j+ 1
n−1∑
`=j
(
`
j
)
s(n,`+ 1)x`−j (14)
in terms of the Stirling numbers s(n,k) of the first kind (see Section 1.6).
The value of ζn(−`,x) can be calculated explicitly for ` ∈ N0. Taking s=−` in
the relation (8), with a replaced by x, we find that
ζn(−`,x)= 0(1+ `)In(−`,x)= `! In(−`,x). (15)
The Zeta and Related Functions 151
Now, from the definition (19) of the generalized Bernoulli polynomials B(α)n (x),
we have
In(−`,x)=−
1
2π i
∫
C
(−z)−`−1e−xz(
1− e−z
)n dz
=−Res
z=0
(−z)−`−1e−xz(
1− e−z
)n
= (−1)`Res
z=0
z−n−`−1
zne(n−x)z
(ez− 1)n
= (−1)`Res
z=0
z−n−`−1
∞∑
k=0
B(n)k (n− x)
zk
k!
= (−1)`
B(n)n+`(n− x)
(n+ `)!
,
which, in view of 1.7(20) and (15), yields the desired relationship:
ζn(−`,x)= (−1)
n `!
(n+ `)!
B(n)n+`(x) (` ∈ N0). (16)
Setting n= 1 in (16), we have the well-known result:
ζ(−`,x)=−
B`+1(x)
`+ 1
(` ∈ N0), (17)
where ζ(s,x) := ζ1(s,x) is the Hurwitz (or generalized) Zeta function (see Sec-
tion 2.2).
The number of solutions of
k1+ ·· ·+ kn = k (k ∈ N0; (k1, . . . ,kn) ∈ N0n)
is equal to the coefficient of xk in the Maclaurin series expansion of (1− x)−n:
(1− x)−n =
∞∑
k=0
(
−n
k
)
(−x)k =
∞∑
k=0
(
k+ n− 1
n− 1
)
xk. (18)
The multiple Hurwitz Zeta function in (5) can, thus, be expressed as a simple series:
ζn(s,x)=
∞∑
k=0
(
k+ n− 1
n− 1
)
(x+ k)−s. (19)
152 Zeta and q-Zeta Functions and Associated Series and Integrals
From 1.6(1) or 1.6(5), we find that
(
k+ n− 1
n− 1
)
=
1
(n− 1)!
n−1∑
j=0
|s|(n, j+ 1)kj, (20)
where |s|(n,k) := (−1)n+k s(n,k) are often called the unsigned or absolute Stirling
numbers of the first kind. Combining (19) and (20), we have
ζn(s,x)=
1
(n− 1)!
∞∑
k=0
n−1∑
j=0
|s|(n, j+ 1)kj
(x+ k)−s,
which, by virtue of the identity:
k` = {(−x)+ (x+ k)}` =
∑̀
j=0
(
`
j
)
(−x)`−j(x+ k)j,
yields
ζn(s,x)=
1
(n− 1)!
n−1∑
`=0
|s|(n,`+ 1)
∑̀
j=0
(
`
j
)
(−x)`−j

( ∞∑
k=0
1
(x+ k)s−j
)
=
1
(n− 1)!
n−1∑
`=0
|s|(n,`+ 1)
∑̀
j=0
(
`
j
)
(−x)`−jζ(s− j,x).
Next, it is easy to show that ζn(s,x) is expressibleas a finite combination of the
generalized Zeta function ζ(s,x) with polynomial coefficients in x:
ζn(s,x)=
n−1∑
j=0
pn,j(x)ζ(s− j,x), (21)
where
pn,j(x)=
1
(n− 1)!
n−1∑
`=j
(−1)n+1−j
(
`
j
)
s(n,`+ 1)x`−j. (22)
We shall now find pn,j(x) in (22) as a polynomial in x of degree n− 1− j with rational
coefficients.
Since ζ(s,x) can be continued analytically to a meromorphic function, having a
simple pole at s= 1 with its residue 1, the representation (21) shows that ζn(s,x) is
The Zeta and Related Functions 153
analytic for all s, except for simple poles only at s= k (k = 1, . . . , n; n ∈ N)with their
respective residues given by
Res
s=k
ζn(s,x)= pn,k−1(x) (k = 1, . . . , n; n ∈ N). (23)
In view of 1.6(3) and 1.6(6), ζn(s,x) can be expressed explicitly for the first few
values of n:
ζ2(s,x)= (1−x)ζ(s,x)+ζ(s−1,x),
ζ3(s,x)=
1
2
(
x2−3x+2
)
ζ(s,x)+
(
3
2
−x
)
ζ(s−1,x)+
1
2
ζ(s−2,x),
ζ4(s,x)=
1
6
{(
−x3+6x2−11x+6
)
ζ(s,x)+
(
3x2−12x+11
)
ζ(s−1,x)
−(3x−6)ζ(s−2,x)+ζ(s−3,x)}.
(24)
Letting s=−` in (21) and applying (16), we obtain the desired formula (14). Also,
upon setting x= n (n ∈ N) in (14) and making use of 1.7(21), we find that
B(n)n+k = (−1)
n+kn
(
n+ k
n
) n−1∑
j=0
(−1)j
Bk+j+1(n)
k+ j+ 1
n−1∑
`=j
(
`
j
)
s(n,`+ 1)n`−j. (25)
Now, we express Ress=k ζn(s,x) in (12) in a more recognizable form:
Res
s=k
ζn(s,x)=
B(n)n−k(n− x)
(n− k)!(k− 1)!
. (26)
Indeed, it follows from 1.7(19) and (12) that
lim
s→k
(s− k)ζn(s,x)=
1
(n− k)!(k− 1)!
lim
z→0
dn−k
dzn−k
∞∑
j=0
B(n)j (n− x)
zj
j!
=
B(n)n−k(n− x)
(n− k)!(k− 1)!
.
The Vardi-Barnes Multiple Gamma Functions
Vardi [1190, p. 498] gave another expression for the multiple Gamma functions 0n(a)
(see Section 1.4), whose general form was also studied by Barnes [97]:
0n(a)=
[
n∏
m=1
R
(−1)m ( am−1)
n−m+1
]
Gn(a) (n ∈ N), (27)
154 Zeta and q-Zeta Functions and Associated Series and Integrals
where
Gn(a) := exp
[
ζ ′n(0,a)
]
with ζ ′n(s,a)=
∂
∂s
ζn(s,a)
and
Rm := exp
(
m∑
k=1
ζ ′k(0,1)
)
with R0 = 1. (28)
In particular, the special cases of (27), when n= 1 and n= 2, give other forms of the
simple and double Gamma functions 01 = 0 and 02:
0(a)= exp
[
−ζ ′(0)+ ζ ′(0,a)
]
=
√
2π exp
[
ζ ′(0,a)
]
,
(29)
where ζ(s) := ζ(s,1) is the Riemann Zeta function (see Section 2.3);
02(a)= A(2π)
1
2−
1
2 a exp
(
−
1
12
+ ζ ′2(0,a)
)
, (30)
where we have used (24) and the known identity (see Voros [1201, p. 462,
Eq. (A.11)]):
logA=
1
12
− ζ ′(−1). (31)
Here we can give another proof of the multiplication formula for 02 (see 1.4(21))
different from that of Barnes [94], by using (30) (see Choi and Quine [278]). We
consider
n−1∑
`=0
n−1∑
j=0
ζ2
(
s,a+
`+ j
n
)
=
n−1∑
`=0
n−1∑
j=0
∞∑
k1,k2=0
(
a+
`+ j
n
+ k1+ k2
)−s
= ns
∞∑
k1,k2=0
n−1∑
`=0
n−1∑
j=0
(na+ `+ j+ nk1+ nk2)
−s
= ns
∞∑
k1,k2=0
(na+ k1+ k2)
−s
= ns ζ2(s,na),
which, upon differentiating with respect to s, yields
n−1∑
`=0
n−1∑
j=0
ζ ′2
(
s,a+
`+ j
n
)
= (logn)ns ζ2(s,na)+ n
s ζ ′2(s,na).
The Zeta and Related Functions 155
By virtue of (30), we readily obtain the following multiplication formula for 02:
n−1∏
`=0
n−1∏
j=0
02
(
a+
`+ j
n
)
= C(n)(2π)−
1
2 n(n−1)a n−
n2 a2
2 +na02(na), (32)
where
C(n) := An
2
−1
·e
1
12 (1−n
2)
·(2π)
1
2 (n−1)·n
5
12 .
An interesting identity is also obtained from (32):
n−1∏
`=0
n−1∏
j=0
′02
(
`+ j
n
)
=
C(n)
n
, (33)
where the prime denotes the exclusion of the case when `= 0= j.
2.2 The Hurwitz (or Generalized) Zeta Function
The Hurwitz (or generalized) Zeta function ζ(s,a) is defined by
ζ(s,a) :=
∞∑
k=0
(k+ a)−s (<(s) > 1; a 6∈ Z−0 ). (1)
It is easy to see that ζ(s,a)= ζ1(s,a) for the case when n= 1 in 2.1(2). Thus, we
can deduce many properties of ζ(s,a) from those of ζn(s,a) in Section 2.1. Indeed,
the series for ζ(s,a) in (1) converges absolutely for <(s)= σ > 1. The convergence
is uniform in every half-plane σ ≥ 1+ δ (δ > 0), so ζ(s,a) is an analytic function of
s in the half-plane <(s)= σ > 1. Setting n= 1 in 2.1(6), we have the integral repre-
sentation:
0(s)ζ(s,a)=
∞∫
0
xs−1 e−ax
1− e−x
dx=
∞∫
0
xs−1 e−(a−1)x
ex− 1
dx
=
1∫
0
xa−1
1− x
(
log
1
x
)s−1
dx (<(s) > 1; <(a) > 0).
(2)
Moreover, ζ(s,a) can be continued meromorphically to the whole complex s-plane
(except for a simple pole at s= 1 with its residue 1) by means of the contour integral
156 Zeta and q-Zeta Functions and Associated Series and Integrals
representation (see Theorem 2.5):
ζ(s,a)=−
0(1− s)
2π i
∫
C
(−z)s−1 e−az
1− e−z
dz, (3)
where the contour C is the Hankel loop of Theorem 2.5. The connection between
ζ(s,a) and the Bernoulli polynomials Bn(x) is also given in 2.1(17).
From the definition (1) of ζ(s,a), it easily follows that
ζ(s,a)= ζ(s,n+ a)+
n−1∑
k=0
(k+ a)−s (n ∈ N); (4)
ζ
(
s,
1
2
a
)
− ζ
(
s,
1
2
a+
1
2
)
= 2s
∞∑
n=0
(−1)n (a+ n)−s. (5)
Hurwitz’s Formula for ζ(s,a)
The series expression ζ(s,a) was originally meaningful for σ > 1 (s= σ + it).
Hurwitz obtained another series representation for ζ(s,a) valid in the half-plane
σ < 0:
ζ(1− s,a)=
0(s)
(2π)s
{
e−
1
2π is L(a,s)+ e
1
2π is L(−a,s)
}
(0< a 5 1, σ =<(s) > 1; 0< a< 1, σ > 0),
(6)
where the function L(x,s) is defined by
L(x,s) :=
∞∑
n=1
e2π inx
ns
(x ∈ R; σ =<(s) > 1), (7)
which is often referred to as the periodic (or Lerch) Zeta function.
We note that the Dirichlet series in (7) is a periodic function of x with period 1
and that L(1,s)= ζ(s), the Riemann Zeta function (see Section 2.3). The series in (7)
converges absolutely for σ > 1. Yet, if x /∈ Z, the series can also be seen to converge
conditionally for σ > 0. So, the formula (6) is also valid for σ > 0, if a 6= 1.
We observe that the function L(x,s) in (7) is a linear combination of the Hurwitz
Zeta functions, when x is a rational number. Indeed, setting x= p/q (1 5 p 5
q; p, q∈N) in (7), the terms in (7) can be rearranged according to the residue classes
mod q, by letting
n= kq+ r (1 5 r 5 q; k ∈ N0),
The Zeta and Related Functions 157
which gives us, for σ > 1,
L
(
p
q
,s
)
=
∞∑
n=1
1
ns
exp
(
2π inp
q
)
=
q∑
r=1
∞∑
k=0
exp
(
2π irp
q
)
(kq+ r)s
=
1
qs
q∑
r=1
exp
(
2π irp
q
) ∞∑
k=0
1(
k+ rq
)s
=
1
qs
q∑
r=1
exp
(
2π irp
q
)
ζ
(
s,
r
q
)
.
Therefore, if we take a= p/q in the Hurwitz formula (6), we obtain:
ζ
(
1− s,
p
q
)
=
20(s)
(2πq)s
q∑
r=1
cos
(
πs
2
−
2πrp
q
)
ζ
(
s,
r
q
)
(1 5 p 5 q; p, q ∈ N),
(8)
which holds true, by the principle of analytic continuation, for all admissible values of
s ∈ C.
Hermite’s Formula for ζ(s,a)
We, first, recall Plana’s summation formula:
n∑
k=m
f (k)=
1
2
[ f (m)+ f (n)]+
n∫
m
f (τ )dτ − 2
∞∫
0
q(m, t)− q(n, t)
e2π t− 1
dt, (9)
where f (z) is a bounded analytic function in m 5 <(z)5 n and
q(λ, t)=
1
2i
[ f (λ+ it)− f (λ− it)] (i=
√
−1).
Choose the function f (z) in (9) as follows:
f (z)= (a+ z)−s (s= σ + it; |arg(a+ z)|< π). (10)
Then, it is easy to see that
q(x,y)=−
[
(a+ x)2+ y2
]− 12 s
sin
[
s arctan
(
y
x+ a
)]
.
In view of the elementary inequality:∣∣∣∣arctan( yx+ a
)∣∣∣∣5 min{π2 , |y|x+ a
}
,
158 Zeta and q-Zeta Functions and Associated Series and Integrals
we obtain
|q(x,y)|5
[
(a+ x)2+ y2
] 1
2−
1
2σ
∣∣∣y−1∣∣∣ sinh(1
2
π |s|
)
(|y|> a);
|q(x,y)|5
[
(a+ x)2+ y2
]− 12σ ∣∣∣∣sinh( y|s|x+ a
)∣∣∣∣ (|y|< a).
(11)
Making use of (11), it is easily seen that the integral:
∞∫
0
q(x,y)
(
e2πy− 1
)−1
dy (σ > 0)
converges when x = 0 and tends to 0 as x→∞. Also, the improper integral:
∞∫
0
(a+ x)−sdx (σ =<(s) > 1)
converges. Therefore, if σ > 1, it is valid to make n→∞ (m= 0) in (9) with the
function f (z) in (10). Thus, we readily obtain Hermite’s formula for ζ(s,a):
ζ(s,a)=
1
2
a−s+
a1−s
s− 1
+ 2
∞∫
0
(
a2+ y2
)− 12 s {
sin
(
s arctan
y
a
)} dy
e2πy− 1
. (12)
We note that the integral involved in (12) converges for all admissible values of
s ∈ C. Moreover, the integral is an entire function of s. A special case of the formula
(12) when a=1 is attributed to Jensen.
Setting s= 0 in (12), we have
ζ(0,a)=
1
2
− a, (13)
which is also obtained from 2.1(17) in view of 1.7(8). If we set z= s in 1.3(30) and
differentiate the resulting equation with respect to s, we find that
ψ(s)= logs−
1
2s
− 2
∞∫
0
t dt(
t2+ s2
) (
e2π t− 1
) (<(s) > 0). (14)
Taking the limit in (12) as s→ 1, by virtue of the uniform convergence of the integral
in (12), we get
lim
s→1
{
ζ(s,a)−
1
s− 1
}
= lim
s→1
a1−s− 1
s− 1
+
1
2a
+ 2
∞∫
0
ydy(
a2+ y2
)(
e2πy− 1
) ,
The Zeta and Related Functions 159
which, in view of (14), yields
lim
s→1
{
ζ(s,a)−
1
s− 1
}
=−
0′(a)
0(a)
=−ψ(a). (15)
Differentiating (12) with respect to s and setting s= 0 in the resulting equation,
we have
{
d
ds
ζ(s,a)
}
s=0
=
(
a−
1
2
)
loga− a+ 2
∞∫
0
arctan
( y
a
)
e2πy− 1
dy, (16)
which, by virtue of 1.2(30), yields
d
ds
ζ(s,a)
∣∣∣∣
s=0
= log 0(a)−
1
2
log(2π), (17)
which is equivalent to the identity 2.1(29). In addition to (17), it is easy to find from
the definition (1) of ζ(s,a) that
∂
∂a
ζ(s,a)=−sζ(s+ 1,a). (18)
The respective special cases of (15) and (17) when a= 1, by means of 1.2(4) and
1.1(13), become
lim
s→1
{
ζ(s,a)−
1
s− 1
}
= lim
�→0
{
ζ(1+ �,a)−
1
�
}
= γ (19)
and
ζ ′(0)=−
1
2
log(2π), (20)
where ζ(s) is the Riemann Zeta function (see Section 2.3).
Further Integral Representations for ζ(s,a)
In addition to (12), some known integral representations of ζ(s,a) are recalled here:
0(s)ζ(s,a)=
∞∫
0
t s−1 e−at
1− e−t
dt =
∞∫
0
t s−1 e−(a−1)t
et− 1
dt
=
1∫
0
ta−1
1− t
(
log
1
t
)s−1
dt
(
<(s) > 1; <(a) > 0
)
,
(21)
160 Zeta and q-Zeta Functions and Associated Series and Integrals
which is the same as (2);
ζ(s,a)=
1
2
a−s−
a1−s
1− s
+
1
0(s)
∞∫
0
(
1
et− 1
−
1
t
+
1
2
)
e−at t s−1 dt
(<(s) >−1; <(a) > 0);
(22)
ζ(s,a)=
π 2s−2
s− 1
∞∫
0
[
t2+ (2a− 1)2
] 1
2 (1−s)
cos
[
(s− 1) arctan
(
t
2a−1
)]
cosh2
(
1
2π t
) dt
(
<(a) >
1
2
)
;
(23)
ζ(s,a)= cos
(
1
2
πs
)
sin(2πa)
∞∫
0
t−s
cosh(2π t)− cosh(2πa)
dt
+ sin
(
1
2
πs
) ∞∫
0
t−s
[
cosh(2πa)− e−2π t
]
cosh(2π t)− cosh(2πa)
dt
(<(s) < 1 when 0< <(a) < 1; <(s) < 0 when a= 1);
(24)
ζ(s,a)= a−s+
n∑
k=0
0(k+ s− 1)
0(s)
Bk
k!
a−k−s+1
+
1
0(s)
∞∫
0
(
1
et− 1
−
n∑
k=0
Bk
k!
tk−1
)
e−at t s−1 dt
(<(s) >−(2n− 1); <(a) > 0; n ∈ N0).
(25)
Some Applications of the Derivative Formula (17)
We begin by giving another proof of the derivative formula (17). Indeed, by the ana-
lytic continuation of ζ(s,a) and the special case of (4) when n= 1, we observe that
ζ(s,a+ 1)= ζ(s,a)− a−s, (26)
which, upon differentiating with respect to s and setting s= 0 in the resulting equation,
yields
ζ ′(0,a+ 1)= ζ ′(0,a)+ log a.
Let f (a) := exp[ζ ′(0,a)]. We then have f (a+ 1)= af (a) (a> 0) and
d2
da2
log f (a)=
d2
da2
d
ds
ζ(s,a)
∣∣∣∣
s=0
=
∞∑
k=0
1
(k+ a)2
> 0 (a> 0),
The Zeta and Related Functions 161
which implies that f (a) is logarithmically convex on (0,∞). Thus, by appealing to the
Bohr-Mollerup theorem (Theorem 1.1), we obtain, for some constant C,
f (a)= C0(a),
which, for a= 1, yields
C = exp[ζ ′(0)]= (2π)−
1
2 .
This completes our second proof of the derivative formula (17).
Many authors gave seemingly different proofs of Stirling’s formula 1.1(52) (see
e.g., Blyth and Pathak [135], Choi [261], Diaconis and Freeman [380] and Patin
[889]). Here, by taking the limit in (16) as a→∞, we have
lim
a→∞
{
ζ ′(0,a)+ a+
1
2
loga− a loga
}
= 0, (27)
which, upon taking the exponential and using (17), immediately yields Stirling’s
formula 1.1(33).
Combining formulas (17) and 1.1(42), we obtain a formula for the Beta function
B(α,β) :
B(α,β)= (2π)
1
2 exp
[
ζ ′(0,α)+ ζ ′(0,β)− ζ ′(0,α+β)
]
, (28)
where ζ ′(s,a)= ∂
∂sζ(s,a). Applying the formula (16) and the following trigonometric
identities:
arctana+ arctanb= arctan
a+ b
1− ab
(ab< 1),
arctana− arctanb= arctan
a− b
1+ ab
(ab>−1)
(29)
to (28), we can readily deduce an integral representation of B(α,β) (cf. Choi and Nam
[276]):
B(α,β)=
αα−
1
2 ββ−
1
2
(α+β)α+β−
1
2
(2π)
1
2 eI(α,β) (α > 0, β > 0), (30)
where, for convenience,
I(α,β) := 2ρ
∞∫
0
arctan
{
(t3+ t)ρ3
αβ(α+β)
}
dt
e2π tρ − 1
(
ρ2 := α2+αβ +β2
)
.
162 Zeta and q-Zeta Functions and Associated Series and Integrals
Differentiating both sides of (22) with respect to s and letting s= 0 in the resulting
equation, we obtain
ζ ′(0,a)=
(
a−
1
2
)
loga− a+
∞∫
0
(
1
et− 1
−
1
t
+
1
2
)
e−at
t
dt
(<(a) > 0).
(31)
In a similar manner, (31) leads us to another integral representation of B(α,β) (cf. Choi
and Nam [276]):
B(α,β)=
αα−
1
2 ββ−
1
2
(α+β)α+β−
1
2
(2π)
1
2 eJ(α,β) (α > 0; β > 0), (32)
where, for convenience,
J(α,β) :=
∞∫
0
(
1
et− 1
−
1
t
+
1
2
)(
e−αt+ e−βt− e−(α+β)t
) dt
t
.
Another Form for 02(a)
From 2.1(24) and 2.1(30), by virtue of (17), we obtain another form for the double
Gamma function 02(a) :
02(a)= A {0(a)}
1−a exp
[
−
1
12
+ ζ ′(−1,a)
]
(a> 0), (33)
where ζ ′(s,a)= ∂
∂sζ(s,a).
In addition to the integral representation 1.4(78), we can express log02(a) as
improper integrals in many ways. For example, we give two integral representations
for log02(a):
log02(a)=−
1
12
+ logA−
a2
4
+
(
1
2
a2−
1
2
a
)
loga+ (1− a) log0(a)
+ 2
∞∫
0
{
1
2
(a2+ t2)
1
2 sin
(
arctan
t
a
)
log(a2+ t2)
+ (a2+ t2)
1
2 cos
(
arctan
t
a
)
arctan
t
a
}
dt
e2π t− 1
(<(a) > 0);
(34)
The Zeta and Related Functions 163
log02(a)= logA−
a2
4
+
(
a2
2
−
a
2
+
1
12
)
loga+ (1− a) log0(a)
−
∞∫
0
(
1
et− 1
−
1
t
+
1
2
−
t
12
)
e−at
t2
dt (<(a) > 0).
(35)
Indeed, by differentiating Hermite’s formula (12) for ζ(s,a), with respect to s, letting
s→−1 in the resulting equation and applying (17) and the identity for ζ ′(−1,a), we
readily obtain (34). Conversely, setting n= 2 in (25), we have
ζ(s,a)=
sa−s−1
12
+
a−s
2
+
a1−s
s− 1
+
1
0(s)
∞∫
0
(
1
et− 1
−
1
t
+
1
2
−
t
12
)
t s−1e−atdt
(36)
(<(s) >−3; <(a) > 0).
Employing the same technique as in getting (34), by making use of (36) and consider-
ing the following identities:
d
ds
1
0(s)
∣∣∣∣
s=−1
=−1 and
1
0(s)
∣∣∣∣
s=−1
= 0,
we obtain (35).
Glaisher [484, p. 47] expressed the Glaisher-Kinkelin constant A given in 1.3(2) as
an integral:
A= 2
7
36π−
1
6 exp
1
3
+
2
3
1
2∫
0
log 0(t+ 1)dt
. (37)
By setting a= 1 in (34) and (35), we can also obtain integral representations of
logA:
logA=
1
3
− 2
∞∫
0
{
1
2
(1+ t2)
1
2 sin(arctan t) log(1+ t2)
+ (1+ t2)
1
2 cos(arctan t) arctan t
} dt
e2π t− 1
(38)
and
logA=
1
4
+
∞∫
0
(
1
et− 1
−
1
t
+
1
2
−
t
12
)
e−t
t2
dt. (39)
164 Zeta and q-Zeta Functions and Associated Series and Integrals
The formula (35) can be used to obtain an asymptotic formula for log02(a) by first
observing that, for some M > 0 and for all a> 0,∣∣∣∣
∞∫
0
(
1
et− 1
−
1
t
+
1
2
−
t
12
)
e−at
t2
dt
∣∣∣∣< Ma .
Thus, by employing 1.1(34), we have
log02(a)= logA+
3a2
4
− a−
1
12
−
(
a2
2
− a+
5
12
)
loga
+
1
2
(1− a) log(2π)+O
(
a−1
)
(a→∞; a> 0),
(40)
which may be compared with 1.3(7).
2.3 The Riemann Zeta Function
The Riemann Zeta function ζ(s) is defined by
ζ(s) :=

∞∑
n=1
1
ns =
1
1−2−s
∞∑
n=1
1
(2n−1)s (<(s) > 1)
1
1−21−s
∞∑
n=1
(−1)n−1
ns (<(s) > 0; s 6= 1).
(1)
It is easy to see from the definitions (1) and 2.2(1) that
ζ(s)= ζ(s,1)=
(
2s− 1
)−1
ζ
(
s,
1
2
)
= 1+ ζ(s,2) (2)
and
ζ(s)=
1
ms− 1
m−1∑
j=1
ζ
(
s,
j
m
)
(m ∈ N \ {1}). (3)
In view of (2), we can deduce many properties of ζ(s) from those of ζ(s,a) given in
Section 2.2. In fact, the series ζ(s)=
∑
∞
n=1 n
−s in (1) represents an analytic function
of s in the half-plane <(s)= σ > 1. Setting a= 1 in 2.2(2), we have an integral rep-
resentation of ζ(s) in the form:
0(s)ζ(s)=
∞∫
0
xs−1 e−x
1− e−x
dx=
∞∫
0
xs−1
ex− 1
dx
=
1∫
0
1
1− x
(
log
1
x
)s−1
dx (<(s) > 1).
(4)
The Zeta and Related Functions 165
Furthermore, just as ζ(s,a), ζ(s)can be continued meromorphically to the whole com-
plex s-plane (except for a simple pole at s= 1 with its residue 1) by means of the
contour integral representation:
ζ(s)=−
0(1− s)
2π i
∫
C
(−z)s−1 e−z
1− e−z
dz, (5)
where the contour C is the Hankel loop of Theorem 2.5.
Now 2.2(19) and (5), together, imply that the Laurent series of ζ(s) in a neighbor-
hood of its pole s= 1 has the form:
ζ(s)=
1
s− 1
+ γ +
∞∑
n=1
an (s− 1)
n, (6)
where γ is the Euler-Mascheroni constant given in 1.1(3) and an is also expressed as
(see Ivić [586, pp. 4–6]):
an = lim
m→∞
{
m∑
k=1
(log k)n
k
−
(log m)n+1
n+ 1
}
(n ∈ N). (7)
The Riemann Zeta function ζ(s) in (1) plays a central rôle in the applications of
complex analysis to number theory. The number-theoretic properties of ζ(s) are exhib-
ited by the following result, known as Euler’s formula, which gives a relationship
between the set of primes and the set of positive integers:
ζ(s)=
∏
p
(
1− p−s
)−1
(<(s) > 1), (8)
where the product is taken over all primes.
From 2.2(4), we have [cf. Equation (2) for the special cases when n= 0 and n= 1]
ζ(s)= ζ(s,n+ 1)+
n∑
k=1
k−s (n ∈ N0). (9)
The connection between ζ(s) and the Bernoulli numbers is given as follows:
ζ(−n)=

−
1
2 (n= 0)
−
Bn+1
n+1 (n ∈ N),
(10)
which is deduced by setting x= 1 in 2.1(17) and using 1.6(5).
166 Zeta and q-Zeta Functions and Associated Series and Integrals
Riemann’s Functional Equation for ζ(s)
The special case of 2.2(8), when p= 1= q, yields Riemann’s functional equation
for ζ(s):
ζ(1− s)= 2(2π)−s0(s) cos
(
1
2
πs
)
ζ(s) (11)
or, equivalently,
ζ(s)= 2(2π)s−10(1− s) sin
(
1
2
πs
)
ζ(1− s). (12)
Taking s= 2n+ 1 (n ∈ N) in (11), the factor cos
(
1
2πs
)
vanishes, and we find that
ζ(−2n)= 0 (n ∈ N), (13)
which are often referred to as the trivial zeros of ζ(s). The equation (13) can also be
proven by combining (10) and 1.7(7).
By using the Legendre duplication formulas 1.1(29) and 1.1(12), it is not difficult
to see that the functional equations (11) or (12) can be written in a simpler form:
8(s)=8(1− s), (14)
where the function 8(s) is defined by
8(s) := π−
1
2 s0
(
1
2
s
)
ζ(s). (15)
The function 8(s) has simple poles at s= 0 and s= 1. According to Riemann, to
remove these poles, we multiply 8(s) by 12 s(1− s) and define
ξ(s) :=
1
2
s(1− s)8(s), (16)
which is an entire function of s and satisfies the functional equation:
ξ(s)= ξ(1− s). (17)
Setting s= 2n (n ∈ N) in (11) and applying (10), we have the well-known identity:
ζ(2n)= (−1)n+1
(2π)2n
2(2n)!
B2n (n ∈ N0), (18)
The Zeta and Related Functions 167
which, in view of 1.7(7), enables us to list the following special values:
ζ(2)=
π2
6
, ζ(4)=
π4
90
, ζ(6)=
π6
945
,
ζ(8)=
π8
9450
, ζ(10)=
π10
93555
, . . . .
(19)
We may recall here a known recursion formula for ζ(2n) (see also Section 4.1):
ζ(2n)=
2
2n+ 1
n−1∑
k=1
ζ(2k)ζ(2n− 2k) (n ∈ N \ {1}), (20)
which can also be used to evaluate ζ(2n) (n ∈ N \ {1}).
We get no information about ζ(2n+ 1) (n ∈ N) from Riemann’s functional
equation, since both members of (11) vanish upon setting s= 2n+ 1 (n ∈ N). In fact,
until now, no simple formula analogous to (18) is known for ζ(2n+ 1) or even for any
special case, such as ζ(3). It is not even known whether ζ(2n+ 1) is rational or irra-
tional, except that the irrationality of ζ(3) was proven recently by Apéry [56]. Instead,
a known integral formula for ζ(2n+ 1) is recalled here:
ζ(2n+ 1)=
(−1)n+1 (2π)2n+1
2(2n+ 1)!
1∫
0
B2n+1(t) cot(π t)dt (n ∈ N). (21)
It is readily seen that ζ(s) 6= 0 (<(s)= σ = 1), and (12) shows that ζ(s) 6= 0
(σ 5 0), except for the trivial zeros in (13). Furthermore, in view of the second series
definition of ζ(s) in (1), we find that ζ(s) < 0 (s ∈ R; 0< s< 1). The assertion that
all the non-trivial zeros of ζ(s) have real part 12 is popularly known as the Riemann
hypothesis, which was conjectured (but not proven) in the memoir of Riemann [977].
This hypothesis is still one of the most challenging mathematical problems today
(see Edwards [398]).
It is easy to derive from (12) and (13) that (cf., e.g., Srivastava [1084, p. 387,
Eq. (1.15)])
ζ ′(−2n)= lim
�→0
ζ(−2n+ �)
�
= (−1)n
(2n)!
2(2π)2n
ζ(2n+ 1) (n ∈ N). (22)
Relationship between ζ(s) and the Mathematical Constants B and C
Just as log A in 2.1(31), we can also express log B and log C introduced in Section 1.4
as special cases of ζ ′(s). Indeed, using the Euler-Maclaurin summation formula
1.4(68), we can obtain a number of analytical representations of ζ(s), such as
168 Zeta and q-Zeta Functions and Associated Series and Integrals
(cf. Hardy [537, p. 333])
ζ(s)= lim
n→∞
{
n∑
k=1
k−s−
n1−s
1− s
−
1
2
n−s
}
(<(s) >−1), (23)
ζ(s)= lim
n→∞
{
n∑
k=1
k−s−
n1−s
1− s
−
1
2
n−s+
1
12
sn−s−1
}
(<(s) >−3), (24)
and
ζ(s)= lim
n→∞
{
n∑
k=1
k−s−
n1−s
1− s
−
1
2
n−s+
1
12
sn−s−1
−
1
720
s(s+ 1)(s+ 2)n−s−3
}
(<(s) >−5).
(25)
Now, it is not difficult to express the mathematical constants B and C as follows:
logB=−ζ ′(−2) (26)
and
logC =−ζ ′(−3)−
11
720
, (27)
respectively.
It is clear from 1.4(70) that logC must be the finite part of the divergent sum∑
k3 logk, according to some regularization; hence, logC must be related to ζ ′(s)
for some special value of s. By differentiating both sides of (25) with respect to s and
letting s=−3, we obtain
−ζ ′(−3)= lim
n→∞
{
n∑
k=1
k3 logk−
(
n4
4
+
n3
2
+
n2
4
−
1
120
)
logn+
n4
16
−
n2
12
}
+
11
720
, (28)
which, when compared with 1.4(70), yields the desired expression (27). The special
case of (22) when n= 1 also shows that, in view of (26),
logB=
ζ(3)
4π2
. (29)
The Zeta and Related Functions 169
Integral Representations for ζ(s)
In addition to (4) and (5), in terms of (2), we find from Section 2.2 that
ζ(s)=
1
0(s)
∞∫
0
t s−1
et− 1
dt (<(s) > 1); (30)
ζ(s)=
1
2
+
1
s− 1
+
1
0(s)
∞∫
0
(
1
et− 1
−
1
t
+
1
2
)
e−t t s−1 dt
(<(s) >−1);
(31)
ζ(s)= sin
(π
2
s
) ∞∫
0
cosh(2π)− e−2π t
cosh(2π t)− cosh(2π)
dt
ts
(<(s) < 0); (32)
ζ(s)= 1+
1
s− 1
+
n∑
k=1
0(k+ s− 1)
0(s)
Bk
k!
+
1
0(s)
∞∫
0
(
1
et− 1
−
n∑
k=0
Bk
k!
tk−1
)
e−t t s−1 dt
(<(s) >−(2n− 1); n ∈ N0);
(33)
ζ(s)=
1
2
+
1
s− 1
+ 2
∞∫
0
sin(s arctan t)(
1+ t2
) 1
2 s
dt
e2π t− 1
. (34)
Now, we recall some summation formulas analogous to the infinite series version
of Plana’s summation formula 2.2(9). To do this, let z= τ + it,
1
e−2π iz− 1
= X(τ, t)+ iY(τ, t)
and
1
e−π iz− eπ iz
= X1(τ, t)+ iY1(τ, t),
from which
X(τ, t)=
cos(2πτ)− e−2π t
e2π t− 2cos(2πτ)+ e−2π t
,
Y(τ, t)=
sin(2πτ)
e2π t− 2cos(2πτ)+ e−2π t
,
X1(τ, t)=
cos(πτ)
(
eπ t− e−π t
)
e2π t− 2cos(2πτ)+ e−2π t
,
170 Zeta and q-Zeta Functions and Associated Series and Integrals
and
Y1(τ, t)=
sin(πτ)
(
eπ t+ e−π t
)
e2π t− 2cos(2πτ)+ e−2π t
.
Conversely, we let
p(τ, t) :=
1
2
[ f (τ + it)+ f (τ − it)]
and
q(τ, t) :=
1
2i
[ f (τ + it)− f (τ − it)],
so that, by appealing to the Reflection Principle, p(τ, t) and q(τ, t) represent, respec-
tively, the real and imaginary parts of the analytic function f (τ + it), whose domain
is symmetric with respect to the τ -axis, in case the function f (τ + i0) is real. Then,
we have the summation formulas (see Lindelöf [769, Chapter III]):
∞∑
n=m
f (n)=
∞∫
α
f (τ )dτ − 2
∞∫
0
Q(α, t)dt, (35)
where, for convenience,
Q(τ, t) := p(τ, t)Y(τ, t)+ q(τ, t)X(τ, t); (36)
∞∑
n=m
f (n)=−
1
2π i
α+i∞∫
α−i∞
(
π
sin(πz)
)2
F(z)dz, (37)
F being a primitive function of f ;
∞∑
n=m
(−1)n f (n)=−
α+i∞∫
α−i∞
f (z)dz
eπ iz− e−π iz
=−2
∞∫
0
Q1(α, t)dt, (38)
where, for convenience,
Q1(τ, t) := p(τ, t)Y1(τ, t)+ q(τ, t)X1(τ, t), (39)
α being any number between m− 1 and m in both (37) and (39).
The Zeta and Related Functions 171
Moreover, the formulas (35), (37) and (38) are valid, if the function f satisfies each
of the following conditions:
(a) f (z) is analytic for <(z) > 0 (z= τ + it);
(b) limt→∞ exp(−2π |t|) f (τ + it)= 0 uniformly for 0≤ τ <∞;
(c)lim
τ→∞
∞∫
−∞
exp(−2π |t|) |f (τ + it)|dt = 0.
Now, we set f (z)= z−s, with m= 1 and α = 12 , in the three summation formulas
just introduced. We, thus, obtain
p(τ, t)= (τ 2+ t2)−
1
2 s cos
(
s arctan
t
τ
)
,
q(τ, t)=−(τ 2+ t2)−
1
2 s sin
(
s arctan
t
τ
)
.
(40)
We also find, from (35), that
ζ(s)=
2s−1
s− 1
− 2s
∞∫
0
(
1+ t2
)− 12 s
sin(s arctan t)
dt
eπ t+ 1
. (41)
Setting z= 12 + it in (37), we have
ζ(s)=
π 2s−2
s− 1
∞∫
0
(
1+ t2
) 1−s
2 cos[(s− 1) arctan t][
cosh
(
1
2π t
)]2 dt. (42)
Starting with the second part of the definition (1) of ζ(s) and applying (38), we also get
ζ(s)=
2s−1
1− 21−s
∞∫
0
(
1+ t2
)− 12 s cos(sarctan t)
cosh
(
1
2π t
) dt. (43)
Formulas (34), (41), (42) and (43) are due to Jensen. It is observed that the
definite integrals appearing in these four expressions represent analytic functions for
all bounded values of s. In fact, any one of these expressions gives the analytic con-
tinuation of ζ(s) in the whole complex s-plane, except for a simple pole at s= 1 with
its residue 1, just as it is with the contour integral representation (5).
Some other integral representations of ζ(s) are recalled here as follows:
ζ(s)=
1
s− 1
+
sin(πs)
π
∞∫
0
[log(1+ t)−ψ(1+ t)]
dt
ts
, (44)
172 Zeta and q-Zeta Functions and Associated Series and Integrals
ζ(s+ 1)=
sin(πs)
πs
∞∫
0
ψ ′(1+ t)]
dt
ts
=
sin(πs)
πs
∞∫
0
[ψ(1+ t)+ γ ]
dt
t1+s
,
(45)
and
ζ(s+m)= (−1)m−1
0(s) sin(πs)
π 0(s+m)
∞∫
0
ψ (m)(1+ t)
dt
ts
(m ∈ N), (46)
all three of which are due to de Bruijn [371] and are valid for 0< <(s) < 1.Moreover,
ζ(s)=
1
s− 1
+
sin(πs)
π(s− 1)
∞∫
0
(
ψ ′(1+ t)−
1
1+ t
)
dt
t1−s
(0< <(s) < 2; s 6= 1);
(47)
ζ(s)=
1
0(s+ 1)
∞∫
0
et ts
(et− 1)2
dt (<(s) > 1); (48)
ζ(s)=
(
1− 21−s
)−1
0(s)
∞∫
0
t s−1
et+ 1
dt (<(s) > 0); (49)
ζ(s)=
(
1− 2−s
)−1
20(s)
∞∫
0
t s−1
sinh t
dt (<(s) > 1); (50)
ζ(s)=
(
1− 21−s
)−1
20(s+ 1)
∞∫
0
ts et
(et+ 1)2
dt (<(s) > 0); (51)
ζ(s)=
π
1
2 s
s(s− 1)0
(
1
2 s
) + π 12 s
0
(
1
2 s
) ∞∫
1
(
t
1
2 (1−s)+ t
1
2 s
) ω(t)
t
dt, (52)
which is due to Riemann and where
ω(t) :=
∞∑
n=1
exp
(
−n2π t
)
. (53)
A Summation Identity for ζ(n)
Sitaramachandrarao and Sivaramsarma [1035] proved the following summation
identity for ζ(n), by using a known transformation formula and some reciprocity
The Zeta and Related Functions 173
relations:
2
∞∑
q=1
1
qn
q−1∑
k=1
1
k
= nζ(n+ 1)− n−2∑
k=1
ζ(n− k)ζ(k+ 1) (n ∈ N \ {1}), (54)
where an empty sum is understood (as usual) to be nil. The formula (54) contains many
(known or new) special cases. Here, we give an elementary proof of (54) (cf. Williams
[1229]). Let us, first, consider the sum:
n−2∑
k=1
ζ(n− k)ζ(k+ 1)= ζ(2)ζ(n− 1)+ ζ(3)ζ(n− 2)+ ·· ·+ ζ(n− 1)ζ(2)
=
∞∑
p=1
∞∑
q=1
{
1
p2
·
1
qn−1
+
1
p3
·
1
qn−2
+ ·· ·+
1
pn−1
·
1
q2
}
= lim
N→∞
SN,
where, for convenience,
SN :=
N∑
p=1
N∑
q=1
{
1
p2
·
1
qn−1
+
1
p3
·
1
qn−2
+ ·· ·+
1
pn−1
·
1
q2
}
(N ∈ N). (55)
Observing that the sum within braces in (55) is a geometric series and taking care of
the exceptional case when p= q, the series (55) becomes
N∑
q=1

N∑
p=1
p6=q
1
p− q
(
q1−n
p
−
p1−n
q
)
+ (n− 2)
1
qn+1
. (56)
Now, consider the first double sum in (56):
N∑
q=1
N∑
p=1
p6=q
1
p− q
(
q1−n
p
−
p1−n
q
)
=
N∑
q=1
N∑
p=1
p6=q
1
p− q
·
q1−n
p
+
N∑
p=1
N∑
q=1
q6=p
1
q− p
·
p1−n
q
.
(57)
174 Zeta and q-Zeta Functions and Associated Series and Integrals
Inverting the order of summation in the second term on the right-hand side of (57), it
is fairly straightforward to write (57) in the form:
N∑
q=1
N∑
p=1
p6=q
q1−n
p(p− q)
+
N∑
p=1
N∑
q=1
q6=p
p1−n
q(q− p)
= 2
N∑
q=1
1
qn−1
N∑
p=1
p6=q
1
p(p− q)
. (58)
Combining (55) through (58), we find that
N∑
q=1
N∑
p=1
{
1
p2
·
1
qn−1
+
1
p3
·
1
qn−2
+ ·· ·+
1
pn−1
·
1
q2
}
= (n− 2)
N∑
q=1
1
qn+1
+ 2
N∑
q=1
1
qn−1
N∑
p=1
p6=q
1
p(p− q)
.
(59)
Now, consider
−q
N∑
p=1
p6=q
1
p(p− q)
=
N∑
p=1
p6=q
(
1
p
−
1
p− q
)
=
q−1∑
p=1
1
q− p
−
N∑
p=q+1
1
p− q
+
N∑
p=1
1
p
−
1
q
=
q−1∑
p=1
1
p
−
N−q∑
p=1
1
p
+
N∑
p=1
1
p
−
1
q
=−
1
q
+
q−1∑
p=1
1
p
+
N∑
p=N−q+1
1
p
,
(60)
which, when substituted into (59), yields
N∑
q=1
N∑
p=1
{
1
p2
·
1
qn−1
+
1
p3
·
1
qn−2
+ ·· ·+
1
pn−1
·
1
q2
}
= n
N∑
q=1
1
qn+1
− 2
N∑
q=1
1
qn
q−1∑
k=1
1
k
− 2 N∑
q=1
1
qn
 N∑
k=N−q+1
1
k
.
(61)
Finally, we consider the inequality:
0 5
N∑
k=N−q+1
1
k
=
1
N− q+ 1
+
1
N− q+ 2
+ ·· ·+
1
N
5
q
N− q+ 1
,
The Zeta and Related Functions 175
so that
0<
N∑
q=1
1
qn
 N∑
k=N−q+1
1
k
< N∑
q=1
1
qn−1
·
1
N− q+ 1
5
N∑
q=1
1
q(N− q+ 1)
(n ∈ N \ {1})
=
1
N+ 1
N∑
q=1
(
1
q
+
1
N− q+ 1
)
=
2
N+ 1
N∑
q=1
1
q
<
2
N+ 1
(1+ logN)→ 0 as N→∞.
(62)
By taking the limit in (61) as N→∞ and applying (62), we complete the proof of
the summation identity (54).
2.4 Polylogarithm Functions
The so-called Polylogarithm functions have been studied rather extensively in the
works of (for example) Lewin [751, 752], who was exceedingly fascinated by the
following integral formula, even during his school days:
1
2 (
√
5−1)∫
0
log(1− t)
t
dt =
{
log
(√
5− 1
2
)}2
−
π2
10
. (1)
Many of the earlier authors who studied these functions include Euler, Abel, Legendre,
Kummer, Spence and so on. These functions have been found useful in physics, elec-
tronics, quantum electrodynamics, etc., and are closely related to many other mathe-
matical functions, such as the Clausen integral (or function) Cl2(x) given by
Cl2(x)= x logπ − x log sin
( x
2
)
+ 2π log
G
(
1− x2π
)
G
(
1+ x2π
) , (2)
where G is the Barnes G-function studied in Section 1.4.
Throughout this work, we choose to follow the notations and conventions used by
Lewin [751, 752].
176 Zeta and q-Zeta Functions and Associated Series and Integrals
The Dilogarithm Function
The Dilogarithm function Li2(z) is defined by
Li2(z) :=
∞∑
n=1
zn
n2
(|z| ≤ 1)
=−
z∫
0
log(1− t)
t
dt.
(3)
It follows from (3) that
d
dx
Li2
(
−
1
x
)
=
log(1+ x)− log x
x
,
the integration of which yields
Li2
(
−
1
x
)
+Li2 (−x)= 2Li2 (−1)−
1
2
(log x)2 , (4)
where the constant of integration is determined by taking x= 1.
The special case of (4) when x=−1= eiπ gives
2Li2(1)= 2Li2(−1)+
π2
2
. (5)
It is easy to see also that
Li2(−1)=−
1
2
Li2(1), (6)
which, in view of (5), yields
Li2(1)=
1
12
+
1
22
+
1
32
+ ·· · =
π2
6
= ζ(2), (7)
where ζ(s) is the Riemann Zeta function (see Section 2.3). It follows from (6) and
(7) that
Li2(−1)=−
π2
12
. (8)
By taking x= yeiπ in (4), we obtain
Li2 (y)+Li2
(
1
y
)
=
π2
3
−
1
2
(log y)2− iπ logy (y> 1). (9)
The Zeta and Related Functions 177
From (3), we also find that
Li2(z)=− logz log(1− z)−
z∫
0
log t
1− t
dt
=− logz log(1− z)−Li2(1− z)+Li2(1),
from which
Li2(z)+Li2(1− z)=
π2
6
− logz log(1− z). (10)
By choosing a suitable argument and then employing differentiation and integra-
tion, we have Landen’s formulas (see Lewin [751, p. 5]):
Li2 (x)+Li2
(
x
x− 1
)
=−
1
2
{log(1− x)}2 (x< 1) (11)
and
Li2 (x)+Li2
(
x
x− 1
)
=
π2
2
− 2iπ logx+ iπ log(x− 1)
−
1
2
{log(x− 1)}2 (x> 1).
(12)
A factorization formula for Li2 (x) is given by
1
n
Li2
(
xn
)
=
n−1∑
k=0
Li2
(
ωk x
) (
ω := exp
(
2π i
n
))
, (13)
which follows from the elementary identity:
1− xn =
n−1∏
k=0
(
1−ωk x
) (
ω := exp
(
2π i
n
))
.
The special case of (13) when n= 2 gives us
1
2
Li2
(
x2
)
= Li2 (x)+Li2 (−x). (14)
By eliminating Li2 (x) between (11) and (14), we get
Li2
(
x
x− 1
)
+
1
2
Li2
(
x2
)
−Li2 (−x)=−
1
2
{log(1− x)}2 . (15)
178 Zeta and q-Zeta Functions and Associated Series and Integrals
Setting x= 3−
√
5
2 and x=
√
5−1
2 in (11) yields the following results:
Li2
(
1−
√
5
2
)
=−
π2
15
+
1
2
{
log
(√
5− 1
2
)}2
(16)
and
Li2
(
−
1+
√
5
2
)
=−
π2
10
+
1
2
{
log
(√
5+ 1
2
)}2
, (17)respectively.
The Legendre’s Chi function χ2(x) is defined by
χ2(x) :=
∞∑
n=1
x2n−1
(2n− 1)2
(−1 5 x 5 1)
=
1
2
{Li2 (x)−Li2 (−x)} ,
(18)
which, in view of (3), immediately becomes
χ2(x)=
1
2
x∫
0
log
(
1+ t
1− t
)
dt
t
. (19)
From (19), one readily finds the Euler-Legendre-Landen identity:
χ2
(
1− x
1+ x
)
+χ2(x)=
π2
8
+
1
2
logx log
(
1+ x
1− x
)
. (20)
Putting (1− x)/(1+ x)= x gives x2+ 2x− 1= 0 or x=
√
2− 1= tan
(
1
8π
)
. We,
thus, find from (20) that
χ2
(
tan
π
8
)
=
π2
16
−
1
4
[
log
(
tan
π
8
)]2
. (21)
Upon integrating by parts twice, it is easily observed that
x∫
0
tα Li2 (t)dt =
xα+1
α+ 1
Li2 (x)+
xα+1−1
(α+ 1)2
log(1− x)−
1
(α+ 1)2
x∫
0
1− tα+1
1− t
dt.
(22)
The Zeta and Related Functions 179
In view of 1.2(3), the integral formula (22) with x= 1 assumes the form:
1∫
0
tα−1 Li2 (t) dt =
π2
6α
−
ψ(α+ 1)+ γ
α2
, (23)
which, for α = n ∈ N0, yields
1∫
0
tn Li2 (t) dt =
π2
6(n+ 1)
−
1
(n+ 1)2
n+1∑
k=1
1
k
. (24)
Nielsen (see Lewin [751, pp. 20–21]) applied (24) to show that
1∫
0
f (tx)Li2 (t) dt =
π2
6x
x∫
0
f (t)dt−
1
x
∞∑
n=0
anxn+1
(n+ 1)2
(
n+1∑
k=1
1
k
)
, (25)
where f (x) :=
∑
∞
n=0 anx
n. Take an = n+ 1, so that f (t)= (1− t)−2. By making use
of the well-known expansion:
1
2
{log(1− x)}2 =
∞∑
n=1
xn+1
n+ 1
(
n∑
k=1
1
k
)
,
we find the following integral equation involving the Dilogarithm function:
1∫
0
Li2 (t)
(1− xt)2
dt =
π2
6(1− x)
−
Li2(x)
x
−
{log(1− x)}2
2x
(−1 5 x 5 1), (26)
whose special cases when x=−1 and x= 12 become
1∫
0
Li2 (t)
(1+ t)2
dt =
1
2
(log 2)2 (27)
and
1∫
0
Li2 (t)
(2− t)2
dt =
π2
24
, (28)
respectively.
180 Zeta and q-Zeta Functions and Associated Series and Integrals
If z is a pure imaginary number in (3), say z= iy (y ∈ R), then
Li2 (iy)=
1
4
Li2
(
−y2
)
+ iTi2 (y) (y ∈ R), (29)
where the new function Ti2(x) is defined by
Ti2 (x) :=
∞∑
n=1
(−1)n+1 x2n−1
(2n− 1)2
(x≤ 1). (30)
From (29), we also see that
Ti2 (y)=
1
2i
[Li2 (iy)−Li2 (−iy)] . (31)
It follows from (18) that
Ti2 (y)=−iχ2(iy). (32)
In view of the following series expansion of tan−1 x:
tan−1 x=
∞∑
n=1
(−1)n+1
x2n−1
2n− 1
(x≤ 1), (33)
the definition (30) can be written in the form:
Ti2 (x)=
x∫
0
tan−1 t
t
dt or Ti2 (x)=
x∫
0
(
tan−1 t
) d(log t)
dt
dt. (34)
If one takes (34) as a definition of Ti2 (x), the domain of Ti2 (x)may be extended to∞.
Some known formulas and special values for Ti2 (x) are recalled here:
Ti2 (x)−Ti2
(
1
x
)
= sgn(x)
π
2
log |x| (x ∈ R), (35)
where sgn(x) is defined by
sgn(x) :=
{
1 (x> 0),
−1 (x< 0);
(36)
Ti2 (1)= G, (37)
where G is the Catalan constant given in 1.3(16);
1
2
Ti2
(
2x
1− x2
)
= Ti2 (x)+Ti2 (−x,1)−Ti2 (x,1), (38)
The Zeta and Related Functions 181
where Ti2 (x,a) is defined by
Ti2 (x,a) :=
x∫
0
tan−1 t
a+ t
dt; (39)
1
3
Ti2
(
3x−x3
1−3x2
)
= Ti2 (x)+Ti2
(
1−
√
3x
√
3+x
)
−Ti2
(
1+
√
3x
√
3−x
)
+
π
6
log
( √
3+x
1−
√
3x
·
1+
√
3x
√
3−x
) (
−
1
√
3
< x<
1
√
3
)
,
(40)
which, upon setting x= tanθ, yields a trigonometric form:
1
3
Ti2 (tan 3θ)= Ti2 (tan θ)+Ti2
[
tan
(π
6
− θ
)]
−Ti2
[
tan
(π
6
+ θ
)]
+
π
6
log
[
tan
(
π
6 + θ
)
tan
(
π
6 − θ
)] . (41)
Clausen’s Integral (or Function)
From the series definition in (3) of the Dilogarithm function Li2(z), we have
Li2
(
eiθ
)
=
∞∑
n=1
cos nθ
n2
+ i
∞∑
n=1
sin nθ
n2
. (42)
From the integral definition in (3) of Li2(z), we also have
Li2
(
eiθ
)
−Li2 (1)=−
exp(iθ)∫
1
log(1− t)
t
dt,
which, upon putting t = eiη, yields
Li2
(
eiθ
)
−Li2 (1)=−i
θ∫
0
log
(
1− eiη
)
dη
=−i
θ∫
0
log
[
2 sin
(
1
2
η
)
exp
(
−
1
2
(π − η)
)
i
]
dη
=−i
θ∫
0
log
[
2 sin
(
1
2
η
)]
dη
+
1
4
(π − θ)2−
1
4
π2 (0 5 θ 5 2π).
(43)
182 Zeta and q-Zeta Functions and Associated Series and Integrals
Comparing the real parts in (42) and (43) gives us a well-known Fourier series:
∞∑
n=1
cos nθ
n2
=
π2
6
−
θ(2π − θ)
4
(0 5 θ 5 2π). (44)
Conversely, if we compare the imaginary parts in (42) and (43), we obtain
Clausen’s integral (or function) Cl2(θ) defined by [cf. Equation (2) above]
Cl2(θ) :=
∞∑
n=1
sin nθ
n2
=−
θ∫
0
log
[
2 sin
(
1
2
η
)]
dη. (45)
Equation (43) can, therefore, be written in the form:
Li2
(
eiθ
)
=
π2
6
−
θ(2π − θ)
4
+ iCl2(θ) (0 5 θ 5 2π). (46)
Some known properties and special values of the Clausen integral (or function) include
the periodic properties:
Cl2(2nπ ± θ)= Cl2(±θ)=±Cl2(θ), (47)
which, for n= 1 and with θ replaced by π + θ , yields
Cl2(π + θ)=−Cl2(π − θ). (48)
From the series definition (45), it is obvious that
Cl2(nπ)= 0 (n ∈ Z), (49)
which, for n= 1, gives
π∫
0
log
(
2 sin
1
2
θ
)
dθ = 0 and
π/2∫
0
log(sinθ) dθ =−
1
2
π log2. (50)
Setting θ = 12π in the series definition (45), and using (48), we find that
Cl2
(
1
2
π
)
= G=−Cl2
(
3
2
π
)
, (51)
The Zeta and Related Functions 183
where G is the Catalan constant given in 1.4(16);
1
2
Cl2(2θ)= Cl2(θ)−Cl2(π − θ). (52)
From (45), by employing integration by parts, we have
Cl2(θ)=−θ log
(
sin
1
2
θ
)
+
θ∫
0
1
2
η cot
(
1
2
η
)
dη. (53)
If we substitute the known formula:
cot
(
1
2
θ
)
=
1
1
2θ
−
∞∑
k=1
θ
k2π2− 14θ
2
(54)
into (53) and evaluate the resulting integral, we obtain
Cl2(θ)=−θ log
(
sin
1
2
θ
)
+ θ +
∞∑
n=1
[
2θ − 2nπ log
(
2nπ + θ
2nπ − θ
)]
(0 5 θ < 2π);
(55)
Ti2(tanθ)= θ log(tanθ)+
1
2
Cl2(2θ)+
1
2
Cl2(π − 2θ), (56)
whose interesting special case when θ = 112π becomes
Cl2
(π
6
)
+Cl2
(
5π
6
)
=
4
3
G. (57)
The Trilogarithm Function
The Trilogarithm function Li3(z) is defined by
Li3(z) :=
∞∑
n=1
zn
n3
(|z|5 1)
=
z∫
0
Li2(t)
t
dt.
(58)
An obvious special case of (58) is
Li3(1)= ζ(3), (59)
where ζ(s) is the Riemann Zeta function (see Section 2.3).
184 Zeta and q-Zeta Functions and Associated Series and Integrals
The following simple functional relationship for Li3(x) would follow, if we divide
both sides of (14) by x and then integrate each term:
1
4
Li3
(
x2
)
= Li3 (x)+Li3 (−x), (60)
which, for x= 1, yields
Li3(−1)=−
3
4
Li3(1). (61)
Similarly, we find from (4) and (9) that
Li3 (−x)−Li3
(
−
1
x
)
=−
π2
6
logx−
1
6
(log x)3 (62)
and
Li3 (y)−Li3
(
1
y
)
=
π2
3
logy−
1
6
(log y)3−
1
2
iπ(log y)2 (y> 1), (63)
respectively.
Landen’s formula for Li3 (x) (see Lewin [751, p. 138]) is recalled here as
follows:
Li3
(
x
x− 1
)
+Li3 (1− x)+Li3 (x)= Li3 (1)+
π2
6
log(1− x)
−
1
2
logx [log(1− x)]2+
1
6
[log(1− x)]3 (0< x< 1),
(64)
whose domain can be extended by using (63) with y= 1− x:
Li3
(
x
x− 1
)
+Li3
(
1
1− x
)
+Li3 (x)= Li3 (1)−
π2
6
log(1− x)
−
1
2
log(−x) [log(1− x)]2+
1
3
[log(1− x)]3 (x< 0).
(65)
Taking x= 12 in (64) or x=−1 in (65), we obtain
2Li3
(
1
2
)
+Li3 (−1)= Li3 (1)−
π2
6
log2+
1
3
(log2)3,
The Zeta and Related Functions 185
which, upon eliminating Li3 (−1) with the aid of (61), yields
Li3
(
1
2
)
=
7
8
Li3 (1)−
π2
12
log2+
1
6
(log 2)3. (66)
Setting 1− x= x/(1− x), we have
(1− x)2 = x or x2− 3x+ 1= 0,
which gives the value:
x=
3−
√
5
2
= 4 sin2
( π
10
)
(x< 1). (67)
Now, writing 1− x for x in (60), and evaluating (64) with x given by (67), we obtain
Li3
(
3−
√
5
2
)
=
4
5
Li3 (1)+
π2
15
log
(
3−
√
5
2
)
−
1
12
[
log
(
3−
√
5
2
)]3
.
(68)
An interesting definite integral of Li3 (1) was given by Hjortnaes [559]:
Li3 (1)= 10
log( 1+
√
5
2 )∫
0
t2 coth t dt, (69)
which, upon writing coth t in exponential form and putting y= 1− e−2t, yields
Li3 (1)=
5
4
√
5−1
2∫
0
[log(1− y)]2
(
2
y
+
1
1− y
)
dy. (70)
Integrating (70) by parts and expressing it in terms of trilogarithms, we arrive, once
again, at (68).
The Polylogarithm Functions
The Polylogarithm function Lin(z) is defined by
Lin(z) :=
∞∑
k=1
zk
kn
(|z|5 1; n ∈ N \ {1})
=
z∫
0
Lin−1(t)
t
dt (n ∈ N \ {1, 2}).
(71)
186 Zeta and q-Zeta Functions and Associated Series and Integrals
Clearly, we have
Lin(1)=ζ(n) (n ∈ N \ {1}), (72)
in terms of the Riemann Zeta function ζ(s) (see Section 2.3). Setting z= iy (y ∈ R) in
(71), we have
Lin(iy)=
1
2n
Lin
(
−y2
)
+ iTin(y) (y ∈ R), (73)
where Tin(x) is the inverse tangent integral of order n defined by
Tin(x) :=
∞∑
k=1
(−1)k+1
x2k−1
(2k− 1)n
(|x|5 1). (74)
It is obvious that
Tin(x)=
x∫
0
Tin−1(t)
t
dt, (75)
which, in conjunction with (34) for Ti2(x), extends the domain of Tin(x).
If z= reiθ , one defines
Lin
(
reiθ
)
:= Lin (r,θ)+ i [Tin(%)−Tin(%, tanθ)], (76)
where % = r sinθ/(1− r cosθ). Then, it is readily seen that
Lin (r,θ)=
r∫
0
Lin−1 (t,θ)
t
dt (77)
and
Tin(%, tanθ)=
%∫
0
(
Tin−1(t)
t+ tanθ
+
Tin−1(t, tanθ)
t
−
Tin−1(t, tanθ)
t+ tanθ
)
dt. (78)
The generalized Clausen function Cln(θ) is defined by
Cln(θ) :=

∞∑
k=1
sinkθ
kn
(n is even),
∞∑
k=1
coskθ
kn
(n is odd).
(79)
The Zeta and Related Functions 187
From these definitions, we have
Cl2n+1(θ)= Li2n+1(1)−
θ∫
0
Cl2n(t)dt (80)
and
Cl2n(θ)=
θ∫
0
Cl2n−1(t)dt. (81)
The Log-Sine integral Lsn(θ) of order n is defined by
Lsn(θ) :=−
θ∫
0
(
log
∣∣∣∣2sin 12 t
∣∣∣∣)n−1 dt (n ∈ N \ {1}), (82)
whose special case only when n= 2 satisfies the relationship (cf. Equation (45)):
Ls2(θ)= Cl2(θ). (83)
The generalized Log-Sine integral of order n and index m is defined by
Ls(m)n (θ) :=−
θ∫
0
tm
(
log
∣∣∣∣2sin 12 t
∣∣∣∣)n−m−1 dt. (84)
The associated Clausen function Gln(θ) of order n is defined by
Gln(θ) :=

∞∑
k=1
coskθ
kn
(n is even),
∞∑
k=1
sinkθ
kn
(n is odd),
(85)
which also satisfies the following obvious relationships analogous to (80) and (81):
Gl2n(θ)= Li2n(1)−
θ∫
0
Gl2n−1(t)dt (86)
and
Gl2n+1(θ)=
θ∫
0
Gl2n(t)dt, (87)
each of which was studied extensively by Glaisher [488].
188 Zeta and q-Zeta Functions and Associated Series and Integrals
Dividing both sides of (4) by x and integrating from 1 to x, we obtain (cf.
Equation (62))
Li3(−x)−Li3
(
−
1
x
)
=−
1
3!
(log x)3+ 2Li2(−1) logx. (88)
By repeating this process n− 2 times, we finally arrive at
Lin(−x)+ (−1)
n Lin
(
−
1
x
)
=−
1
n!
(log x)n
+ 2
[
1
2 n
]∑
k=1
(log x)n−2k
(n−2k)!
Li2k(−1) (n∈N \ {1}),
(89)
which was attributed to Jonquiére by Nielsen [861], although equivalent formulas
were given much earlier by Spence and Kummer (see Lewin [751, p. 173]).
Taking x= eiπ in (89), we find that
[
1+ (−1)n
]
Lin(1)=−
inπn
n!
+ 2
[
1
2 n
]∑
k=1
in−2kπn−2k
(n− 2k)!
Li2k(−1). (90)
Since the left-hand side of (90) vanishes when n is odd, the evaluation of Li2n+1(1)
is not possible from (90). Yet, using the following known relationship between Lin(1)
and Lin(−1):
Lin(−1)=−
(
1− 21−n
)
Lin(1) (91)
and writing 2n for n in (90), we readily get
2(−1)n−1
(
22n− 1
)
(2n)!
L2n =
1
(2n)!
− 2
n−1∑
k=1
(−1)k−1
(
22k−1− 1
)
(2k)!(2n− 2k)!
L2k, (92)
where, for the sake of brevity,
L2k := (2k)!π
−2k21−2k Li2k(1). (93)
By using the generating function 1.6(2) of the Bernoulli numbers, one can also obtain
Li2n(−1)= (−1)
n
(
22n−1− 1
)
B2n
π2n
(2n)!
(n ∈ N) (94)
and
Li2n(1)= (−1)
n+1 (2π)
2n
2(2n)!
B2n (n ∈ N), (95)
which is precisely the same as 2.3(18).
The Zeta and Related Functions 189
Although there is no result like (94) when the order of the Polylogarithm func-
tion is odd, there is an analogous relation involving the Euler numbers En given in
Section 1.7. Dividing (35) by x and integrating from 1 to x would yield
Ti3(x)+Ti3
(
1
x
)
= 2Ti3(1)+
1
4
π(log x)2 (x> 0). (96)
Repeated applications of this process lead us to Spence’s formula:
Tin (x)+(−1)
n−1 Tin
(
1
x
)
=
1
2
π
(log x)n−1
(n−1)!
+ 2
[
1
2 (n−1)
]∑
k=1
(log x)n−2k−1
(n− 2k− 1)!
Ti2k+1 (1) (x>0).
(97)
Next, we find from (73) that
Li2n+1(i)−Li2n+1(−i)= 2iTi2n+1(1). (98)
Setting x= exp
(
1
2π i
)
and writing 2n+ 1 for n in (89), we obtain
Li2n+1(−i)−Li2n+1(i)
= (−1)ni
[
−
π2n+1
(2n+ 1)!22n+1
+ 2π2n+1
n∑
k=1
B2k
(
22k−1− 1
)
(2k)!(2n− 2k+ 1)!22n−2k+1
]
.
(99)
Consider
2e
1
2 z
ez+ 1
= 2
e
1
2 z− e−
1
2 z
2
(
z
ez− 1
−
1
2
2z
e2z− 1
)
, (100)
which can be expanded by 1.6(2) and 1.6(40) in powers of z. Upon equating the coef-
ficients of z2n, we find that
E2n
(2n)!22n+1
=
[
1
(2n+ 1)!22n+1
− 2
n∑
k=1
B2k
(
22k−1− 1
)
(2k)!(2n− 2k+ 1)!22n−2k+1
]
. (101)
Now, a combination of (98), (99) and (101) leads us to the well-known Euler series:
Ti2n+1(1)=
∞∑
k=1
(−1)k+1
1
(2k− 1)2n+1
=
(−1)nπ2n+1
(2n)!22n+2
E2n, (102)
190 Zeta and q-Zeta Functions and Associated Series and Integrals
which, in view of 1.6(59), gives
Ti3(1)=
π3
32
, Ti5(1)=
5π5
1536
, Ti7(1)=
61π7
184320
, . . . . (103)
Starting with the known formula for Li2(x,θ):
Li2(r,θ)+Li2
(
1
r
,θ
)
= 2Gl2(θ)−
1
2
(logr)2 (104)
and following a procedure analogous to that in proving (89), one obtains
Lin(r,θ)+ (−1)
n Lin
(
1
r
,θ
)
=−
1
n!
(log r)n
+ 2
[
1
2 n
]∑
k=1
(log r)n−2k
(n− 2k)!
Gl2k(θ) (n∈N\{1}),
(105)
which, upon setting r = eiπ , noting that Lin(−r,θ)= Lin(r,π − θ) and writing 2n
for n, gives
Gl2n(π − θ)=
(−1)n−1π2n
2(2n)!
+
n∑
k=1
(−1)n−kπ2n−2k
(2n− 2k)!
Gl2k(θ). (106)
The following relationship between Gl2n(θ) and Gl2n(π − θ) can readily be obtained
from the series definition (85):
Gl2n(π − θ)+Gl2n(θ)=
1
22n−1
Gl2n(2θ). (107)
The generating function 1.7(1) for the Bernoulli polynomials can be rewritten in
the form:
text
et− 1
= 1+ t
∞∑
n=1
2t cos(2nπx)− 4nπ sin(2nπx)
t2+ 4n2π2
. (108)
Making use of the expansion:
1
t2+ 4n2π2
=
1
4n2π2
∞∑
k=0
(−1)k
(
t
2nπ
)2k
(|t|< 2nπ; n ∈ N)
The Zeta and Related Functions 191
and rearranging the double series in (108) in powers of t, the coefficients are easily
expressible in terms of Gln(2πx), defined by (85). Thus, employing 1.7(1) on the left-
hand side of (108) and equating the coefficients of the same powers on both sides, we
find that
Gln(2πx)= (−1)
1+
[
1
2 n
]
2n−1πn
Bn(x)
n!
(0 5 x 5 1; n ∈ N \ {1}), (109)
which gives us the following special cases:
Gl2(θ)=
1
4
(π − θ)2−
π2
12
,
Gl3(θ)=
1
12
θ(π − θ)(2π − θ),
Gl4(θ)=
1
90
π4−
1
12
θ2
(
π2−πθ +
1
4
θ2
)
,
Gl5(θ)=
1
720
θ(π − θ)(2π − θ)
(
4π2+ 6πθ − 3θ2
)
.
(110)
The Log-Sine Integrals
A recurrence relationship for Lsn(π), defined by (82), is given by
(−1)m
m!
Lsm+2(π)= π
(
1− 2−m
)
ζ(m+ 1)
+
m−1∑
k=2
(−1)k+1
k!
(
1− 2k−m
)
ζ(m− k+ 1)Lsk+1(π) (m ∈ N).
(111)
To derive the recurrence relation (111), we let
I :=
π∫
0
exp
[
x log
(
2sin
1
2
θ
)]
dθ
=
∞∑
n=0
π∫
0
xn
n!
[
log
(
2sin
1
2
θ
)]n
dθ
=−
∞∑
n=0
xn
n!
Lsn+1(π).
(112)
192 Zeta and q-Zeta Functions and Associated Series and Integrals
Alternatively, in view of 1.1(44), we have
I =
π∫
0
[
exp
{
log
(
2sin
1
2
θ
)}]x
dθ
=
π∫
0
2x sinx
(
1
2
θ
)
dθ
= 2x
√
π
0
(
1
2 +
1
2 x
)
0
(
1+ 12 x
) .
By the Legendre duplication formula 1.1(29) for 0, we, thus, obtain
I = π y, (113)
where, for convenience,
y :=
0(1+ x)[
0
(
1+ 12 x
)]2 .
Making use of the following notations:
Dn f (x)=
dn
dxn
f (x)= f (n)(x) and Dn0 f (x)=
dn
dxn
f (x)
∣∣∣∣
x=0
= f (n)(0),
differentiating (112) n times and setting x= 0, we find that
Lsn+1(π)=−D
n
0 I =−πD
n
0 y, (114)
which, by virtue of 1.3(53), yields
Dn0 logy= (−1)
n (n− 1)!
(
1− 21−n
)
ζ(n) (n ∈ N \ {1}), (115)
so that we have
logy=: f (x)=
∞∑
n=2
xn
n
(−1)n
(
1− 21−n
)
ζ(n), (116)
since
log y|x=0 = 0= D
1
0 log y.
The Zeta and Related Functions 193
From (114), we get Lsn+1(π)=−πDn0 exp(f (x)). Let ym := D
m e f . Then,
ym+1 = D
m+1 e f = Dm
(
De f
)
= Dm
(
f ′ e f
)
.
Hence, using Leibniz’s rule for differentiation, we have
ym+1 =
m∑
k=0
(
m
k
)
f (m−k+1) yk, (117)
which, upon considering
ym|x=0 =−
1
π
Lsm+1(π)
in (114), yields
Lsm+2(π)=−π f
(m+1)
0 +
m∑
k=1
(
m
k
)
f (m−k+1)0 Lsk+1(π). (118)
Now, setting
Ls2(π)= 0, f
′
0 = 0 and f
(m)
0 = (−1)
m (m− 1)!(
1− 21−m
)
ζ(m) (m ∈ N \ {1})
in (118), we immediately arrive at the desired recurrence relation (111).
Some simple consequencesof (111) are presented below:
Ls2(π)=−
π∫
0
log
(
2 sin
1
2
θ
)
dθ = 0,
Ls3(π)=−
π∫
0
[
log
(
2 sin
1
2
θ
)]2
dθ =−
1
12
π3,
Ls4(π)=−
π∫
0
[
log
(
2 sin
1
2
θ
)]3
dθ =
3
2
π ζ(3),
Ls5(π)=−
π∫
0
[
log
(
2 sin
1
2
θ
)]4
dθ =−
19
240
π5,
194 Zeta and q-Zeta Functions and Associated Series and Integrals
Ls6(π)=−
π∫
0
[
log
(
2 sin
1
2
θ
)]5
dθ =
45
2
π ζ(5)+
5
4
π3 ζ(3),
Ls7(π)=−
π∫
0
[
log
(
2 sin
1
2
θ
)]6
dθ =−
45
2
π {ζ(3)}2−
275
1344
π7,
Ls8(π)=−
π∫
0
[
log
(
2 sin
1
2
θ
)]7
dθ=
2835
4
π ζ(7)+
315
8
π3 ζ(5)+
133
20
π5 ζ(3),
Ls9(π)=−
π∫
0
[
log
(
2 sin
1
2
θ
)]8
dθ
=−
24177
26880
π9− 1890π ζ(3)ζ(5)−
105
2
π3 {ζ(3)}2.
2.5 Hurwitz–Lerch Zeta Functions
The Hurwitz–Lerch Zeta function 8(z,s,a) is defined by
8(z,s,a) :=
∞∑
n=0
zn
(n+ a)s(
a ∈ C \Z−0 ; s ∈ C when |z|< 1; <(s) > 1 when |z| = 1
)
,
(1)
which satisfies the obvious functional relation:
8(z,s,a)= zn8(z,s,n+ a)+
n−1∑
k=0
zk
(k+ a)s
(
n ∈ N; a ∈ C \Z−0
)
. (2)
By writing the Eulerian integral 1.1(1) in the form:
0(z)= sz
∞∫
0
e−st tz−1 dt (<(z) > 0; <(s) > 0), (3)
we can deduce the following integral representation from (1):
8(z,s,a)=
1
0(s)
∞∫
0
t s−1 e−at
1− ze−t
dt =
1
0(s)
∞∫
0
t s−1 e−(a−1)t
et− z
dt
(<(a) > 0; |z|5 1, z 6= 1, <(s) > 0; z= 1, <(s) > 1),
(4)
The Zeta and Related Functions 195
by noting that
zn
(n+ a)s
=
1
0(s)
∞∫
0
e−at t s−1
(
ze−t
)n
dt (<(a) > 0; <(s) > 0). (5)
If use is made of the infinite-series version of Plana’s summation formula 2.2(9)
(cf. Lindelöf [769, p. 61]; see also Erdélyi et al. [421, p. 22]) and the definition (1),
another definite integral representation of 8(z,s,a) is obtained in the form:
8(z,s,a)=
1
2as
+
∞∫
0
zt
(t+ a)s
dt
− 2
∞∫
0
sin
[
t log z− s tan−1
(
t
a
)](
t2+ a2
)− 12 s dt
e2π t− 1
(<(a) > 0),
(6)
which, for z= 1, immediately reduces to Hermite’s formula 2.2(12) for ζ(s,a).
By setting z= eiθ in (4) and using (3), we get Lipschitz’s formula:
20(s)
∞∑
n=1
einθ
(n+ a)s
=
∞∫
0
e−at t s−1
eiθ − e−t
cosh t− cosθ
dt
(0< θ < 2π; <(a) >−1; <(s) > 0).
(7)
Several contour integral and series representations of8(z,s,a) include (see Erdélyi
et al. [421, pp. 28–31]):
8(z,s,a)=−
0(1− s)
2π i
∫
C
(−t)s−1 e−at
1− ze−t
dt
(<(a) > 0; |arg(−t)|5 π),
(8)
where the contour C is the Hankel loop of Theorem 2.5, which, obviously, does not
enclose any pole of the integrand, that is, t = logz± 2nπ i (n ∈ N0);
8(z,s,a)=
0(1− s)
za
∞∑
n=−∞
(− logz+ 2nπ i)s−1 e2nπ ia
(0< a 5 1; <(s) < 0; |arg(− logz+ 2nπ i)|5 π);
(9)
Lerch’s transformation formula:
8(z,s,a)=
i(2π)s−10(1− s)
za
{
e−
1
2 iπs L
(
−a,1− s,
logz
2π i
)
−e
iπ
(
1
2 s+2a
)
L
(
a,1− s,1−
logz
2π i
)}
,
(10)
196 Zeta and q-Zeta Functions and Associated Series and Integrals
where the so-called Lipschitz-Lerch transcendent L(ξ,s,a) is defined by
φ(ξ,a,s) :=
∞∑
n=0
e2nπ iξ
(n+ a)s
=8
(
e2π iξ ,s,a
)
=: L (ξ,s,a)(
a ∈ C \Z−0 ; R(s) > 0 when ξ ∈ R \Z; R(s) > 1 when ξ ∈ Z
)
,
(11)
which was first studied by Rudolf Lipschitz (1832–1903) and Matyáš Lerch (1860–
1922) in connection with Dirichlet’s famous theorem on primes in arithmetic progres-
sions.
Replacing s by 1− s in 2.2(6), we obtain another equivalent form of Hurwitz’s
formula 2.2(6):
ζ(s,a)= 2(2π)s−10(1− s)
∞∑
n=1
ns−1 sin
(
2nπa+
πs
2
)
(<(s) < 0; 0< a 5 1).
(12)
By applying the binomial theorem to (− logz± 2nπ i)s−1 in (9) and making use
of (12), 1.1(12) and 1.1(21), we readily obtain the following formulas (cf. Erdélyi
et al. [421, p. 29, Eq. (8)]):
8(z,s,a)=
0(1− s)
za
(
log
1
z
)s−1
+
1
za
∞∑
k=0
ζ(s− k,a)
(logz)k
k!
(|logz|< 2π; s 6∈ N; a 6∈ Z−0 );
(13)
8(z,m,a)
=
1
za

∞∑
k=0
k 6=m−1
ζ(m− k,a)
(logz)k
k!
+
(logz)m−1
(m− 1)!
[
ψ(m)−ψ(a)− log
(
log
1
z
)]
(m ∈ N \ {1}; |logz|< 2π; a 6∈ Z−0 ).
(14)
By setting s=−m (m ∈ N) in (13) and using 2.1(17),8 can also be expressed in terms
of the Bernoulli polynomials as follows:
8(z,−m,a)=
m!
za
(
log
1
z
)−m−1
−
1
za
∞∑
k=0
Bm+k+1(a)(logz)k
k!(m+ k+ 1)
(|logz|< 2π).
(15)
The Zeta and Related Functions 197
The special case of (1) when s= 1 yields
8(z,1,a)=
∞∑
n=0
zn
n+ a
= a−1 2F1(1, a ; 1+ a ; z) (|z|< 1), (16)
and, more generally, we have
8(z,k,a)= a−k k+1Fk(1, a, . . . , a ; a+ 1, . . . ,a+ 1 ; z) (k ∈ N0), (17)
where pFq denotes the generalized hypergeometric function (see Section 1.5).
It follows from (13) and (16) that
lim
z→1
{
(1− z)1−s8(z,s,a)
}
= 0(1− s) (<(s) < 1) (18)
and
lim
z→1
{
8(z,1,1)
log(1− z)
}
=−1. (19)
The Jonquière function F(z,s) is defined by
F(z,s) :=
∞∑
k=1
zk
ks
= z8(z,s,1), (20)
whose many properties can be deduced from those of the 8-function. For example,
Lerch’s transformation formula (10) readily yields Jonquière’s formula:
F(z,s)+ eisπ F
(
1
z
,s
)
=
(2π)s
0(s)
e
1
2 iπs ζ
(
1− s,
logz
2π i
)
. (21)
Moreover, we obtain
F(z,−m)= (−1)m+1 F
(
1
z
,−m
)
(m ∈ N) (22)
and
F(z,m)+ (−1)m F
(
1
z
,m
)
=−
2π i
m!
Bm
(
logz
2π i
)
(m ∈ N \ {1}), (23)
both of which provide the analytic continuation of F(z,s) in (20) outside its circle of
convergence |z| = 1.
198 Zeta and q-Zeta Functions and Associated Series and Integrals
By taking a= 1 in (15) and using 1.7(5), we get
F(z,−m)= m!
(
log
1
z
)−m−1
−
∞∑
k=0
Bm+k+1
k!(m+ k+ 1)
(logz)k
(m ∈ N; |logz|< 2π).
(24)
In addition to the Jonquière function F(z,s) in (20), various special cases of
8(z,s,a) are listed below:
ζ(s)=8(1,s,1); (25)
ζ(s,a)=8(1,s,a); (26)
Li2(z)= z8(z,2,1); (27)
Lis(z) :=
∞∑
k=1
zk
ks
= z8(z,s,1)= F(z,s)
(s ∈ C and |z|< 1; <(s) > 1 and |z| = 1).
(28)
The Taylor Series Expansion of the Lipschitz-Lerch Transcendent L(x,s,a)
The function L(x,s,a), defined by (11), was first studied by Lipschitz [770, 771] and
Lerch [745] in connection with Dirichlet’s famous theorem on primes in arithmetic
progressions. For x ∈ Z, (11) reduces immediately to the Hurwitz Zeta function ζ(s,a)
(see Section 2.2). Since L(x,s,a) is a special case of8(z,s,a),many properties of this
function can be obtained from those of 8(z,s,a).
The Lerch functional equation for L(x,s,a) can be deduced from (10):
L(x,1− s,a)=
0(s)
(2π)s
{
e
π i
(
1
2 s−2ax
)
L(−a,s,x)+ e
π i
(
−
1
2 s+2a(1−x)
)
L(a,s,1− x)
}
(0< x< 1; 0< a< 1).
(29)
Several interesting proofs of (29) were given by Apostol [58] and Berndt [117].
The function L(x,s,a) satisfies the differential-difference equations:
∂L(x,s,a)
∂a
=−sL(x,s+ 1,a) (30)
and
∂L(x,s,a)
∂x
+ 2π iaL(x,s,a)= 2π iL(x,s− 1,a). (31)
The Zeta and Related Functions 199
Klusch [673] gave a Taylor series expansion of the function L(x,s,a+ ξ) in the
neighborhood of ξ = 0 in the form: For fixed a ∈ R+,
∞∑
k=0
(−1)k
(s)k
k!
L(x,s+ k,a)ξ k = L(x,s,a+ ξ) (|ξ |< a), (32)
which provides the analytic continuation of L(x,s,a) to the whole complex s-plane.
Klusch [673] also applied (32) to derive several summation formulas which will be
investigated systematically in Chapter 3.
By setting ξ =−t in (32), we get
∞∑
k=0
(s)k
k!
L(x,s+ k,a) tk = L(x,s,a− t) (|t|< a), (33)
which, upon replacing k by k+ 2 and s by s− 1, dividing the resulting equation by t2
and differentiating both sides with respect to t yields the following corrected version
of one of Klusch’s results (Klusch [673, p. 517, Eq. (3.3)]):
∞∑
k=1
k
(k+ 2)!
0(s+ k+ 1)
r0(s)
L(x,s+ k+ 1,a) tk−1
= t−2 {L(x,s,a− t)+L(x,s,a)}−
2t−3
rs− 1
{L(x,s− 1,a− t)−L(x,s− 1,a)}
(|t|< a; t 6= 0).
(34)
Evaluation of L(x,−n,a)
Since L(x,s,a)= ζ(s,a) (x ∈ Z), whose properties are studied in Section 2.2, in what
follows, we assume that x is not an integer. By making use of the classical method
presented in Section 2.1, L(x,s,a) can be extended to the whole complex s-plane by
means of the contour integral (see Apostol [58]):
I(x,s,a)=
1
2π i
∫
C1
zs−1 eaz
1− ez+2π ix
dz, (35)
where the contour C1 is a loop which begins at −∞, encircles the origin once in the
positive (counter-clockwise)direction and returns to −∞. Since I(x,s,a) is an entire
function of s and
L(x,s,a)= 0(1− s) I(x,s,a), (36)
we can use (36) to continue L(x,s,a) analytically to the whole complex s-plane as an
entire function of s (x 6∈ Z).
200 Zeta and q-Zeta Functions and Associated Series and Integrals
Apostol [58] computed recursively the values of L(x,−n,a) (n ∈ N0; x 6∈ Z) by
applying (31) and (35). Thus, he obtained (cf. Apostol [58])
L(x,0,a)=
1
1− exp(2π ix)
=
i
2
cot(πx)+
1
2
, (37)
L(x,−1,a)=
a
2
[icot(πx)+ 1]−
1
4
csc2(πx), (38)
L(x,−2,a)=
a2
2
[
icot(πx)+
1
4
]
−
a
2
csc2(πx)−
i
4
cot(πx) csc2(πx) (39)
and
L(x,−n,a)=−
Bn+1
(
a;e2π ix
)
n+ 1
(n ∈ N0), (40)
where the Bn(a;α) are defined by 1.8(1).
2.6 Generalizations of the Hurwitz–Lerch Zeta Function
Lin and Srivastava [765] introduced and investigated the following generalization of
the Hurwitz–Lerch Zeta function 8(z,s,a), defined by 2.5(1):
8(ρ,σ )µ,ν (z,s,a) :=
∞∑
n=0
(µ)ρ n
(ν)σ n
zn
(n+ a)s(
µ ∈ C; a,ν ∈ C \Z−0 ; ρ,σ ∈ R
+
; ρ < σ when s,z ∈ C;
ρ = σ and s ∈ C when |z|< δ := ρ−ρ σ σ ; ρ = σ and <(s−µ+ ν) > 1
when |z| = δ
)
,
(1)
where (λ)n denotes the Pochhammer symbol, defined by 1.1(5).
Clearly, we have
8(σ,σ )ν,ν (z,s,a)=8
(0,0)
µ,ν (z,s,a)=8(z,s,a) (2)
and
8
(1,1)
µ,1 (z,s,a)=8
∗
µ(z,s,a) :=
∞∑
n=0
(µ)n
n!
zn
(n+ a)s
, (3)
where, as also noted by Lin and Srivastava [765],8∗µ(z,s,a) is a generalization of the
Hurwitz–Lerch Zeta function considered by Goyal and Laddha [504, p. 100, Eq. (1.5)].
The Zeta and Related Functions 201
For further results involving these classes of generalized Hurwitz–Lerch Zeta func-
tions, see the recent works by Garg et al. [467] and Lin et al. [767]. Recently, among
other things, Choi et al. [273] presented and investigated the following multiple-series
generalization of the Hurwitz–Lerch Zeta function8(z,s,a), by calling it the multiple
(or, simply, n-tuple) Hurwitz–Lerch Zeta function 8n(z,s,a), defined by
8n(z,s,a) :=
∞∑
m1,...,mn=0
zm1+···+mn
(m1+ ·· ·+mn+ a)s(
a ∈ C \Z−0 ; s ∈ C when |z|< 1; <(s) > n when |z| = 1
)
.
(4)
In its special case when z= 1, the definition (4) yields the following familiar mul-
tiple (or, simply, n-tuple) Hurwitz Zeta function ζn(s,a) given in 2.1(2):
8n(1,s,a)= ζn(s,a)
(
<(s) > n; a ∈ C \Z−0
)
. (5)
The special case of (4) when n= 1 reduces to the Hurwitz–Lerch Zeta function
8(z,s,a), defined by 2.5(1), as follows:
81(z,s,a)=8(z,s,a). (6)
Furthermore, when n= 1 and z= 1, the definition (4) yields the Hurwitz (or general-
ized) Zeta function ζ(s,a), defined by 2.2(1):
81(1,s,a)= ζ(s,a)
(
a ∈ C \Z−0 ; <(s) > 1
)
. (7)
See also a recent investigation by Srivastava and Attiya [1092], using the Hurwitz–
Lerch Zeta function in Geometric Function Theory.
We recall several interesting properties of the multiple Hurwitz–Lerch Zeta func-
tion 8n(z,s,a) in (4) investigated by Choi et al. [267, 273].
First of all, the multiple Hurwitz–Lerch Zeta function 8n(z,s,a) satisfies the fol-
lowing functional relation:
8n(z,s,a)= z
`8n(z,s,`+ a)+
∑
m1,...,mn=0
m1+···+mn5`−1
zm1+···+mn
(m1+ ·· ·+mn+ a)s
(` ∈ N; a ∈ C \Z−0 ),
(8)
which, for n= 1, yields a known functional relation for 8(z,s,a) given by 2.5(2).
202 Zeta and q-Zeta Functions and Associated Series and Integrals
It is easy to derive the following summation formula:
∞∑
k=0
(s)k
k!
8n(z,s+ k,a)t
k
=8n(z,s,a− t) (|t|< |a|; s 6= 1), (9)
where (λ)m denotes the Pochhammer symbol defined by 1.1(5). Indeed, we see that
8n(z,s,a− t)=
∑
m1,...,mn=0
zm1+···+mn
(m1+ ·· ·+mn+ a− t)s
=
∑
m1,...,mn=0
zm1+···+mn
(m1+ ·· ·+mn+ a)s
(
1−
t
m1+ ·· ·+mn+ a
)−s
=
∑
m1,...,mn=0
zm1+···+mn
(m1+ ·· ·+mn+ a)s
∞∑
k=0
(s)k
k!
(
t
m1+ ·· ·+mn+ a
)k
(|t|< |m1+ ·· ·+mn+ a|; mj ∈ N0; j= 1, . . . ,n)
=
∞∑
k=0
(s)k
k!
 ∑
m1,...,mn=0
zm1+···+mn
(m1+ ·· ·+mn+ a)s+k
 tk
=
∞∑
k=0
(s)k
k!
8n(z,s+ k,a)t
k (|t|< |a|; s 6= 1),
which proves (9). The interchange of the order of summation can be justified by the
absolute convergence of the series involved.
Now, from the Eulerian integral form of the Gamma function 0 (see Section 1.1),
we have
0(s)= us
∞∫
0
e−ut t s−1 dt
(
<(s) > 0; <(u) > 0
)
. (10)
By setting
u= m1+ ·· ·+mn+ a
(
(m1, . . . ,mn) ∈ N0n
)
,
multiplying each side of the resulting equation by zm1+···+mn and summing up for each
mj from 0 to∞, ( j= 1, . . . ,n), we can get the following integral representation for
The Zeta and Related Functions 203
8n(z,s,a):
8n(z,s,a)=
1
0(s)
∞∫
0
t s−1 e−at(
1− ze−t
)n dt = 1
0(s)
∞∫
0
t s−1 e−(a−n)t
(et− z)n
dt
(
n ∈ N; <(a) > 0; |z|5 1 (z 6= 1) and <(s) > 0; z= 1 and <(s) > 1
)
.
(11)
In its special case when n= 1, (11) yields a known integral representation for the
Hurwitz–Lerch Zeta function 8(z,s,a) (see 2.5(4)).
We can get a particular infinite-series version of Plana’s summation formula 2.2(9)
in the form:
∞∑
k=0
f (k)=
1
2
f (0)+
∞∫
0
f (τ )dτ − 2
∞∫
0
q(0, t)
e2π t− 1
dt, (12)
where f (z) is a bounded analytic function in 0 5 <(z) <∞, the integral
∞∫
0
f (τ )dτ
is assumed to be convergent, and
q(n, t)→ 0 as n→∞ or t→∞.
If we apply (12) to the function
f (u) :=
zu+m1+···+mn−1
(u+m1+ ·· ·+mn+ a)s
,
then we get
∞∑
k=0
zk+m1+···+mn−1
(k+m1+ ·· ·+mn−1+ a)s
=
zm1+···+mn−1
2(m1+ ·· ·+mn−1+ a)s
+
∞∫
0
zτ+m1+···+mn−1
(τ +m1+ ·· ·+mn−1+ a)s
dτ
− 2
∞∫
0
zm1+···+mn−1
[
(m1+ ·· ·+mn−1+ a)
2
+ t2
]− s2
·sin
[
t log z− s arctan
(
t
m1+ ·· ·+mn−1+ a
)]
dt
e2π t− 1
.
204 Zeta and q-Zeta Functions and Associated Series and Integrals
Now, summing up for each mj from 0 to∞, ( j= 1, . . . ,n− 1), we finally obtain the
following integral representation for 8n(z,s,a):
8n(z,s,a)=
1
2
8n−1(z,s,a)+
∞∫
0
zt8n−1(z,s,a+ t)dt
− 2
∞∑
m1,...,mn−1=0
zm1+···+mn−1
∞∫
0
[
(m1+ ·· ·+mn−1+ a)
2
+ t2
]−s2
·sin
[
t log z− s arctan
(
t
m1+ ·· ·+mn−1+ a
)]
dt
e2π t− 1
(n ∈ N),
(13)
provided that the involved integrals and series converge.
The special case of (13) when n= 1 yields the known integral representation for
8(z,s,a) (see 2.5(6)).
It is not difficult to deduce the following contour integral representation of
8n(z,s,a), by applying a procedure that is used to get a contour integral represen-
tation of the multiple Hurwitz Zeta function ζn(s,a) (see, for details, Section 2.1):
8n(z,s,a)=−
0(1− s)
2π i
∫
C
(−t)s−1 e−at(
1− ze−t
)n dt
(
n ∈ N; <(a) > 0; |arg(−t)|5 π
)
,
(14)
where the contour C is the Hankel loop given as in 2.1(7).
The special cases of (14) when z= 1 and n= 1 would immediately yield the known
integral representations of the multiple Hurwitz Zeta function ζn(s,a) in 2.1(8) and the
Hurwitz–Lerch Zeta function 8(z,s,a) in 2.5(8), respectively.
The special case of 8n(z,s,a) in (14) when
s=−` (` ∈ N0)
can be expressed in terms of the Apostol-Bernoulli polynomials B(α)k (a;λ) of order α,
defined by means of the following generating functions (see Section 1.8):
8n(z,−`,a)= (−1)
n `!
(n+ `)!
B(n)n+` (a;z) (` ∈ N0). (15)
Indeed, upon setting
s=−` (` ∈ N0)
The Zeta and Related Functions 205
in (14), we find, by using 1.8(13), that
8n(z,−`,a)=−
0(1+ `)
2π i
∫
C
(−t)−`−1 e−at(
1− ze−t
)n dt
= (−1)` `! Res
t=0
t−n−`−1
(
t
1− ze−t
)n
e−at
= (−1)` `! Res
t=0
t−n−`−1
∞∑
k=0
B(n)k (a;z)
(−1)k tk
k!
= (−1)n
`!
(n+ `)!
B(n)n+` (a;z).
The special case of (15) when z= 1, together with 1.8(14), would reduce immediately
to the known formula (see 2.1(16)).
The multiple Hurwitz–Lerch Zeta function 8n(z,s,a) can, thus, be expressed as
follows, as a simple series (see 2.1(19)):
8n(z,s,a)=
∞∑
m=0
(
m+ n− 1
n− 1
)
zm
(m+ a)s(
a ∈ C \Z−0 ; s ∈ C when |z|< 1; <(s) > n when |z| = 1
)
.
(16)
We find from (16), as in getting 2.1(21), that 8n(z,s,a) is expressible as a finite com-
bination of the Hurwitz–Lerch Zeta function8(z,s,a) with polynomial coefficients in
a as follows:
8n(z,s,a)=
n−1∑
j=0
pn,j(a)8(z,s− j,a), (17)
where
pn,j(a)=
1
(n− 1)!
n−1∑
`=j
(−1)n+1−j
(
`
j
)
s(n,`+ 1)a`−j. (18)
We now find pn,j(a) in(18) as a polynomial in a of degree n− 1− j with rational
coefficients.
Since
∞∑
`=0
f (`)=
k−1∑
j=0
∞∑
`=0
f (k`+ j) (k ∈ N), (19)
206 Zeta and q-Zeta Functions and Associated Series and Integrals
we get
8n(z,s,a)= k
−s
k−1∑
j=0
8n
(
zk,s,
a+ j
k
)
zj (k ∈ N). (20)
Thus, by combining (11) and (20), we immediately obtain the following sum-integral
representation:
8n(z,s,a)=
k−1∑
j=0
zj
0(s)
∞∫
0
t s−1 e−(a+j)t(
1− zk e−kt
)n dt
(
k, n∈N; <(a)>0; <(s)>0 when |z|51 (z 6=1); <(s)>1 when z=1
)
,
(21)
which is a straightforward generalization of the following sum-integral representation
given by Lin and Srivastava [765, p. 727, Eq. (7)]:
8(z,s,a)=
k−1∑
j=0
zj
0(s)
∞∫
0
t s−1 e−(a+j)t(
1− zk e−kt
) dt
(
k∈N; <(a)>0; <(s)>0 when |z|51 (z 6=1); <(s)>1 when z=1
)
.
(22)
For further special cases of (21) or (22), just as noted also by Lin and Srivastava
[765, p. 726], see Yen et al. [1244, p. 100, Theorem] and Nishimoto et al. [864, p. 94,
Theorem 4].
In view of (17), we can deduce some properties of the multiple Hurwitz–Lerch Zeta
function 8n(z,s,a) from those (see, e.g., [447]) of the Hurwitz–Lerch Zeta function
8(z,s,a). Combining the integral formula 2.5(6) for 8(z,s,a) with (17), we obtain
another sum-integral representation for 8n(z,s,a):
8n(z,s,a)
=
(−1)n+1
2as (n− 1)!
n−1∑
j=0
(−1) j
n−1∑
`=j
(
`
j
)
s(n,`+ 1)a`+
n−1∑
j=0
pn,j(a)
∞∫
0
zt
(t+ a)s−j
dt
− 2
n−1∑
j=0
pn, j(a)
∞∫
0
sin
[
t logz+ ( j− s) tan−1
(
t
a
)](
t2+ a2
) 1
2 ( j−s) dt
e2π t− 1
(
n ∈ N; <(a) > 0; s ∈ C
)
.
(23)
The Zeta and Related Functions 207
By applying Apostol’s formula 2.5(40) (see [58, p. 164]; see also [467, p. 806,
Eq. (19)]):
φ(ξ,a,1− `)=8
(
e2π iξ,1− `,a
)
=−
B`
(
a;e2π iξ
)
`
(` ∈ N; ξ ∈ R) (24)
to (17), we obtain
8n
(
e2π iξ,1− `,a
)
=−
n−1∑
j=0
pn,j(a)
B`+j
(
a;e2π iξ
)
`+ j
(n, ` ∈ N). (25)
The following known summation formula for 8(z,s,a), expressed in terms of the
Hurwitz (or generalized) Zeta function ζ(s,a) (see [788, p. 298, Eq. (38)]):
8
(
e2π i
p
q,s,a
)
= q−s
q∑
`=1
ζ
(
s,
a+ `− 1
q
)
exp
(
2(`− 1)pπ i
q
)
(p ∈ Z; q ∈ N),
(26)
when applied to (17), would readily yield
8n
(
e2π i
p
q,s,a
)
= q−s
n−1∑
j=0
pn,j(a)q
j
q∑
`=1
ζ
(
s− j,
a+ `− 1
q
)
exp
(
2(`− 1)pπ i
q
)
(n, q ∈ N; p ∈ Z).
(27)
Lin and Srivastava [765, p. 729, Eq. (17)] presented the following integral repre-
sentation for 8∗µ(z,s,a):
8∗µ(z,s,a)=
1
0(s)
∞∫
0
t s−1 e−at(
1− ze−t
)µ dt (28)
(
<(a) > 0; <(s) > 0 when |z|5 1 (z 6= 1); <(s) > 1 when z= 1
)
,
which was given earlier by Goyal and Laddha [504, p. 100, Eq. (1.6)], together with
the seemingly unnecessary constraint µ= 1 (see, for details, [765, p. 729]). Here, in
view of (11) and (28), it is seen that
8n(z,s,a)=8
∗
n(z,s,a) (n ∈ N). (29)
208 Zeta and q-Zeta Functions and Associated Series and Integrals
Lin and Srivastava [765] also recalled the Riemann-Liouville fractional derivative
operator Dµz , defined by (cf. [422, p. 181 et seq.]; see also [896] and [648])
Dµz { f (z)} :=

1
0(−µ)
z∫
0
(z− t)−µ−1 f (t)dt
(
R(µ) < 0
)
dm
dzm
Dµ−mz { f (z)}
(
m− 1 5 R(µ) < m (m ∈ N)
)
,
(30)
to, first, notice that8∗µ(z,s,a) is essentially a Riemann-Liouville fractional derivative
of the classical Hurwitz–Lerch function 8(z,s,a) of order µ− 1 as follows:
8∗µ (z,s,a)=
1
0(µ)
Dµ−1z
{
zµ−1 8(z,s,a)
} (
R(µ) > 0
)
. (31)
By using their remarkable observation (31), together with the Eulerian integrals of
the first and second kind, Hankel’s integral representation for the Gamma function
[421, p. 13, Eq. 1.6(2)] and the theory of the Mellin-Barnes contour integration (cf.
[421, p. 49]), Lin and Srivastava [765] presented various integral representations for
8∗µ(z,s,a). In view of (29), as well as for the sake of completeness, we recall the
known integral representations for 8∗µ(z,s,a) given by Lin and Srivastava (see [765,
Eqs. (27), (28), (34), and (37)]) in their following special forms for 8n(z,s,a):
8n(z,s,a)=
1
(n− 1)!
∞∫
0
tn−1 e−t8(0,1)n,1 (zt,s,a)dt (n ∈ N), (32)
8n(z,s,a)=
1
(n− 1)!
1∫
0
(
log
1
t
)n−1
8
(0,1)
n,1
(
z log
1
t
,s,a
)
dt (n ∈ N), (33)
8n(z,s,a)=
1
2π i
(+0)∫
−∞
w−1 ew8(0,1)n,1
(
zw−1,s,a
)
dw (n ∈ N; |arg(w)|5 π),
(34)
where the contour of integration in the complex w-plane is Hankel’s loop (cf., e.g.,
[1225, p. 245]), which starts from the point at−∞ along the lower side of the negative
real axis, encircles the origin once in the positive (counter-clockwise) direction and
then returns to the point at −∞ along the upper side of the negative real axis.
8n(z,s,a)=
1
2π i
τ+i∞∫
τ−i∞
w−1 ew 8(1,0)n,1
(
zw−1,s,a
)
dw
(
n ∈ N; τ ∈ R+
)
, (35)
where the contour of integration is a Mellin-Barnes contour (cf. [421, p. 49]).
The Zeta and Related Functions 209
Garg et al. [467, p. 809, Theorem 2; p. 811, Theorem 3] gave transformation for-
mulas for 8∗n (z,s,a) and Lin et al. [767, p. 825, Eqs. (39) and (40)] presented several
expansion and transformation formulas for the general case 8∗µ (z,s,a). By means of
(29), we choose to rewrite these known results in their special cases applicable to
8n (z,s,a) as follows:
8n (z,s,a)=
i z−a 0(1− s)
(n− 1)!
n−1∑
k=0
(
n− 1
k
)
B(n)n−k−1
·
k∑
j=0
(−1)n−k+j−1
(
k
j
)(
s− 1
j
)
j! (−a)k−j (2π)s−j−1
·
[
exp
(
−
1
2
(s− j)π i
)
8
(
e−2πai,1− s+ j,
logz
2π i
)
−exp
[(
2a+
1
2
(s− j)
)
π i
]
8
(
e2πai,1− s+ j,1−
logz
2π i
)]
(n ∈ N),
(36)
provided that each side of (36) exists;
8n (z,s,a)= i z
−a 0(1−s)
n−1∑
j=0
( j−a+1)n−j−1
j!(n− j− 1)!
j∑
k=0
(
j− 1
k− 1
)
(1−s)kB
( j)
j−k(2π)
s−k−1
·
[
exp
(
−
1
2
(s− k)π i
)
8
(
e−2πai,1− s+ k,
logz
2π i
)
−exp
[(
2a+
1
2
(s− k)
)
π i
]
8
(
e2πai,1− s+ k,1−
logz
2π i
)]
(n ∈ N),
(37)
provided that each side of (37) exists;
8n (z,s,a)= i z
−a 0(1− s)
n−1∑
j=0
( j− a+ 1)n−j−1
j! (n− j− 1)!
·
j∑
k=0
(
j− 1
k− 1
)
(1− s)k B
( j)
j−k·(2π)
s−k−1
·
[
exp
(
−
1
2
(s− k)π i
)
8
(
e−2πai,1− s+ k,
logz
2π i
)
−exp
[(
2a+
1
2
(s− k)
)
π i
]
8
(
e2πai,1− s+ k,1−
logz
2π i
)]
(n ∈ N),
(38)
210 Zeta and q-Zeta Functions and Associated Series and Integrals
and
8n
(
e2π iξ,n− s,
p
q
)
=
i 0(1− n+ s)
(n− 1)!
n−1∑
j=0
(
n− 1
j
)(
j−
p
q
+ 1
)
n−j−1
·
j∑
k=0
(
j− 1
k− 1
)
(1− n+ s)k B
( j)
j−k·(2πq)
n−s−k−1
·
[ q∑
r=1
ζ
(
1−n+ s+ k,
ξ+r−1
q
)
exp
[
−
(
1
2
(n−s−k)+
2(ξ + r−1)p
q
)
π i
]
−
q∑
r=1
ζ
(
1− n+ s+ k,
r− ξ
q
)
exp
[(
1
2
(n− s− k)+
2(r− ξ)p
q
)
π i
]]
(p ∈ Z; q, n ∈ N).
(39)
Garg et al. [467, p. 813, Eqs. (51) and (52)] applied the definition 2.5(11) and the
summation formula (26) to (36) and (37), by setting
z= e2π iξ and a=
p
q
(p ∈ Z; q ∈ N; ξ ∈ R),
and, thereby, expressed the function:
8∗n
(
e2π iξ,n− s,
p
q
)
in terms of the Hurwitz Zeta function ζ(s,a) and the generalized Apostol-Bernoulli
polynomials B( j)k (x;λ). We rewrite these known results of Garg et al. [467, Eqs. (51)
and (52)] in their following special forms for 8n
(
e2π iξ,n− s, pq
)
):
8n
(
e2π iξ,n− s,
p
q
)
=
i (−1)n−1 0(1− n+ s)
(n− 1)!
n−1∑
k=0
(
n− 1
k
)
B(n)n−k−1 q
n−s−k−1
·
k∑
j=0
(
k
j
)(
n− s− 1
j
)
j! pk−j (2π)n−s−j−1
·
[ q∑
r=1
ζ
(
1−n+ s+ j,
ξ + r− 1
q
)
exp
[
−
(
1
2
(n−s−j)+
2(ξ + r− 1)p
q
)
π i
]
−
q∑
r=1
ζ
(
1− n+ s+ j,
r− ξ
q
)
exp
[(
1
2
(n− s− j)+
2(r− ξ)p
q
)
π i
]]
(n ∈ N)
(40)
The Zeta and Related Functions 211
and
8n
(
e2π iξ,n− s,
p
q
)
=
i 0(1− n+ s)
(n− 1)!
n−1∑
j=0
(
n− 1
j
)(
j−
p
q
+ 1
)
n−j−1
·
j∑
k=0
(
j− 1
k− 1
)
(1− n+ s)k B
( j)
j−k·(2πq)
n−s−k−1
·
[ q∑
r=1
ζ
(
1−n+ s+ k,
ξ + r− 1
q
)
exp
[
−
(
1
2
(n−s−k)+
2(ξ+r−1)p
q
)
π i
]
−
q∑
r=1
ζ
(
1− n+ s+ k,
r− ξ
q
)
exp
[(
1
2
(n− s− k)+
2(r− ξ)p
q
)
π i
]]
(n ∈ N).
(41)
In a recent paper, Srivastava et al. [1107] introduced and investigated further
generalizations of the above-defined Lin-Srivastava model 8(ρ,σ )µ,ν (z,s,a), first, in the
followingform:
8
(ρ,σ,κ)
λ,µ;ν (z,s,a) :=
∞∑
n=0
(λ)ρn(µ)σn
(ν)κn · n!
zn
(n+ a)s(
λ,µ ∈ C; a,ν ∈ C \Z−0 ; ρ,σ,κ ∈ R
+
; κ − ρ− σ >−1 when s,z ∈ C;
κ − ρ− σ =−1 and s ∈ C when |z|< δ∗ := ρ−ρ σ−σ κκ ;
κ − ρ− σ =−1 and <(s+ ν− λ−µ) > 1 when |z| = δ∗
)
(42)
and, then, in a substantially more general form given by
8
(ρ1,...,ρp,σ1,...,σq)
λ1,...,λp;µ1,...,µq
(z,s,a)=
∞∑
n=0
p∏
j=1
(λj)nρj
n!
q∏
j=1
(µj)nσj
zn
(n+ a)s
(
p,q ∈ N0; λj ∈ C ( j= 1, . . . ,p); a,µj ∈ C \Z−0 ( j= 1, . . . ,q);
ρj,σk ∈ R+ ( j= 1, . . . ,p; k = 1, . . . ,q); 1>−1 when s,z ∈ C;
1=−1 and s∈C when |z|<∇; 1=−1 and <(4)>
1
2
when |z| = ∇
)
,
(43)
212 Zeta and q-Zeta Functions and Associated Series and Integrals
where, for convenience,
∇ :=
 p∏
j=1
ρ
−ρj
j
 q∏
j=1
σ
σj
j
,
1 :=
q∑
j=1
σj−
p∑
j=1
ρj
and
4 := s+
q∑
j=1
µj−
p∑
j=1
λj+
p− q
2
.
The special case of the definition (43) when p− 1= q= 1 would obviously cor-
respond to the above-investigated (Lin-Srivastava) generalized Hurwitz–Lerch Zeta
function 8(ρ,σ,κ)λ,µ;ν (z,s,a), defined by (42).
The following further special case of the generalized Hurwitz–Lerch Zeta function
8
(ρ,σ,κ)
λ,µ;ν (z,s,a) (which corresponds to the definition (43) with p− 2= q= 1) when
ρ = σ = κ = 1:
8λ,µ;ν(z,s,a) :=
∞∑
n=0
(λ)n(µ)n
(ν)n · n!
zn
(n+ a)s
=:8(1,1,1)λ,µ;ν (z,s,a) (44)(
λ,µ ∈ C; ν,a ∈ C \Z−0 ; s ∈ C when |z|< 1; <(s+ ν− λ−µ) > 1
when |z| = 1
)
was investigated earlier by Garg et al. [466, p. 313, Eq. (1.7)].
For various interesting properties and results involving the families of generalized
Hurwitz–Lerch Zeta functions:
8
(ρ,σ,κ)
λ,µ;ν (z,s,a) and 8
(ρ1,...,ρp,σ1,...,σq)
λ1,...,λp;µ1,...,µq
(z,s,a),
which are defined by (42) and (43), respectively, as well as for the numerous spe-
cial cases and consequences of each of these general models, we choose to refer the
interested reader to the above-mentioned paper by Srivastava et al. [1107], as well as
to a subsequent investigation by Srivastava et al. [1106] in which several two-sided
bounding inequalities for the extended Hurwitz–Lerch Zeta function
8
(ρ1,...,ρp,σ1,...,σq)
λ1,...,λp;µ1,...,µq
(z,s,a)
were given (see also Problems 51, 52 and 53 at the end of this chapter).
The Zeta and Related Functions 213
2.7 Analytic Continuations of Multiple Zeta Functions
The multiple Hurwitz zeta function ζn(s,a) (n ∈ N), defined by 2.1(2), is obviously
a generalization of the Riemann zeta function ζ(s) given in 2.3(1). Another very
useful and widely-investigated generalization of ζ(s) is the multiple zeta functions
ζd (s1, . . . ,sd) of depth d (d ∈ N), defined by
ζd (s1, . . . ,sd) :=
∑
0<n1<n2<···<nd
1
n1s1 n2s2 · · ·ndsd
(1)
(
(s1, . . . ,sd) ∈ Cd; <(sd) > 1;
d∑
j=1
<
(
sj
)
> d
)
,
which is often referred to as Euler-Riemann-Zagier zeta functions of depth d.
If (s1, . . . ,sd) ∈ Nd, ζd (s1, . . . ,sd) is also called the multiple zeta values (or Euler-
Zagier sums) (see [1246]). Since these values have turned out to be connected with
a variety of research subjects, such as topology, theoretical physics and the theory of
mixed Tate motives, an extensive research has been made of those values (see, e.g.,
[151], [182], [734], [1245], [1246], [1247]). Ohno [872] presented a unified algebraic
relationship among the multiple zeta values that generalizes the so-called sum formula
(or sum conjecture) (see [560]) and another remarkable identity, referred to as the dual-
ity theorem (see [1246]). Here, we introduce two totally different methods of analytic
continuation of ζd (s1, . . . ,sd), defined by (1), which were given by Zhao [1260] and
Akiyama et al. [16]. An analytic continuation of ζ2(s1,s2)was proven by, for example,
Apostol and Vu [68], Atkinson [78], Motohashi [849] and Katsurada and Matsumoto
[639]. Arakawa and Kaneko [70] gave the analytic continuation of ζd (s1, . . . ,sd) by
considering a function of one variable sd, while s1, . . . ,sd−1 are positive integers.
Generalized Functions of Gel’fand and Shilov
We remark in proceeding this subsection that most of the arguments given here are
due to Zhao [1260].
Theorem 2.7 The infinite sum in (1) converges absolutely when
<(sd) > 1 and
d∑
j=1
<
(
sj
)
> d.
Proof. The case d = 1 is obvious, since ζ (s1) becomes the Riemann zeta function,
defined by 2.3(1), and converges absolutely for <(s1) > 1. So, let us assume d = 2
and write σi =<(si). If σ1 > 1, then, being also σ2 > 1, the result is clear:
|ζ2 (s1,s2)|5
∑
0<n1<n2
1
n1σ1 n2σ2
5
∞∑
n1=1
1
n1σ1
·
∞∑
n2=1
1
n2σ1
<∞.
214 Zeta and q-Zeta Functions and Associated Series and Integrals
If σ1 = 1, then the theorem is proven by using the following well-known asymptotic
formula (see 1.2(68)):
n∑
n1=1
1
n1
= log n+O(1) (n→∞). (2)
Since σ2 > 1, write σ2 = δ+µ for some δ > 1 and µ > 0. Then, we have, as n→∞,
n∑
n1=1
1
n1·nσ2
=
1
nσ2
n∑
n1=1
1
n1
=
1
nσ2
(log n+O(1))
=
1
nδ
·
1
nµ
[log n+O(1)] 5
L
nδ
for some L> 0, the second term in the third equality tending to as n→∞. We, there-
fore, find that
|ζ2 (s1,s2)|5 L
∞∑
n=1
1
nδ
<∞.
If σ1 < 1, then one has the following estimate:
n−1∑
n1=1
1
n1σ1 ·nσ2
=
[log2 n]∑
k=1
∑
n
2k
5n1<
n
2k−1
1
n1σ1 nσ2
< 2max{0,−σ1}
∞∑
k=1
n/2k(
n/2k
)σ1 nσ2
=
2max{0,−σ1}
nσ1+σ2−1
∞∑
k=1
1
2(1−σ1)k
5
M
nσ1+σ2−1
for some M > 0, being 1− σ1 > 0. So, in this case, being σ1+ σ2− 1> 1,
|ζ2 (s1,s2)|5 M
∞∑
n=1
1
nσ1+σ2−1
<∞.
The general case is seen to be easily verified by following the argument of case
d = 2. �
Preliminary on Generalized Functions
For any n ∈ N, Zhao [1260] introduces the space K of test functions consisting of all
the complex-valued smooth functions on Rn, which decrease to zero faster than any
The Zeta and Related Functions 215
negative power of
|x| =
√√√√ n∑
i=1
xi2 as |x| →∞
(
x= (x1, . . . ,xn) ∈ Rn
)
(for details of test functions, see [478, Chapter 1]). Throughout this subsection, we
use the abbreviations x= (x1, . . . ,xn) ∈ Rn and s= (s1, . . . ,sn) ∈ Cn. We say that a
generalized function g(x;s) (for this definition, see [478, Chapter 1]) is entire in s, if
the inner product< g(x;s),ϕ(x) > is an entire function of s for any test function ϕ(x).
As an example of the generalized functions, let us suppose, for x ∈ R, that
xs+ =
x
s (x> 0)
0 (x 5 0),
(3)
whose value on a test function ϕ(x) is given by
< xs+,ϕ(x) >:=
∞∫
0
xsϕ(x)dx.
The following lemmas play dominant rôles in this subsection.
Lemma 2.8 ([478, p. 48, Eq. (3)]) We have, for any test function ϕ,
∞∫
0
xs−1ϕ(x)dx=
1∫
0
xs−1
ϕ(x)− n∑
j=0
ϕ( j)(0)
j!
xj
 dx
+
∞∫
1
xs−1ϕ(x)dx+
n∑
j=0
ϕ( j)(0)
j!(s+ j)
(4)
(<(s) >−n− 1; s 6= −1,−2, . . . ,−n; n ∈ N0).
As an easy consequence of Lemma 2.8, one finds the following result.
Lemma 2.9 ([478, p. 57]) The generalized function xs−1+ /0(s) has an analytic con-
tinuation to an entire function in s, such that
xs−1+
0(s)
∣∣∣∣∣
s=−n
= δ(n)(x) (n ∈ N0),
where δ is the delta function (see [478, Section 1.3]) and, for any test function ϕ,
< δ(n)(x),ϕ(x) >:= (−1)n δ(n)(0) (n ∈ N0).
216 Zeta and q-Zeta Functions and Associated Series and Integrals
Lemma 2.10 The generalized function
f (x;s,u)=
(1− x)s−1+ x
u−1
+
0(s)0(u)
can be extended to an entire function in complex variables s and u.
Proof. We need to show that, for any test function ϕ on [0, 1], the function of two
complex variables s and u defined by
1∫
0
(1− x)s−1 xu−1
0(s)0(u)
ϕ(x)dx (<(u) > 0; <(s) > 0),
can be analytically continued to an entire function on all of C2. This follows easily
from Lemma 2.8, since the interval [0, 1] can be separated into [0, 1/2]∪ [1/2, 1]. �
Remark 2 The Riemann zeta function ζ(s), defined by 2.3(1), can be obtained by the
Mellin-transformation (see 2.3(4)):
ζ(s)=
1
0(s)
∞∫
0
xs−2·x
ex− 1
dx=
1
s− 1
∞∫
0
xs−2
0(s− 1)
·
x
ex− 1
dx (<(s) > 1). (5)
In view of Lemma 2.9, one finds that (s− 1)ζ(s) can be considered as the value of the
generalized function xs−2/0(s− 1) (which is entire) on the test function x/(ex− 1).
In thisway, one immediately recovers the analytic continuation of ζ(s). Zhao [1260]
used this idea to give an analytic continuation the multiple zeta functions of any depth
as follows:
Theorem 2.11 The multiple zeta function ζd (s1, . . . ,sd) of depth d (d ∈ N) defined by
(1), can be continued meromorphically to the whole space Cd with possible simple
poles at sd = 1 and sd( j) := sj+ ·· ·+ sd = d− j+ 2− ` ( j, ` ∈ N; 1 5 j< d).
Proof. If d = 1, then ζ(s1) is the Riemann zeta function, defined by 2.3(1). So, one
may assume d ∈ N \ {1}. By the identities∑
0<n1<n2<···<nd
e−n1 t1−n2 t2−···−nd td
=
d∑
j=1,nj∈N
e−n1 t1−(n1+n2)t2−···−(n1+···+nd)td
=
d∏
j=1
∑
nj∈N
e−nj (tj+···+td) =
d∏
j=1
(
etj+···+td − 1
)−1
The Zeta and Related Functions 217
and
∞∫
0
e−α t t s−1 dt =
0(s)
αs
(<(s) > 0; <(α) > 0), (6)
it is easy to find that
0(s1) · · ·0(sd) ζd (s1, . . . ,sd)=
∞∫
0
∞∫
0
· · ·
∞∫
0
ts1−11 · · · t
sd−1
d dt1 · · ·dtd∏d
j=1
(
etj+···+td − 1
) . (7)
Now, put xd+1 := 0 and make the change of variables
x1 · · ·xj = tj+ ·· ·+ td, if and only if tj = x1 · · ·xj
(
1− xj+1
)
(8)
( j ∈ N; 1 5 j 5 d).
It is not difficult to get the Jacobian of (8):
∂(t1, . . . , td)
∂(x1, . . . ,xd)
= det

∂t1
∂x1
∂t1
∂x2
· · ·
∂t1
∂xd
...
... · · ·
...
∂td
∂x1
∂td
∂x2
· · ·
∂td
∂xd
= x
d−1
1 ·x
d−2
2 · · ·x
2
d−2·xd−1. (9)
It follows from (8) and (9) that
0(s1) · · ·0(sd) ζd (s1, . . . ,sd)
=
1∫
0
· · ·
1∫
0
∞∫
0
d∏
j=1
xsd( j)−d+j−2j
d∏
j=2
(
1− xj
)sj−1−1 ϕ(x)dx1 · · ·dxd, (10)
where the function ϕ(x) is given by
ϕ(x) :=
xd1·x
d−1
2 · · ·x
2
d−1·xd∏d
j=1 (e
x1···xj − 1)
, (11)
which is seen to be a test function in K. Letting uj := sd( j)− d+ j− 1, we can apply
the generalized function
xu1−11+
0(u1)
·
d∏
j=2
(
1− xj
)sj−1−1
+
·x
uj−1
j+
0
(
sj−1
)
0
(
uj
)
218 Zeta and q-Zeta Functions and Associated Series and Integrals
on ϕ(x) and get exactly (sd − 1) ζd (s1, . . . ,sd)/
∏d−1
j=1 0
(
uj
)
, since 0(sd)=
(sd − 1) 0 (ud).
By Lemma 2.9 and Lemma 2.10, all of the following generalized functions
f1(x;s)=
xu1−11+
0(u1)
and fj(x;s)=
(
1− xj
)sj−1−1
+
·x
uj−1
j+
0
(
sj−1
)
0
(
uj
) ( j ∈ N; 2 5 j 5 d)
can be extended to entire functions in s. Furthermore, fj depends only on xj as a func-
tion of x. Hence, one can set the entire function
ξ (s1, . . . ,sd) :=
〈 d∏
j=1
fj(x;s), ϕ(x)
〉
(12)
on Cd and define an analytic continuation of ζd (s1, . . . ,sd) on Cd by
ζd (s1, . . . ,sd) :=
∏d−1
j=1 0
(
uj
)
sd − 1
ξ (s1, . . . ,sd). (13)
This completes the proof of the theorem, because Gamma functions have simple poles
only at nonpositive integers (see 1.1(10) and 1.1(12)). �
Zhao [1260, Theorem 6 and Theorem 7] evaluates the residues of ζd (s1, . . . ,sd) at
the simple poles given in Theorem 2.11 as follows:
Theorem 2.12 The residue of ζd (s1, . . . ,sd) at sd = 1 is 1 or ζd−1 (s1, . . . ,sd−1) ,
according to whether d = 1 or d ∈ N \ {1}.
For depth d ∈ N \ {1} and i, ` ∈ N with 1 5 i 5 d− 1, the residue of ζd (s1, . . . ,sd)
on the hyperplane
sd(i)= d− i+ 2− `
in Cd is equal to
ζi−1 (s1, . . . ,si−1)
∑
ad(i+1)=`−1
ai+1,...,ad=0
 d∏
j=i+1
Baj 0
(
ad( j)+ uj
)
aj!0
(
ad( j+ 1)+ uj+ 1
)
, (14)
where ζ(s0) := 1, ad( j) := aj+ ·· ·+ ad, ad(d+ 1) := 0, and uj := sd( j)− d+ j− 1.
Proof. The case d = 1 is obvious, since ζ(s1) is the Riemann zeta function. So, we
may assume d ∈ N\{1}. Let ϕ(x) and ξ (s1, . . . ,sd) be as in (11) and (12). In view
The Zeta and Related Functions 219
of (13), we find that
Res
sd=1
ζd (s1, . . . ,sd)= lim
sd→1
(sd − 1) ζd (s1, . . . ,sd)
=
 limsd→1
d−1∏
j=1
0
(
uj
)
{
lim
sd→1
ξ (s1, . . . ,sd)
}
,
where the two involved limits are readily seen to be
lim
sd→1
d−1∏
j=1
0
(
uj
)
= 0(sd−1− 1) ·
d−2∏
j=1
0(sd−1( j)− d+ j)
and
lim
sd→1
ξ (s1, . . . ,sd)=
ξ (s1, . . . ,sd−1)
0 (sd−1)
.
Now, it is easy to see that
Res
sd=1
ζd (s1, . . . ,sd)=
∏d−2
j=1 0(sd−1( j)− d+ j)
sd−1− 1
·ξ (s1, . . . ,sd−1)= ζd−1 (s1, . . . ,sd−1).
The proof of the remaining part, that is (14), is left to an interested reader. �
Remark 3 By the power series definition, one easily sees that
ζ2 (0, s2)= ζ (s2− 1)− ζ (s2), (15)
whereas it is known (see 3.3(35)) that
ζ2 (s1, s2)+ ζ2 (s2, s1)+ ζ (s1+ s2)= ζ (s1) ζ (s2). (16)
Setting s2 = 0 in (16) and using (15) yields
ζ2 (s1, 0)= ζ (s1) ζ (0)− ζ (s1− 1). (17)
Now, setting s2 = 0 in (15) and s1 = 0 in (17), together with 2.3(10), gives
lim
s2→0
lim
s1→0
ζ2 (s1, s2)=
5
12
and lim
s1→0
lim
s2→0
ζ2 (s1, s2)=
1
3
. (18)
It is observed that the limits of a function at some point along different routes in the
multivariable space may be different.
220 Zeta and q-Zeta Functions and Associated Series and Integrals
From (10) and the identity (use 1.7(2))
x2 y
(ex− 1)(exy− 1)
= 1−
1
2
(1+ y)x+
1
12
(
y2+ 3y+ 1
)
x2+O
(
x3
)
(x→ 0),
(19)
we readily obtain (see Zhao [1260, p. 1280]) that
ζ2 (s1, s2)=
4s1+ 5s2
12(s1+ s2)
+R(s1, s2), (20)
whereR(s1, s2) represents an analytic function of (s1, s2) in a neighborhood of (0, 0)
and lim(s1,s2)→(0,0) R(s1, s2)= 0. It is seen that (20) also gives the same limit values
as in (18).
Euler-Maclaurin Summation Formula
We begin by remarking that most of this subsection is due to Akiyama et al. [16] and
recalling another version of Euler-Maclaurin summation formula (cf. [3, p. 806] and
1.3(68)): Let N ∈ N and a, b ∈ R with a< b. Suppose that f ∈ CN([a, b]) denotes (as
usual) the set of functions having continuous derivatives of Nth order on the interval
[a, b]. Then, we have
∑
a<n≤b
f (n)=
b∫
a
f (x)dx+
K∑
k=1
(−1)k
k!
B̃k(b)f
(k−1)(b)
−
K∑
k=1
(−1)k
k!
B̃k(a)f
(k−1)(a)−
(−1)K
K!
b∫
a
B̃K(x)f
(K)(x)dx,
(21)
where B̃k(x) is the periodic Bernoulli polynomial, defined by B̃k(x) := Bk(x− [x]) ([x]
denotes the greatest integer 5 x).
Setting a= 1, b= m and K = `+ 1 in (21) yields
m∑
n=1
f (n)=
m∫
1
f (x)dx+
1
2
(f (1)+ f (m))+
∑̀
k=1
Bk+1
(k+ 1)!
(
f (k)(m)− f (k)(1)
)
−
(−1)`+1
(`+ 1)!
m∫
1
B̃`+1(x) f
(`+1)(x)dx,
(22)
where f ∈ C`+1([1, m]), m ∈ N and ` ∈ N0.
The Zeta and Related Functions 221
Consider f (x)= x−s in (22). We get, by subtracting the resulting identity of∑m
n=1 n
−s from that of
∑
∞
n=1 n
−s
= ζ(s), the following formula:
φ`(m,s)=
m∑
n=1
1
ns
−
{
m1−s− 1
1− s
+
1
2ms
−
∑̀
k=1
(s)k ak
ms+k
+ ζ(s)−
1
s− 1
}
, (23)
where (s)k is the Pochhammer symbol given in 1.1(5), ak := Bk+1/(k+ 1)!, and
φ`(m,s) :=
(s)`+1
(`+ 1)!
∞∫
m
B̃`+1(x)x
−s−`−1 dx
= O
(
m−<(s)−`
)
,
(24)
where, for example, see Hardy [537, p. 332, 13.10.3]. We rewrite (23) in the following
equivalent form:
∞∑
n=m+1
1
ns
=−φ`(m,s)+
m1−s
s− 1
−
1
2ms
+
∑̀
k=1
(s)k ak
ms+k
(<(s) > 1). (25)
Consider the following multiple Zeta function in two variables:
ζ2 (s1, s2)=
∑
0<n1<n2
1
ns11 n
s2
2
(<(si) > 1; i= 1, 2).
We make use of (25) to get
ζ2 (s1, s2)=
∞∑
n1=1
1
ns11
∞∑
n2=n1+1
1
ns22
=
∞∑
n1=1
1
ns11
{
−φ`(n1,s2)+
n1−s21
s2− 1
−
1
2ns21
+
∑̀
k=1
(s2)k ak
ns2+k1
}
=
ζ (s1+ s2− 1)
s2− 1
−
ζ (s1+ s2)
2
+
∑̀
k=1
(s2)k ak ζ (s1+ s2+ k)−
∞∑
n1=1
φ`(n1,s2)
ns11
(26)
for <(si) > 1 (i= 1, 2). The terms on the right hand side, except the last one, have
meromorphic continuations. The last sum is absolutely convergent and, hence, holo-
morphic in <(s1+ s2+ `− 1) > 0. Thus, we now have a meromorphic continuation
of ζ2 (s1, s2) to C2, which is holomorphic in{
(s1, s2) ∈ C2 | s2 6= 1, s1+ s2 /∈ {2, 1, 0,−2,−4,−6, . . .}
}
.
222 Zeta and q-Zeta Functions and Associated Series and Integrals
It is easy to see that this manipulation can be employed to a multiple zeta function
with d variables. Indeed, we have
ζd (s1, . . . ,sd)
=
∞∑
n1=1
1
ns11
∞∑
n2=n1+1
1
ns22
· · ·
∞∑
nd−1=nd−2+1
1
nsd−1d−1
∞∑
nd=nd−1+1
1
nsdd
=
∞∑
n1=1
1
ns11
∞∑
n2=n1+1
1
ns22
· · ·
∞∑
nd−1=nd−2+1
1
nsd−1d−1
·
{
−φ`(nd−1,sd)+
n1−sdd−1
sd − 1
−
1
2nsdd−1
+
∑̀
k=1
(sd)k ak
nsd+kd−1
}
=
ζd−1 (s1, . . . ,sd−2, sd−1+ sd − 1)
sd − 1
−
ζd−1(s1, . . . ,sd−2, sd−1+ sd)
2
+
∑̀
k=1
(sd)k ak ζd−1 (s1, . . . ,sd−2,sd−1+ sd + k)
−
∑
0<n1<···<nd−1
φ`(nd−1,sd)
ns11 · · ·n
sd−1
d−1
(<(si) > 1; i= 1, . . . , d).
(27)
Since
∑
0<n1<···<nd−1
∣∣∣∣∣φ`(nd−1,sd)ns11 · · ·nsd−1d−1
∣∣∣∣∣5 ∑
nd−1
n−`−<(sd)+d−2d−1
nLd−1
with
L :=<(sd−1)+
∑
15i5d−2
<(si)50
<(si),
the last summation is absolutely convergent in the set
`− d+ 1+<(sd)+<(sd−1)+
∑
15i5d−2
<(si)50
<(si) > 0. (28)
Since ` can be taken arbitrarily large, we get an analytic continuation of ζd (s1, . . . ,sd)
to Cd. Now, we consider the set singularities. It is seen that the singular part of
ζ2 (s1,s2) may be formally given as follows:
ζ(s1+ s2− 1)
s2− 1
+
∑
k1=0
ak1 (s2)k1
s1+ s2+ k1− 1
,
The Zeta and Related Functions 223
since, by analytic continuation, ζ(s) can be written as ζ(s)= 1/(s− 1)+ τ(s), where
τ(s) is an entire function. We find from this expression that
s2 = 1, s1+ s2 ∈ {2, 1, 0,−2,−4,−6, . . .}
forms the set of all singularities of ζ2 (s1,s2). For the case ζ3 (s1,s2,s3), by using the
singular part of ζ2, we see that singularities lie on
s3 = 1, s2+ s3 ∈ {2, 1, 0,−2,−4,−6, . . .}
and
s1+ s2+ s3 ∈ {3, 2, 1, 0,−1,−2,−3, . . .}.
To show these are all singularities of ζ3 (s1,s2,s3), it suffices to prove that no singulari-
ties, defined by one of the above equations, will identically vanish. This can be shown
by changing variables:
u1 = s1, u2 = s2+ s3, u3 = s3.
In fact, it is seen that the singular part of ζ3 (u1,u2− u3,u3) is given by
1
u3− 1
ζ2 (u1,u2− 1)+
∑
k2=0
ak2 (u3)k2 ζ2 (u1,u2+ k2).
It is found from this expression that the singularities of ζ2 (u1,u2+ k2) are summed
with functions of u3 of different degrees. Thus, these singularities, as a weighted sum
by another variable u3, will not vanish identically. Similarly, we can prove Theo-
rem 2.13 below.
Theorem 2.13 The multiple zeta function ζd (s1, . . . ,sd) continues meromorphically
to Cd and has singularities on
sd = 1, sd−1+ sd = 2, 1, 0,−2,−4, . . . ,
and
j∑
i=1
sd−i+1 ∈ Z5j ( j= 3, 4, . . . , d),
where Z5j is the set of integers less than or equal to j.
We conclude this section by remarking the following facts: Akiyama et al. [16]
defines the multiple zeta values at nonpositive integers by
ζd (−r1, . . . ,−rd) := lim
s1→−r1
· · · lim
sd→−rd
ζd (s1, . . . ,sd)
(
rj ∈ N0; j= 1, . . . ,d
)
.
(29)
224 Zeta and q-Zeta Functions and Associated Series and Integrals
If, for simplicity, we write
(s)−1 :=
1
s− 1
,
then it follows from (24), (27) and (29) that
ζd (−r1, . . . ,−rd)=
rd∑
k=−1
(−rd)k ak ζd−1 (−r1, . . . ,−rd−2,−rd−1− rd + k), (30)
where we have used the fact that, in view of (24),
φ`(nd−1,−rd)= 0 (`= rd).
Akiyama et al. [16] and Akiyama and Tanigawa [17] made a basic use of (30) to give
the multiple zeta values at nonpositive integers as asserted by Theorem 2.13 below.
Theorem 2.14 [17, p. 330, Theorem 2]
ζd (−n, 0, . . . ,0)=
(−1)n
d!
d∑
j=1
s(d, j)
jBn+j
n+ j
(n ∈ N0), (31)
where s(d, j) denotes the Stirling numbers of the first kind (see Section 1.6) and Bj the
Bernoulli numbers (see Section 1.7).
Problems
1. Show that
ζ ′
(
−2k+ 1,
p
q
)
=
1
2k
[ψ(2k)− log(2πq)] B2k
(
p
q
)
−
q−2k
2k
[ψ(2k)− log(2π)] B2k
+
(−1)k+1π
(2πq)2k
q−1∑
n=1
sin
(
2πpn
q
)
ψ (2k−1)
(
n
q
)
+
(−1)k+1 2 · (2k− 1)!
(2πq)2k
q−1∑
n=1
cos
(
2πpn
q
)
ζ ′
(
2k,
n
q
)
+
ζ ′(−2k+ 1)
q2k
(p< q; p, q, k ∈ N).
(Miller and Adamchik [829, p. 203, Proposition 1])
2. Prove that there exist constants Dk, defined by
logDk := lim
n→∞
(
n∑
m=1
mk logm− p(n,k)
)
(k ∈ N0),
The Zeta and Related Functions 225
where the definition of p(n,k) in Adamchik [6, p. 198, Eq. (20)] is corrected here as fol-
lows:
p(n,k) :=
nk
2
logn+
nk+1
k+ 1
(
logn−
1
k+ 1
)
+ k!
k∑
j=1
nk−j Bj+1
( j+ 1)!(k− j)!
logn+ (1− δkj) j∑
`=1
1
k− `+ 1

and δkj is the Kronecker symbol defined by 1.6(25).
(Bendersky [114, pp. 273–275]; Adamchik [6, p. 198])
3. For the constants Dk (k ∈ N0), defined in Problem 2, show that
D0 = (2π)
1
2 D1 = A, D2 = B and D3 = C
and
log Dk =
Bk+1 Hk
k+ 1
− ζ ′(−k) (k ∈ N0),
where Bn are the Bernoulli numbers and Hn are the harmonic numbers, and the mathemati-
cal constants A, B, and C are given as in Section 1.4.
(Adamchik [6, pp. 198–199])
4. Prove that
π∫
0
(
log
[(
2sin 12 t
)(
2cos 12 t
)p])3
dt =−2πζ(3)
[
3
4
(
1+ p3
)
−
3
8
(
p+ p2
)]
.
(Lewin [752, p. 220])
5. The Barnes double Zeta function ζ2(v;α,w) is the meromorphic continuation of the series
defined by
ζ2(v;α,w) :=
∞∑
m=0
∞∑
n=0
(α+m+ nw)−v (α > 0; w> 0; v ∈ C).
The functions ρ2(1,w) and 02(α,(1,w)) are defined by
logρ2(1,w) :=− lim
α→0
[
ζ ′2(0;α,w)+ logα
]
and
02(α,(1,w)) := ρ2(1,w) exp
[
ζ ′2(0;α,w)
]
,
respectively.
(a) Prove that, for any N ∈ N and <(v) >−N+ 1 (v ∈ C \ [Z−0 ∪ {1, 2}]),
ζ2(v;α,w)= ζ(v,α)+
ζ(v− 1)
v− 1
w1−v
+
N−1∑
n=0
(
−v
n
)
ζ(−n,α)ζ(v+ n)w−v−n+RN(v;α,w)
226 Zeta and q-Zeta Functions and Associated Series and Integrals
with the estimate:
RN(v;α,w)= O
(
w−<(v)−N
)
,
where the O-constant depends on v, N and α.
(b) Show also that, for N ∈ N \ {1, 2},
ζ ′2(0;α,w)=−
1
12
w logw+
[
1
12
− ζ ′(−1)
]
w
+
1
2
(
1
2
−α
)
logw+ log0(α)+
(
1
2
α−
3
4
)
log(2π)
+ ζ(−1,α)w−1 (logw− γ )
+
N−1∑
n=2
(−1)n
n
ζ(−n,α)ζ(n)w−n+R′N(0;α,w)
and
log02(α,(1,w))=
1
2
α log
(
2πw−1
)
+ log0(α)
+ [ζ(−1,α)− ζ(−1)]w−1 (logw− γ )
+
N−1∑
n=2
(−1)n
n
[ζ(−n,α)− ζ(n)]ζ(n)w−n+ SN(α,(1,w))
with the estimates:
R′N(0;α,w)= O
[
w−N(| logw| + 1)
]
and
SN(α,(1,w))= O
(
w−N(| logw| + 1)
]
,
where the O-constants depend on N and α and the prime in ζ ′2(v;α,w) denotes differ-
entiation with respect to v.
(Matsumoto [804, 805])
6. For the Bernoulli numbers and polynomials, show that
(a)
B2n = 4n(−1)
n+1
∞∫
0
x2n−1
e2πx− 1
dx (n ∈ N);
(b)
|B2n(x)| ≤ |B2n| (n ∈ N; 0 5 x 5 1).
7. Show that
ζ(s)=
1(
1− 21−s
)
0(s)
∞∫
0
xs−1
ex+ 1
dx (<(s) > 0; s 6= 1).
(See, e.g., Chen [240, p. 263])
The Zeta and Related Functions 227
8. Prove that
m+n+ 12∫
m
log 0(t)dt =−
m−1∑
k=1
k log k+
1
2
m+n−1∑
k=1
(2k+ 1) log(2k+ 1)
−
1
2
(m+ n)2 (1+ log2)+
m(m− 1)
2
−
3
2
ζ ′(−1)
+
(
−
3
2
+
1
2
log2
)
ζ(−1)+
(
n+
1
2
)
log
√
2π (m∈N; n∈N0).
(Elizalde et al. [412, p. 19])
9. Prove that
z∫
0
tnψ(t)dt = (−1)n−1 ζ ′(−n)+
(−1)n
n+ 1
Bn+1 Hn
−
n∑
k=0
(−1)k
(
n
k
)
zn−k
k+ 1
Bk+1(z)Hk +
n∑
k=0
(−1)k
(
n
k
)
zn−k ζ ′(−k,z)
(n ∈ N0; <(z) > 0).
(Adamchik [6, p. 197])
10. Prove that
za
0(1− s)
8(z,s,a)=
(
log
1
z
)s−1
+
∞∑
k=0
(
s− 1
2k
)
(logz)2k
ζ(s− 2k,a)
0(2k+ 1− s)
−
∞∑
k=0
(
s− 1
2k+ 1
)
(logz)2k+1
ζ(s− 2k− 1,a)
0(2k+ 2− s)
(| logz|< 2π; s 6∈ N; a 6∈ Z−0 ).
Also deduce the relatively more familiar result 2.5(13).
11. Prove that
E2n−1(0)=
4(−1)n
(2π)2n
(2n− 1)!
(
22n− 1
)
ζ(2n) (n ∈ N),
where En(x) denotes the Euler polynomials (see Section 1.6).
(Cf. Srivastava [1084, p. 390])
12. Let f : N×N→ R be a function. For a fixed n ∈ N, let
Tn := {(k, j) ∈ N×N | 1 5 k, j 5 n; k+ j = n+ 1}
and
Sn :=
∑
(x,y)∈Tn
f (x,y),
the sum being taken over all pairs (x,y) ∈ Tn. Show that
Sn =
n∑
k=1
f (k,k)+
n∑
k=2
k−1∑
j=1
{ f ( j,k)+ f (k, j)− f ( j,k− j)}.
(Sitaramachandrarao and Sivaramsarma [1035, p. 603])
228 Zeta and q-Zeta Functions and Associated Series and Integrals
13. Prove that
2
∞∑
q=1
1
qn
 q−1∑
k=1
(k,q)=1
1
k
= n+ 2− 1ζ(n+ 1)
n−2∑
j=1
ζ(n− j)ζ( j+ 1) (n ∈ N \ {1}),
where an empty sum is understood (as usual) to be nil and (k,q)= 1 denotes that k and q
are relatively prime.
(Cf. Equation (54); see also Sitaramachandrarao and Sivaramsarma [1035, p. 602])
14. Prove that
∞∑
k=1
Hk
k2n+1
=
1
2
2n∑
j=2
(−1)j ζ( j)ζ(2n− j+ 2) (n ∈ N),
and
∞∑
k=1
Hk
kn
=
(
1+
n
2
)
ζ(n+ 1)−
1
2
n−1∑
j=2
ζ( j)ζ(n− j+ 1) (n ∈ N \ {1}),
where Hk are the harmonic numbers defined by 3.2(36).
(Georghiou and Philippou [479, p. 29])
15. For bounded maps f , g : N→ C, let
Sf ,g(u,v;w)=
∞∑
r,k=1
f (r+ k)g(k)
ru ku (r+ k)w
(
u, v ∈ N; w ∈ R+)
and
Cf ,g(x,y)=
∞∑
r=2
f (r)
rx
r−1∑
k=1
g(k)+ g(r− k)
ky
(x> 1; y ∈ N).
Show that
Sf ,g(u,v;w)+ Sf ,g(v,u;w)=
u−1∑
j=0
(
v+ j− 1
v− 1
)
Cf ,g(v+w+ j,u− j)
+
v−1∑
j=0
(
u+ j− 1
u− 1
)
Cf ,g(u+w+ j,v− j).
(Subbarao and Sitaramachandrarao [1133, p. 246])
16. Prove that
∞∑
r,k=1
(−1)r+k
rk (r+ k)
=
1
4
ζ(3)
and
∞∑
r,k=1
(−1)k−1
rk (r+ k)
=
5
8
ζ(3).
(Subbarao and Sitaramachandrarao [1133, p. 247])
The Zeta and Related Functions 229
17. Prove that
∞∑
k=1
H(2)k
k2n+1
= ζ(2)ζ(2n+ 1)−
(n+ 2)(2n+ 1)
2
ζ(2n+ 3)
+ 2
n+1∑
j=2
( j− 1)ζ(2j− 1)ζ(2n+ 4− 2j) (n ∈ N),
where H(m)n denotes the generalized harmonic number, defined by
H(m)0 = 0 and H
(m)
n =
n∑
`=1
`−m (m, n ∈ N).
(Georghiou and Philippou [479, p. 35])
18. Prove that
∞∑
m,n=0
′
1
m6
(
m2+ n2
) = 13π8
28350
and
∞∑
m,n=0
′
1
m2
(
m2−mn+ n2
) = √3π4
30
,
where the prime denotes that the summations are taken over all pairs of integers,
except (0,0).
(Smart [1042, p. 10])
19. Prove that
∞∑
k=1
1
k
(H2k−1−Hk − log2)= (log2)
2
−
π2
6
.
(Knuth [679, p. 138])
20. Let a, b, c ∈ R with a> 0 and d = b2− 4ac< 0. The Epstein Zeta function is given by
Z(s) :=
1
2
∑
′
(
am2+ bmn+ cn2
)−s
(<(s) > 1),
where the prime denotes that the summation is to be taken over all pairs (m,n) of integers
other than the pair (0,0). Let k ∈ R+ be a number satisfying
k2 =
c
a
−
(
b
2a
)2
(a, b, c ∈ R; a> 0)
and suppose that the Bessel function Kν(z) is defined by
Kν(z)=
1
2
∞∫
0
e−
z
2
(
u+ 1u
)
uν−1 du=
1
2
∞∫
−∞
e−zcosh t eνt dt
=
∞∫
0
e−zcosh t cosh(νt)dt
(
ν ∈ C; |argz|<
π
2
)
.
230 Zeta and q-Zeta Functions and Associated Series and Integrals
Also, let
σν(n) :=
∑
d|n
dν =
∑
d|n
(n
d
)ν
(ν ∈ C; n ∈ N).
Show that
as Z(s)= ζ(2s)+ k1−2s ζ(2s− 1)
0
(
s− 12
)
0
(
1
2
)
0(s)
+
π s
0(s)
k
1
2−s H(s) (<(s) > 1),
where
H(s)= 4
∞∑
n=1
ns−
1
2 σ1−2s(n) cos
(
nπb
a
)
Ks− 12
(2πkn).
Show, also, that H(s) is an entire function of s, such that
H(s)= H(1− s).
(Bateman and Grosswald [101, p. 366])
21. Show that
(cz+ d)2n A
(
az+ b
cz+ d
,−2n
)
= A(z,−2n)+
1
2
{
1− (cz+ d)2n
}
ζ(2n+ 1)
+
(−1)n (2π)2n+1 i
2(2n+ 2)!
c∑
j=1
2n+2∑
k=0
(
2n+ 2
k
)
Bk
(
j
c
)
B̄2n−k+2
(
jd
c
)
{−(cz+ d)}k−1
(c, n ∈ N),
where A(z,−2n) denotes a Lambert series in the variable e2π iz, defined by
A(z,−2n) :=
∞∑
k=1
k−2n−1
e2π ikz
1− e2π ikz
and B̄n(x) := Bn (x− [x]) ([x] being, as usual, the greatest integer in x).
(Cf. Berndt [119, p. 505]; see also Apostol [57, p. 153])
22. Show that
2k−s
{
ζ
(
s,
a
2
)
− ζ
(
s,
a+ 1
2
)}
=
k∑
`=1
(−1)`−1
`−1∑
j=0
(−1)j
(
k
j
)
(`− 1− j+ a)−s
+
∞∑
n=0
(−1)n+k
k∑
j=0
(−1)j
(
k
j
)
(n+ k− j+ a)−s (k ∈ N; <(s) >−k),
where ζ(s,a) (a> 0) is the Hurwitz (or generalized) Zeta function, defined by 2.2(1).
(Balakrishnan [91, p. 205])
The Zeta and Related Functions 231
23. Show that
π
3∫
0
m−2∑
k=0
(−1)k
2k+ 1
(
2m− 2
2k
)(
θ
2
)2k+1 {
log
(
2 sin
θ
2
)}2m−2k−2
dθ
=
(−)mπ2m
8m(2m− 1)
[(
1
6
)2m−1
− 2
(
1−
1
22m−1
)
B2m
]
(m ∈ N \ {1})
and
π
3∫
0
m−1∑
k=0
(−1)k
(
2m
2k
)(
θ
2
)2k {
log
(
2 sin
θ
2
)}2m−2k
dθ
=
(−)mπ2m+1
22m+2
(
E2m−
1
(2m+ 1)32m
)
(m ∈ N),
where Bn and En are the Bernoulli and Euler numbers, respectively, given in Section 1.7.
(Zhang and Williams [1253, p. 282])
24. Show that
∞∑
n=1
1
nm+2
(2n
n
) = (−1)m 2m
m!
π
3∫
0
θ
{
log
(
2 sin
θ
2
)}m
dθ (m ∈ N0)
and
∞∑
n=1
(−1)n−1
nm+2
(2n
n
) = (−1)m 2m
m!
2 logτ∫
0
θ
{
log
(
2 sinh
θ
2
)}m
dθ (m ∈ N0).
(Zhang and Williams [1253, pp. 286 and 288])
25. Show that
π
3∫
0
[{
log
(
2 sin
θ
2
)}4
−
3θ2
2
{
log
(
2 sin
θ
2
)}2]
dθ =
253π5
23·34·5
and
π
3∫
0
[
θ
{
log
(
2 sin
θ
2
)}4
−
θ3
2
log
(
2 sin
θ
2
)]
dθ =
313π6
24·36·5·7
.
(Zucker [1267, pp. 98 and 99])
26. Prove each of the following claims:
(a) For all a1, a2, . . .
∞∑
k=1
a1 a2 · · · ak−1
(x+ a1) · · ·(x+ ak)
=
1
x
.
232 Zeta and q-Zeta Functions and Associated Series and Integrals
(b)
ζ(3)=:
∞∑
n=1
1
n3
=
5
2
∞∑
n=1
(−1)n−1
n3
(2n
n
) .
(c) Consider the recursion:
n3 un+ (n− 1)
3 un−2 =
(
34n3− 51n2+ 27n− 5
)
un−1 (n ∈ N \ {1}).
Let {bn} be the sequence defined by b0 = 1, b1 = 5 and bn = un for all n ∈ N \ {1};
then the bn are all integers.
Let {an} be the sequence defined by a0 = 0, a1 = 6 and an = un for all n ∈ N \
{1}; then the an are rational numbers with denominator dividing 2[1, 2, . . . , n]3 (here,
[1, 2, . . . , n] is the lowest common multiple of 1, . . . , n).
(d) an/bn→ ζ(3) as n→∞; indeed, the convergence is so fast as to prove that ζ(3)
cannot be rational. To be precise, for all integers p, q with q sufficiently large rela-
tive to �>0,∣∣∣∣ζ(3)− pq
∣∣∣∣> 1qθ+� and θ = 13.41782 · · · ,
which proves the irrationality of ζ(3).
(Cf. van der Poorten [1180, p. 195]; see also Apéry [56])
27. Prove the following inequalities: For n, p ∈ N,
0≤
(
p+
1
2
) n∑
k=1
1
k2p
−
p−1∑
j=1
(
n∑
k=1
1
k2p−2j
)(
n∑
k=1
1
k2j
)
<
8
n
and ∣∣∣∣∣∣
n∑
k=1
1
(2k− 1)2
− 2
(
n∑
k=1
(−1)k+1
2k− 1
)2∣∣∣∣∣∣< 6n .
(Hovstad [571, p. 93])
28. Show that the first inequality of Problem 27 implies that
∞∑
k=1
1
k2p
=Rp
(
∞∑
k=1
1
k2
)p
(p ∈ N),
where Rp is a rational number, satisfying the recurrence formula:
R1 = 1 and
(
p+
1
2
)
Rp =
p−1∑
j=1
RjRp−j (p ∈ N \ {1}).
(Hovstad [571, p. 93])
The Zeta and Related Functions 233
29. Prove that ζ(s) has the factorization
ζ(s)=
ebs
2(s− 1)0
(
1+ 12 s
) ∏
ρ
[(
1−
s
ρ
)
e
s
ρ
]
,
where
b=−1−
γ
2
+ log(2π);
γ is the Euler-Mascheroni constant, and the product is taken over all the so-called non-
trivial zeros of ζ(s).
(Cf. Titchmarsh [1151, pp. 30–31]; see also Melzak [821, p. 111])
30. Prove that
ζ
(
1
2
,a+ 1
)
=
∞∑
n=0
(2n)!
22n (n!)2
ζ
(
n+
1
2
)
(−a)n (|a|< 1)
and
ζ
(
1
2
,na
)
= n−
1
2
n−1∑
k=0
ζ
(
1
2
,a+
k
n
)
(n ∈ N).
(Powell [911, p. 117])
31. Let
(s− 1)ζ(s,a)= 1+
∞∑
n=0
γn(a)(s− 1)
n+1 (0< a≤ 1; n ∈ N0).
Show that
γn(a)=
(−1)n
n!
lim
m→∞
(
m∑
k=0
{log(k+ a)}n
k+ a
−
{log(m+ a)}n+1
n+ 1
)
,
which, for a= 1, gives the coefficients in the Laurent expansion of ζ(s) about s= 1
(cf. Equation 2.3(7)).
(Berndt [118, p. 152])
32. Prove that
limsup
t→∞
|ζ(1+ it)|
log(log t)
≥ eγ,
where γ is the Euler-Mascheroni constant defined by 1.1(3).
(Titchmarsh [1150, p. 79])
33. Prove that
Lim
(
ez
)
=
∞∑
n=0
n6=m−1
ζ(m− n)
zn
n!
+
(
1+
1
2
+ ·· ·+
1
m− 1
− log(−z)
)
zm−1
(m− 1)!
(m ∈ N; |z|< 2π).
(Cohen et al. [336, p. 26])
234 Zeta and q-Zeta Functions and Associated Series and Integrals
34. Prove that
ζ(5)=
5
2
∞∑
n=1
(
n−1∑
k=1
1
k2
−
4
5
1
n2
)
(−1)n
n3
(2n
n
) .
(van der Poorten [1181, p. 274])
35. Let K be a normal extension field of degree n over the rational number field Q. Denote by
OK the integer ring of K. Let I(K) be the set of all nonzero ideals of OK , Na the absolute
norm of an ideal a ∈ I(K) and Trα the trace of α ∈ K over Q. We assume that [K : Q]=
n> 1, let
OK,0 = {α ∈ OK | Trα = 0};
TK =min{Trα | α ∈ OK, Trα > 0}.
For a ∈ I(K) and a0 = OK,0 ∩ a, define
T(a)=Min
{
Trα
T(K)
| α ∈ a, Trα > 0
}
;
N0(a)= #{OK,0/a0},
where OK,0/a0 is the quotient of Z-module OK,0 by submodule a0. For any x ∈ R+, let
jK(x)= #{a ∈ I(K) | Na≤ x};
jK,0(x)= #{a ∈ I(K) | N0(a)≤ x}
= #
{
a ∈ I(K) |
Na
T(a)
≤ x
}
,
and q≡ 1 (mod n). Suppose that K is the subfield of the qth cyclotomic field Cq, such that
K/Q is a tamely ramified cyclic extension of degree n. Show that
ζ(2k+ 1)=
2qn− qn−1− q
qn− 1
ζ(2k) lim
x→+∞
jk(x)
jK,0(x)
.
(Lan [728, p. 273])
36. Prove that
∞∑
n=1
(Hn)3
n5
−
11
4
∞∑
n=1
H(2)n
n6
=
469
32
ζ(8)− 16ζ(3)ζ(5)+
3
2
ζ(2) {ζ(3)}2
and
∞∑
n=1
(Hn)3
n7
−
13
4
∞∑
n=1
H(2)n
n8
=
561
20
ζ(10)−
47
4
{ζ(5)}2
−
49
2
ζ(7)ζ(3)+ 3ζ(2)ζ(3)ζ(5)+
15
4
{ζ(3)}2 ζ(4).
(Flajolet and Salvy [454, p. 27])
The Zeta and Related Functions 235
37. Prove that, for an odd weight m= p+ q,
∞∑
n=1
H(p)(n)nq
= ζ(m)
(
1
2
−
(−1)p
2
(
m− 1
p
)
−
(−1)p
2
(
m− 1
q
))
+
1− (−1)p
2
ζ(p)ζ(q)+ (−1)p
[p/2]∑
k=1
(
m− 2k− 1
q− 1
)
ζ(2k)ζ(m− 2k)
+ (−1)p
[q/2]∑
k=1
(
m− 2k− 1
p− 1
)
ζ(2k)ζ(m− 2k),
where ζ(1) should be interpreted as 0 wherever it occurs.
(Cf. Borwein et al. [146, p. 278]; see also Flajolet and Salvy [454, p. 22])
38. Prove that
∞∑
n=1
2n−1∑
k=1
(−1)n+k
n2 k
= πG−
27
16
ζ(3)
and
∞∑
n=1
2n−1∑
k=1
(−1)n−1
n2 k
= πG−
29
16
ζ(3),
where G denotes the Catalan constant defined by 1.3(16).
(Sitaramachandrarao [1034, p. 13])
39. Prove that, for m ∈ N and <(a) > 0,
ζ ′(−m,a)=
∂
∂z
ζ(z,a)
∣∣∣∣
z=−m
=
1
m+ 1
am+1 log a−
1
(m+ 1)2
am+1−
1
2
am log a
+
m
12
am−1 log a+
1
12
am−1+
∞∑
k=1
α2k a
−(2k−m+1),
where
α2k :=

B2k+2
2k+ 2
( m
2k+ 1
)
loga+
2k∑
j=0
(
m
j
)
(−1)j
2k− j+ 1
 (2k 5 m− 1),
B2k+2
2k+ 2
m∑
j=0
(
m
j
)
(−1)j
2k− j+ 1
(2k = m).
(Elizalde [410, p. 349])
40. Define constants 8k, for k > 1 an odd integer, by
8k =−
2
π
k−2∑
d=1
d odd
(−1)
d−1
2
πd
d!
ζ(k− d+ 1).
236 Zeta and q-Zeta Functions and Associated Series and Integrals
Show that, for real r ≥ 1 and s> 1,
ζ(r,s)=−
1
2
ζ(r+ s)+
∞∑
k=3
k odd
8k
k−1∑
j=0
jeven
(
k
j
)
η(r− j)η(s− k+ j),
where the Eta function is defined by
η(s) :=
(
1− 21−s
)
ζ(s).
(Crandall and Buhler [344, p. 279])
41. Let f (s) be a function defined by the Dirichlet series as follows:
f (s) :=
∞∑
n=2
n−s
∑
k<n
k−1 (<(s)= σ > 1).
Show that f (s) is analytic in the whole complex s-plane, except at simple poles s= 0 and
s= 1− 2a (a ∈ N) with residues given by
Res
s=0
f (s)=−
1
2
and
Res
s=1−2a
f (s)=−
B2a
2a
(a ∈ N),
and a pole of order 2 at s= 1 with residue given by
Res
s=1
f (s)= γ,
where B2a are the Bernoulli numbers, defined by 1.6(2), and γ is the Euler-Mascheroni
constant, defined by 1.1(3).
(Matsuoka [813, p. 399])
42. By using the definitions and restrictions as in Problem 20 of Chapter 2 with an additional
condition c> 0, show that
Z(s)= ζ(2s)a−s+
22s−1 as−1
√
π
0(s)(−d)s−
1
2
ζ(2s− 1)0
(
s−
1
2
)
+Q(s),
where
Q(s)=
2π s·2s−
1
2
√
a0(s)·(−d)
s
2−
1
4
∞∑
n=1
ns−
1
2 σ1−2s(n) cos
(
nπb
a
)
·
∞∫
0
us−
3
2 exp
(
−
πn
√
−d
2a
(
u+ u−1
))
du.
(Selberg and Chowla [1017, p. 87])
The Zeta and Related Functions 237
43. Let r, s ∈ N0 with r > s. Show that
1∫
0
1∫
0
xr ys
1− xy
dxdy
is a rational number whose denominator is a divisor of d2r , where dr denotes the lowest
common multiple of 1, 2, . . . , r.
(Beukers [126, p. 268])
44. Prove that
ζ(3)=
1
4
∞∑
n=1
(−1)n−1
56n2− 32n+ 5
(2n− 1)2
1(3n
n
)(2n
n
)
n3
and
ζ(3)=
∞∑
n=0
(−1)n
72
(4n
n
)(3n
n
) 5265n4+ 13878n3+ 13761n2+ 6120n++1040
(4n+ 1)(4n+ 3)(n+ 1)(3n+ 1)2 (3n+ 2)2
.
(Amdeberhan [34, p. 2])
45. Prove that
∞∑
n=1
(−1)n
n2
n∑
k=1
1
k2
=−
1
6
[
24Li4
(
1
2
)
+ 21ζ(3) log2+ (log2)4
]
+
π2
6
(log2)2+
17
480
π4,
where Li4(z) denotes the Tetralogarithm (see Section 2.4).
(Daudé et al. [367, p. 421])
46. Prove that
∞∑
n=1
(−1)n
(log(2n+ 1))2
2n+ 1
=
π
4
2[ζ ′′ (0, 14)− ζ ′′ (0, 34)]
−4[γ + log(2π)] log
0
(
1
4
)
0
(
3
4
)
+ [γ + log(2π)]2+ π2
12

and
∞∫
0
(log t)2
cosh t
dt =
π
2
2[ζ ′′ (0, 14)− ζ ′′ (0, 34)]
−4 log(2π) log
0
(
1
4
)
0
(
3
4
)
+ [log(2π)]2+ 1
4
π2
.
(Shail [1021])
238 Zeta and q-Zeta Functions and Associated Series and Integrals
47. For an integer a =−1, prove the following asymptotic expansion:
Ga(x) :=
∞∑
n=a+1
ζ(n− a)
(−x)n
n!
=−
(−x)a+1
(a+ 1)!
{logx−ψ(a+ 2)+ γ }
+
a∑
n=0
(−x)n
(
n∑
n=0
(a−k)!
(a+1)!(n−k)!
−
1
n!(a+ 1− n)
)
+O
(
1
x
)
(x→∞).
(Buschman and Srivastava [195, p. 296])
48. For any multi-index k= (k1, k2, . . . , kr)) (ki ∈ N), the weight wt(k) and depth dep(k) of k
are defined by |k| = k1+ k2+ ·· ·+ kr and r, respectively. The height of the index k is also
defined by ht k= #
{
j | kj = 2
}
.
Denote, by I(k, r), the set of multi-indices k of weight k and depth r and, by I0(k, r),
the subset of I(k, r) with admissible indices, that is indices with the additional requirement
that k1 = 2. For (k1, . . . , kr) ∈ I0(k, r), the multiple zeta value (MZV) and the non-strict
multiple zeta value (MZSV) can often be defined, respectively, as follows:
ζ (k1, k2, . . . , kr) :=
∑
n1>···>nr>0
1
nk11 · · ·n
kr
r
and
ζ ∗ (k1, k2, . . . , kr) :=
∑
n1=···=nr=1
1
nk11 · · ·n
kr
r
.
Prove the following formulas:
(a) Sum Formula. For r < k (r, k ∈ N), there hold∑
k∈I0(k,r)
ζ(k)= ζ(k) and
∑
k∈I0(k,r)
ζ ∗(k)=
(
k− 1
r− 1
)
ζ(k).
(b) Cyclic Sum Formula. For (k1, . . . , kr) ∈ I0(k, r),
r∑
i=1
ki−2∑
j=0
ζ ∗ (ki− j, ki+1, . . . ,kr,k1, . . . ,ki−1, j+ 1)= kζ(k+ 1),
where the empty sum means zero.
(Ohno and Okuda [873, p. 3030])
49. Kamano [624] investigated the following multiple zeta function:
ζn (s1, . . . ,sn ; a)=
∑
05m1<···<mn
1
(m1+ a)s1 · · ·(mn+ a)sn(
a> 0; (m1, . . . ,mn) ∈ Zn; (s1, . . . ,sn) ∈ Cn
)
,
(a)
where Z denotes the set of integers. The special case n= 1 of ζn (s1, . . . ,sn ; a) in
(a) reduces to the Hurwitz (or generalized) zeta function ζ(s,a). Also ζn (s1, . . . ,sn ; 1)
The Zeta and Related Functions 239
becomes the Euler-Zagier multiple zeta function denoted by ζn (s1, . . . ,sn ). Matsumoto
[806] proved the analytic continuation of a more general class of multiple zeta functions,
including (a) as a special case. Kamano [624] presented three kinds of limiting values of
ζn (s1, . . . ,sn ; a) in (a) at nonpositive integers.
Show that
ζ ′n (0 ; a)=
(−1)n−1
(n− 1)!
n−1∏
k=1
(
k+ a−
1
2
)
log
0(a)
√
2π
(n ∈ N),
where ζ ′n(s ; a)=
∂
∂s ζn(s ; a), ζn(s ; a) := ζn (s, . . . ,s ; a) in (a) and an empty sum is under-
stood to be nil.
(Kamano [624, Theorem 3])
50. Show that
zγ (z)=
∞∑
k=2
(−1)k
Li(z)
k
(|z|5 1),
where γ (z) denotes the generalized-Euler-constant function, defined by
γ (z)=
∞∑
n=1
zn−1
(
1
n
− log
n+ 1
n
)
(|z|5 1)
=
1∫
0
1∫
0
1− x
(1− xyz)(− log xy)
dxdy (C \ [1,∞)).
(Sondow and Hadjicostas [1053, Theorem 1])
51. Let {αn}n∈N0 be a positive sequence, such that the following infinite series:
∞∑
n=0
e−αnt
converges for any t ∈ R+. Then, for the generalized Hurwitz–Lerch Zeta function
8
(ρ1,...,ρp,σ1,...,σq)
λ1,...,λp;µ1,...,µq
(z,s,a),
defined by 2.6(44), show that
8
(ρ1,...,ρp,σ1,...,σq)
λ1,...,λp;µ1,...,µq
(z,s,a)=
1
0(s)
∞∑
n=0
∞∫
0
t s−1 e−(a−α0+αn)t
(
1− e−(αn+1−αn)t
)
· p9
∗
q
 (λ1,ρ1), . . . , (λp,ρp);
(µ1,σ1), . . . , (µq,σq);
ze−t
 dt (min{<(a),<(s)}> 0),
provided that each member exists, p9∗q being the Fox-Wright hypergeometric function
defined in Problem 64 (Chapter 1).
(Srivastava et al. [1107])
240 Zeta and q-Zeta Functions and Associated Series and Integrals
52. For the generalized Hurwitz–Lerch Zeta function
8
(ρ1,...,ρp,σ1,...,σq)
λ1,...,λp;µ1,...,µq
(z,s,a),
defined by 2.6(44), derive the following extension of the fractional derivative formulas,
such as the one given by 2.6(32):
Dν−τz
{
zν−1 8
(ρ1,...,ρp,σ1,...,σq)
λ1,...,λp;µ1,...,µq
(zκ ,s,a)
}
=
0(ν)
0(τ)
zτ−1 8
(ρ1,...,ρp,κ,σ1,...,σq,κ)
λ1,...,λp,ν;µ1,...,µq,τ
(zκ ,s,a)
(
<(ν) > 0; κ > 0
)
.
(Srivastava et al. [1107])
53. Show that each of the following integral representations holds true:
8
(ρ1,...,ρp,σ1,...,σq)
λ1,...,λp;µ1,...,µq
(z,s,a)=
1
0(s)
∞∫
0
t s−1 e−at p9
∗
q
 (λ1,ρ1), . . . , (λp,ρp);
(µ1,σ1), . . . , (µq,σq);
ze−t
dt
(
min{<(a),<(s)}> 0
)
and
8
(ρ1,...,ρp,σ1,...,σq)
λ1,...,λp;µ1,...,µq
(z,s,a)=
q∏
j=1
0
(
µj
)
p∏
j=1
0
(
λj
)
·
1
2π i
∫
L
0(−ξ) {0(ξ + a)}s
p∏
j=1
0
(
λj+ ρjξ
)
{0(ξ + a+ 1)}s
q∏
j=1
0
(
µj+ σjξ
) (−z)ξ dξ (|arg(−z)|< π),
where the path of integration L is a Mellin-Barnes type contour in the complex
ξ -plane, which starts at the point −i∞ and terminates at the point i∞
(
i :=
√
−1
)
with
indentations, if necessary, in such a manner as to separate the poles of 0(−ξ) from the
poles of 0
(
λj+ ρjξ
)
( j= 1, . . . ,p).
(Srivastava et al. [1107])
54. For a given sequence {an}n∈N0 , let theoperator 1 be defined by
10 an := an and 1
j an :=1
j−1 an−1
j−1 an+1 =
j∑
k=0
(−1)k
(
j
k
)
an+k ( j ∈ N).
Then derive the following analytic continuation formula for the Riemann Zeta function
ζ(s) for all s ∈ C \ {1}:
ζ(s)=
1
1− 21−s
∞∑
j=0
1j 1−s
2j+1
.
The Zeta and Related Functions 241
Also, deduce the following special case, which was used in Hardy’s celebrated proof of the
Riemann functional equation 2.3(11):
ζ(s)=
1
1− 21−s
[
1
2
+
1
2
∞∑
n=1
(−1)n−1
(
1
ns
−
1
(n+ 1)s
)] (
<(s) >−1; s 6= 1
)
.
(Cf. Sondow [1048])
55. For the extended Fermi-Dirac function 2v(s;x), defined in terms of the Weyl fractional
integral operator Wsx+ by
2v(s;x) :=Wsx+[ϑ(t;v)]=
1
0(s)
∞∫
0
t s−1 ϑ(x+ t;v)dt
=
1
0(s)
∞∫
x
(t− x)s−1 ϑ(t;v)dt
(
ϑ(t;v) :=
e−vt
et + 1
; <(s) > 0; x = 0; <(v) >−1
)
,
show that
2v(µ+ ν;x)=
1
0(µ)
∞∫
0
tµ−1 2v(ν;x+ t)dt =
1
0(ν)
∞∫
0
tν−1 2v(µ;x+ t)dt
(
min{<(µ),<(ν)}> 0; <(v) >−1
)
and
2v(s;x)=
∞∑
n=0
2v(s− n;0)
(−x)n
n!
(
s 6= n+ 1 (n ∈ N0); x = 0; <(v) >−1
)
,
provided that each member of these assertions exist.
(See, for details, Srivastava et al. [1093])
56. For the extended Bose-Einstein function 2v(s;x), defined in terms of the Weyl fractional
integral operator Wsx+, by
9v(s;x) :=Wsx+[ψ(t;v)]=
1
0(s)
∞∫
0
t s−1 ψ(x+ t;v)dt
=
1
0(s)
∞∫
x
(t− x)s−1 ψ(t;v)dt
(
ψ(t;v) :=
e−vt
et − 1
; <(s) > 1; x = 0; <(v) >−1
)
,
242 Zeta and q-Zeta Functions and Associated Series and Integrals
show that
9v(s;x)= q
−s
q∑
j=1
9 v+j−q
q
(s;qx)
(
<(s) > 1; x = 0; <(v) >−1
)
.
(See, for details, Srivastava et al. [1093])
57. For the extended Fermi-Dirac function 2v(s;x) and extended Bose-Einstein function
2v(s;x) (see Problems 55 and 56 above), derive each of the following identities:
22v(s;x)=92v(s;x)− 2
1−s 9v(s,2x)
(
<(s) > 1; x = 0; <(v) >−1
)
,
2v+1(s,x)= 2
−s
[
9 v
2
(s;2x)−9 v+1
2
(s;2x)
] (
<(s) > 1; x = 0; <(v) >−1
)
and
2v+1(s;x)+2v(s;x)= (v+ 1)
−s e−(v+1)x
(
<(s) > 0; x = 0; <(v) >−1
)
.
(See, for details, Srivastava et al. [1093])
58. For the Apostol-Genocchi polynomials G(α)n (x;λ) (λ ∈ C), defined by 1.8(58), derive each
of the following explicit series representations:
Gn
(
p
q
;e2π iξ
)
=
2 · n!
(2qπ)n
{ q∑
j=1
ζ
(
n,
2ξ + 2j− 1
2q
)
exp
[(
n
2
−
(2ξ + 2j− 1)p
q
)
π i
]
+
q∑
j=1
ζ
(
n,
2j− 2ξ − 1
2q
)
exp
[(
−
n
2
+
(2j− 2ξ − 1)p
q
)
π i
]}
(
n ∈ N \ {1}; p ∈ Z; q ∈ N; ξ ∈ R
)
,
G(α)n
(
p
q
;e2π iξ
)
=
2n
e2π iξ + 1
G(α−1)n−1
(
e2π iξ
)
+
n∑
k=2
2 · k!
(2qπ)k
(
n
k
)
G(α−1)n−k (e
2π iξ )
·
{ q∑
j=1
ζ
(
k,
2ξ + 2j− 1
2q
)
exp
[(
k
2
−
(2ξ + 2j− 1)p
q
)
π i
]
+
q∑
j=1
ζ
(
k,
2j− 2ξ − 1
2q
)
exp
[(
−
k
2
+
(2j− 2ξ − 1)p
q
)
π i
]}
(
n ∈ N \ {1}; p ∈ Z; q ∈ N; ξ ∈ R \3
(
3 :=
{
k+
1
2
: k ∈ Z
})
; α ∈ C
)
The Zeta and Related Functions 243
and
G(α)n
(
p
q
;e2π i
)
=−
n (−2)α
e2π iξ + 1
B(α−1)n−1
(
−e2π iξ)
)
−
n∑
k=2
k!
(2qπ)k
(
n
k
)
B(α−1)n−k
(
−e2π iξ
)
·

q∑
j=1
ζ
(
k,
2ξ + 2j− 1
2q
)
exp
[(
k
2
−
(2ξ + 2j− 1)p
q
)
π i
]
+
q∑
j=1
ζ
(
k,
2j− 2ξ − 1
2q
)
exp
[(
−
k
2
+
(2j− 2ξ − 1)p
q
)
π i
](
n ∈ N \ {1}; p ∈ Z; q ∈ N; ξ ∈ R \3
(
3 :=
{
k+
1
2
: k ∈ Z
})
; α ∈ C
)
in terms of the Hurwitz (or generalized) Zeta function ζ(s,a).
(See Luo and Srivastava [791, p. 5713, Theorem 4; p. 5715, Theorems 6 and 7])
This page intentionally left blank 
3 Series Involving Zeta Functions
The main purpose of this chapter is to present a rather extensive collection of closed-
form sums of series involving the Zeta functions. Many of these summation formulas
will find their applications in Chapter 5 (especially Section 5.3) in the evaluations of
the determinants of the Laplacians for the n-dimensional sphere Sn with the standard
metric.
We begin this chapter by presenting an interesting historical introduction to the
remarkably widely investigated subject of closed-form summation of series involv-
ing the Zeta functions. Among the various methods and techniques used in the vast
literature on the subject, we give here reasonably detailed accounts of those using
the binomial theorem, generating functions, multiple Gamma functions and hyper-
geometric identities. The last section (Section 3.6) deals with other methods, based
(for example) on the Weierstrass canonical product form for the Gamma function and
higher-order derivatives of the Gamma function.
3.1 Historical Introduction
A rather classical (over two centuries old) theorem of Christian Goldbach (1690–
1764), which was stated in a letter dated 1729 from Goldbach to Daniel Bernoulli
(1700–1782), was revived in 1986 by Shallit and Zikan [1022] as the following
problem:∑
ω∈S
(ω− 1)−1 = 1, (1)
where S denotes the set of all nontrivial integer kth powers, that is,
S :=
{
nk | n,k ∈ N \ {1}
}
. (2)
In terms of the Riemann Zeta function ζ(s), defined by 2.3(1), Goldbach’s theorem (1)
assumes the elegant form (cf. Shallit and Zikan [1022, p. 403]):
∞∑
k=2
{ζ(k)− 1} = 1 (3)
Zeta and q-Zeta Functions and Associated Series and Integrals. DOI: 10.1016/B978-0-12-385218-2.00003-7
c© 2012 Elsevier Inc. All rights reserved.
http://dx.doi.org/10.1016/B978-0-12-385218-2.00003-7
246 Zeta and q-Zeta Functions and Associated Series and Integrals
or, equivalently,
∞∑
k=2
F (ζ(k))= 1, (4)
where F(x) := x− [x] denotes the fractional part of x ∈ R. As a matter of fact, it is
fairly straightforward to observe also that
∞∑
k=2
(−1)kF (ζ(k))=
1
2
, (5)
∞∑
k=1
F (ζ(2k))=
3
4
and
∞∑
k=1
F (ζ(2k+ 1))=
1
4
. (6)
Another remarkable result involving the Riemann’s ζ -function is the following
series representation for ζ(3):
ζ(3)=
π2
7
{
1− 4
∞∑
k=1
ζ(2k)
(2k+ 1)(2k+ 2)22k
}
(7)
or, equivalently,
ζ(3)=−
4π2
7
∞∑
k=0
ζ(2k)
(2k+ 1)(2k+ 2)22k
, (8)
since ζ(0)=− 12 (see Equation 2.3(10)). The series representation (7) is contained in
a 1772 paper by Leonhard Euler (1707–1783) (see, e.g., Ayoub [81, pp. 1084–1085]).
It was rediscovered by Ramaswami [966] and (more recently) by Ewell [436]. (See
also Srivastava [1072, p. 7, Equation (2.23)]), where Euler’s result (7) was reproduced
actually from the work of Ramaswami [966]). Numerous further series representations
for ζ(3), which are analogous to (7) or (8), can be found in the works of Wilton [1233],
Zhang and Williams [1250, 1251], Cvijović and Klinowski [351], Chen and Srivastava
[252], Srivastava [1082–1084, 1086], Srivastava and Tsumura [1111, 1112] and others
(cf., e.g., Tsumura [1169, p. 384] and Ewell [438, p. 1004]); see also Berndt [122]).
A considerably large variety of methods were used in the aforementioned works,
as well as in the works of (among others) Jensen [607], Dinghas [385], Srivastava
[1068, 1069, 1072, 1073], Klusch [673], Choi et al. [308], Choi and Srivastava [288,
289, 292, 294, 295], Choi and Seo [285] and Kanemitsu et al. [628], dealing with
summation of series involving the Riemann ζ -function and its such extensions as the
(Hurwitz’s) generalized Zeta function ζ(s,a), defined by 2.2(1).
Series Involving Zeta Functions 247
3.2 Use of the Binomial Theorem
Landau’s formula (cf. Landau [729, p. 274, Eq. (3)]; see also Titchmarsh [1151, p. 33,
Eq. (2.14.1)]):
ζ(s)= 1+
1
s− 1
−
∞∑
k=1
(s)k
(k+ 1)!
{ζ(s+ k)− 1} (1)
is capable of providing the analytic continuation of ζ(s) over the whole complex
s-plane; here, (s)k denotes the Pochhammer symbol, defined by 1.1(5).
Another formula, which can also be used in a similar way, is attributed to
Ramaswami [966, p. 166] (see also Titchmarsh [1151, p. 33, Eq. (2.14.2)]):
(1− 21−s)ζ(s)=
∞∑
k=1
(s)k
k!
ζ(s+ k)
2s+k
. (2)
Motivated by these well-known results (1) and (2) in the theory of the Riemann zeta
function ζ(s), Singh and Verma [1033] derived the following infinite series involving
ζ(s):
ζ(s)=
1
2
+
1
s− 1
+
1
2
∞∑
k=1
(−1)k−1
k · (s)k+1
(k+ 2)!
ζ(s+ k+ 1) (<(s) < 1) (3)
and
ζ(s)= 1+
1
2s+1
s+ 3
s− 1
+
1
2
∞∑
k=1
(−1)k−1
k ·(s)k+1
(k+ 2)!
{ζ(s+ k+ 1)− 1}. (4)
Theproofs of (3) and (4) by Singh and Verma [1033, Sections 2 and 3] depend
rather heavily on the integral representation (see Titchmarsh [1151, p. 14, Eq. (2.1.4)]):
ζ(s)=
1
2
+
1
s− 1
− s
∞∫
1
(
x− [x]−
1
2
)
dx
xs+1
(<(s) >−1). (5)
Srivastava [1069] gave relatively simple proofs of (3) and (4), without using the
integral representation (5). He was, thus, led naturally to an interesting unification (and
generalization) of (3) and (4), involving the Hurwitz (or generalized) Zeta function
ζ(s,a) (see Section 2.2). The elementary techniques employed in Srivastava [1069] are
shown to apply also to the derivation of numerous other results for ζ(s,a), including,
for example, some useful analogues of (1) and (2).
248 Zeta and q-Zeta Functions and Associated Series and Integrals
The derivation of Srivastava’s unification (and generalization) of (3) and (4) is
based simply upon the familiar binomial expansion (see Equation 1.5(22)):
∞∑
k=0
(λ)k
k!
zk = (1− z)−λ (|z|< 1). (6)
Indeed, it follows readily from (6) and the definition 2.2(1) of the generalized Zeta
function ζ(s,a) that (cf., e.g., Wilton [1233, p. 90, Eq. (1)])
∞∑
k=0
(λ)k
k!
ζ(λ+ k,a) tk = ζ(λ,a− t) (|t|< |a|), (7)
which holds true, by the principle of analytic continuation of ζ(s,a), for all values of
λ 6= 1. Observe that (7) is the special case of 2.5(33) when x ∈ Z.
Now, replace the summation index in (7) by k+ 2, set λ= s− 1 and divide both
sides of the resulting equation by t2. We, thus, observe from (7) that
∞∑
k=0
(s− 1)k+2
(k+ 2)!
ζ(s+ k+ 1,a)tk = {ζ(s− 1,a− t)− ζ(s− 1,a)}t−2
− (s− 1)ζ(s,a)t−1 (0< |t|< |a|).
(8)
Differentiating both sides of (8) with respect to t and noticing from the definition
1.1(5) and 2.2(18) that
(s− 1)k+2 = (s− 1)(s)k+1 and
∂
∂t
{ζ(s− 1,a− t)} = (s− 1)ζ(s,a− t), (9)
we have
∞∑
k=1
k ·(s)k+1
(k+ 2)!
ζ(s+ k+ 1,a)tk−1 = {ζ(s,a− t)+ ζ(s,a)}t−2
−
2
s− 1
{ζ(s− 1,a− t)− ζ(s− 1,a)}t−3 (0< |t|< |a|),
(10)
which can also be obtained as the special case of 2.5(34) when x ∈ Z.
For t =−1, (10) readily yields the desired unification (and generalization) of (3)
and (4) in the form (see Srivastava [1069, p. 49, Eq. (2.6)]):
ζ(s,a)= a−s
(
1
2
+
a
s− 1
)
+
1
2
∞∑
k=1
(−1)k−1
k ·(s)k+1
(k+ 2)!
ζ(s+ k+ 1,a), (11)
provided that the series converges.
Series Involving Zeta Functions 249
In the special case of (11) when a= 1, the series converges, if <(s) < 1, and we
immediately obtain (3). Furthermore, in view of 2.3(2) and the special case of 2.3(9)
when n= 1, the formula (11) with a= 2 is precisely the same as the known result (4).
It may be of interest to remark here that alternative proofs of the well-known results
(1) and (2), based upon the integral representation 2.3(30), were given by Menon
[823]. As a matter of fact, both (1) and (2) can also be deduced fairly easily from (7).
Replacing the summation index n in (7) by n+ 1 and setting λ= s− 1, we have
∞∑
k=0
(s)k
(k+ 1)!
ζ(s+ k,a)tk+1
=
1
s− 1
{ζ(s− 1,a− t)− ζ(s− 1,a)} (|t|< |a|),
(12)
where we have made use of the first identity in (9).
By virtue of the definition 2.2(1), (12) with t = 1 assumes the form:
ζ(s,a)=
(a− 1)1−s
s− 1
−
∞∑
k=1
(s)k
(k+ 1)!
ζ(s+ k,a), (13)
which, in view of 2.3(9) when n= 1, yields Landau’s formula (1) for a= 2. Con-
versely, (12) with t = 12 (and s replaced by s+ 1) or (7) with t =
1
2 (and λ= s) simi-
larly yields
ζ(s,2a− 1)− 21−sζ(s,a)=
∞∑
k=1
(s)k
k!
ζ(s+ k,a)
2s+k
, (14)
which leads us immediately to Ramaswami’s formula (2) upon setting a= 1.
For t =−1, (12) yields
ζ(s,a)=
a1−s
s− 1
+
∞∑
k=1
(−1)k−1
(s)k
(k+ 1)!
ζ(s+ k,a), (15)
and (12) with t =− 12 (and s replaced by s+ 1) or (7) with t =−
1
2 (and λ= s) gives
ζ(s,2a)− 21−sζ(s,a)=−
∞∑
k=1
(−1)k−1
(s)k
k!
ζ(s+ k,a)
2s+k
. (16)
Formulas (15) with a= 2 and (16) with a= 1 provide interesting analogues of (1)
and (2), respectively; in fact, this indicated analogue of (2) [that is, (16) with a= 1]
was also given by Ramaswami [966, p. 166]. It is not difficult to deduce (11) as a
natural consequence of (15).
Numerous other consequences of the general results (7), (11) and (12) can be
deduced by suitably specializing the parameter t in a manner detailed above.
250 Zeta and q-Zeta Functions and Associated Series and Integrals
In terms of the binomial coefficient 1.1(40), (6) and (7) with a= 2 are written in
the following equivalent forms:
∞∑
k=0
(
λ+ k− 1
k
)
tk = (1− t)−λ (|t|< 1) (17)
and
∞∑
k=0
(
λ+ k− 1
k
)
{ζ(λ+ k)− 1}tk = ζ(λ,2− t) (|t|< 2), (18)
respectively. This last identity (18) is, in fact, equivalent to (cf. Ramanujan [964, p. 78,
Eq. (15)]); Apostol [61, p. 240, Eq. (7)])
∞∑
k=0
(
λ+ k− 1
k
)
ζ(λ+ k)tk = ζ(λ,1− t) (|t|< 1). (19)
For fixed λ 6= 1, the series in (18) and (19) converge absolutely for |t|< 2 and
|t|< 1, respectively. Thus, by the principle of analytic continuation, the formulas (18)
and (19) are valid for all values of λ 6= 1.
Formula (18) provides a unification (and generalization) of 3.1(4) and 3.1(5) and,
indeed, also of a fairly large number of other summation formulas scattered in the
literature. For example, in view of the relationships 2.3(1) and 2.2(4), (18) with t = 1
gives us (cf. Hansen [531, p. 356, Eq. (54.4.1)])
∞∑
k=1
(
λ+ k− 1
k
)
{ζ(λ+ k)− 1} = 1, (20)
which generalizes 3.1(4), and a special case of (18) when t =−1 yields (cf. Hansen
[531, p. 356, Eq. (54.4.2)])
∞∑
k=1
(−1)k−1
(
λ+ k− 1
k
)
{ζ(λ+ k)− 1} = 2−λ, (21)
which generalizes 3.1(5).
Several additional consequences of the general summation formulas (18) and (19)
are worthy of note. First, replace the summation index k in (18) by k+ 1 and set
λ= s− 1, so that
∞∑
k=0
(
s+ k− 1
k+ 1
)
{ζ(s+ k)− 1}tk+1 = ζ(s− 1,2− t)− ζ(s− 1)+ 1 (|t|< 2),
(22)
Series Involving Zeta Functions 251
which, for t = 1, reduces immediately to the following alternative form of (20)
∞∑
k=0
(
s+ k− 1
k+ 1
)
{ζ(s+ k)− 1} = 1. (23)
Now, it follows from the definition 1.1(20) of the binomial coefficient that(
s+ k− 1
k+ 1
)
=
(s− 1)(s)k
(k+ 1)!
(k ∈ N0). (24)
Thus, the formula (23) can easily be rewritten as Landau’s formula (1).
For t =−1, (22) readily yields (see Srivastava [1073, p. 131, Eq. (2.4)]):
ζ(s)= 1+
1
2s−1
1
s− 1
+
∞∑
k=1
(−1)k−1
(s)k
(k+ 1)!
{ζ(s+ k)− 1}, (25)
which provides an interesting companion of Landau’s formula (1).
Setting t = 12 in (22) and making use of 2.2(4) with a=
1
2 and n= 1, we obtain
another series representation for ζ(s) (see Srivastava [1072, p. 5, Eq. (2.13)]):
ζ(s)=
2s− 1
2s− 2
+
1
2s− 2
∞∑
k=1
(s)k
k!2k
{ζ(s+ k)− 1}. (26)
In their special cases when s= 2, (1) and (25) reduce simply to the summation
formulas (4) and (5), respectively, whereas (26) similarly yields the elegant sum:
∞∑
k=2
k− 1
2k
{ζ(k)− 1} =
π2
8
− 1. (27)
Next, we turn to the summation formula (19), which (for λ= s− 1 and with k
replaced by k+ 1) assumes the form:
∞∑
k=0
(
s+ k− 1
k+ 1
)
ζ(s+ k)tk+1 = ζ(s− 1,1− t)− ζ(s− 1) (|t|< 1). (28)
In view of the identities in 2.3(1) and (24), a special case of (28) when t = 12 readily
yields Ramaswami’s result (2).
In case we add (19) to itself (with t replaced by −t), we obtain the summation
formula (cf. Hansen [531, p. 357, Eq. (54.6.3)]):
∞∑
k=0
(
λ+ 2k− 1
2k
)
ζ(λ+ 2k)t2k =
1
2
{ζ(λ− 1,1− t)+ ζ(λ,1+ t)} (|t|< 1),
(29)
252 Zeta and q-Zeta Functions and Associated Series and Integrals
whereas a similar subtraction yields
∞∑
k=0
(
λ+ 2k
2k+ 1
)
ζ(λ+ 2k+ 1)t2k+1 =
1
2
{ζ(λ,1− t)− ζ(λ,1+ t)} (|t|< 1). (30)
Various interesting special cases of (29) and (30) are given in the literature. In
particular, the special cases of (29) when t = 12 , t =
1
3 and t =
1
6 were considered by
Ramaswami [966, p. 167, Eqs. (1), (3), and (4)], who also gave a special case of (30)
when t = 12 (cf. Ramaswami [966, p. 167, Eq. (2)]), and by Apostol [61], who proved
various generalizations of Ramaswami’s results.
By assigning suitable numerical values to the variable s in some of the aforemen-
tioned special cases of (29) and(30), Ramaswami [966] also evaluated a number of
special sums, including various formulas in Section 3.4 and 3.1(7). Wilton [1233,
p. 92] showed that
∞∑
k=1
(2k− 1)!
(2k+ 3)!
ζ(2k)
22k
=
ζ(3)
2π2
+
1
12
logπ −
11
72
(31)
and
∞∑
k=1
(2k)!
(2k+ 3)!
ζ(2k+ 1)
22k+1
=
3
8
−
1
6
logπ −
1
4
γ +
1
12
log2+
ζ ′(2)
π2
. (32)
Many closed-form evaluations of series involving the Zeta function were investi-
gated systematically by Srivastava [1072], who gave a rather detailed discussion about
their derivations and relationships (if any) among them. As a matter of fact, most of
the series identities considered by Srivastava [1072] will be included in our extensive
list given in Section 3.4.
We record, here, the following identity, involving the Bernoulli polynomials:
Bn+1(a+ t)
n+ 1
=
n∑
k=0
(
n
k
)
Bk+1(a)
k+ 1
tn−k+
tn+1
n+ 1
(n ∈ N0), (33)
which is, in fact, an immediate consequence of the known formula 1.7(13).
It follows from 2.2(15) that the Laurent series expansion of ζ(s,a) at s= 1 is of the
form
ζ(s,a)=
1
s− 1
−ψ(a)+
∞∑
n=1
cn (s− 1)
n, (34)
where the coefficients cn are constants to be determined and the Psi (or Digamma)
function ψ(z) is given in Section 1.3.
Series Involving Zeta Functions 253
Start with the following known identity for the Hurwitz-Lerch Zeta function
8(z,s,a), defined by 2.5(1) (see 2.5(33) and 2.6(9)):
∞∑
k=0
(s)k
k!
8(z,s+ k, a) tk =8(z,s, a− t) (|t|< |a|; s 6= 1), (35)
where (λ)n denotes the Pochhammer symbol, defined by 1.1(5). The general series
identity, which is to be proven in this section, is contained in Theorem 3.1 below (see
[297]).
Theorem 3.1 For every nonnegative integer n,
∞∑
k=2
8(z,k, a)
tn+k
(k)n+1
=
(−1)n
n!
[8′(z,−n, a− t)−8′(z,−n, a)]
+
n∑
k=1
(−1)n+k
n!
(
n
k
)[
(Hn−Hn−k) 8(z,k− n, a)−8
′(z,k− n, a)
]
tk
+ [Hn L1(z,a)−L2(z,a)]
tn+1
(n+ 1)!
(|t|< |a|; |z|< 1; n ∈ N0),
(36)
where Hn denotes the harmonic numbers, defined by
Hn :=
n∑
j=1
1
j
, (37)
it being understood (as elsewhere in this paper) that an empty sum is nil,
8′(z,s,a)=
∂
∂s
8(z,s,a),
and L1(z,a) and L2(z,a) are defined by
L1(z,a) := lim
s→−n
{(s+ n)8(z,s+ n+ 1,a)} (38)
and
L2(z,a) := lim
s→−n
{
8(z,s+ n+ 1,a)+ (s+ n)8′(z,s+ n+ 1,a)
}
. (39)
Before proceeding to prove Theorem 3.1, we show that the assertion (36) of Theo-
rem 3.1 is equivalent to that of Theorem 3.2 below, by making use of the elementary
identity:
1
(k)n+1
=
1
k(k+ 1) · · ·(k+ n)
:=
n∑
j=0
Anj
k+ j
, (40)
254 Zeta and q-Zeta Functions and Associated Series and Integrals
where
Anj =
(−1)j
n!
(
n
j
)
(0 5 j 5 n; j, n ∈ N0);
the combinatorial identities:
n∑
j=0
(−1)j
j+ 1
(
n
j
)
=
1
n+ 1
(n ∈ N0), (41)
n∑
j=1
(−1)j+1
j+ 1
(
n
j
)
Hj =
Hn
n+ 1
(n ∈ N0), (42)
and
n∑
j=n−k+1
(−1)j+1
j− (n− k)
(
n
j
)
= (−1)n+k
(
n
k
)
(Hn−Hn−k) (0 5 k 5 n; n, k ∈ N0),
(43)
the special case k = n of which is recorded in [505, p. 5, Entry 0.155]; the familiar
result:
n∑
j=k
(−1)j
(
j
k
)(
n
j
)
= 0 (0 5 k 5 n− 1; k ∈ N0; n ∈ N), (44)
and some rather simple manipulations, using such elementary series identities as the
ones involved in
n∑
j=0
j−1∑
k=0
Aj,k =
n−1∑
k=0
n∑
j=k+1
Aj,k (45)
and
n∑
j=0
j∑
k=0
Aj,k =
n∑
k=0
n∑
j=k
Aj,k, (46)
where {Aj,k}( j,k ∈ N0) is a suitably bounded double sequence.
Theorem 3.2 For every nonnegative integer n,
∞∑
k=2
8(z,k,a)
k+ n
tk+n =
n∑
k=0
(
n
k
)
8′(z,−k,a− t) tn−k−
n−1∑
k=0
8(z,−k,a)
n− k
tn−k
(47)
− [Hn L1(z,a)+L2(z,a)]
tn+1
n+ 1
−8′(z,−n,a) (|t|< |a|; |z|< 1; n ∈ N0),
where L1(z,a) and L2(z,a) are given as in (38) and (39), respectively.
Series Involving Zeta Functions 255
It follows easily from the definitions 1.1(19) and 1.3(1) that
d
dz
{(z)n} = (z)n[ψ(z+ n)−ψ(z)]. (48)
Proof of Theorem 3.1 Upon transposing the first n+ 2 terms from k = 0 to k = n+ 1
in (35) to the right-hand side, if we divide both sides of the resulting equation by s+ n,
we get
∞∑
k=n+2
(s)n (s+ n+ 1)k−n−18(z,s+ k, a)
tk
k!
=
gn(z,s, t,a)
s+ n
(|t|< |a|; s 6= 1; n ∈ N0),
(49)
where, for convenience,
gn(z,s, t,a) :=8(z,s, a− t)−
n+1∑
k=0
(s)k
k!
8(z,s+ k, a) tk. (50)
Now we shall show that
lim
s→−n
gn(z,s, t,a)= 0. (51)
Indeed, we observe that
8(z,−n, a− t)=
∞∑
k=0
(k+ a− t)n zk
=
∞∑
k=0

n∑
j=0
(
n
j
)
(k+ a)n−j (−t)j
 zk
=
n∑
j=0
(
n
j
)
(−t)j
∞∑
k=0
zk
(k+ a)−n+j
=
n∑
j=0
(−n)j
j!
8(z,−n+ j, a) tj,
from which (51) follows easily. Thus, by l’Hôpital’s rule, we have
lim
s→−n
gn(z,s, t,a)
s+ n
= lim
s→−n
∂
∂s
{gn(z,s, t,a)}. (52)
256 Zeta and q-Zeta Functions and Associated Series and Integrals
Next, by appealing to (50) and (48), we find that
∂
∂s
{gn(z,s, t,a)} =8
′(s, a− t)−8′(s, a)
−
n∑
k=1
h(z,s,a,k)
tk
k!
− h(z,s,a,n+ 1)
tn+1
(n+ 1)!
,
(53)
where, for convenience,
h(z,s,a,k) := (s)k [{ψ(s+ k)−ψ(s)}8(z,s+ k, a)+8
′(z,s+ k, a)]. (54)
In view of 1.3(7), (54) yields
lim
s→−n
h(z,s,a,k)=−(−n)k
[
(Hn−Hn−k) 8(z,k− n, a)−8
′(z,k− n, a)
]
(k = 1, . . . ,n)
(55)
and
lim
s→−n
h(z,s,a,n+ 1)
= lim
s→−n
 n∑
j=0
(s)n
s+ j
(s+ n)8(z,s+ n+ 1, a)+ (s)n+18′(z,s+ n+ 1, a)
,
(56)
which, upon writing
n∑
j=0
(s)n
s+ j
=
n−1∑
j=0
(s)n
s+ j
+
(s)n
s+ n
,
reduces to the following form:
lim
s→−n
h(z,s,a,n+ 1)= (−n)n [−Hn L1(z,a)+L2(z,a)], (57)
where L1(z,a) and L2(z,a) are given by (38) and (39), respectively.
Conversely, upon taking the limit on the left-side of (49) as s→−n, we have
lim
s→−n
∞∑
k=n+2
(s)n (s+ n+ 1)k−n−18(z,s+ k,a)
tk
k!
= (−n)n
∞∑
k=2
8(z,k, a)
tk+n
(k)n+1
.
(58)
Finally, upon taking the limit on both sides of (49) as s→−n and considering (52),
(53), (54), (57) and (58), we are led immediately to the desired series identity (36).
This evidently completes our proof of Theorem 3.1.
Series Involving Zeta Functions 257
Applications of Theorems 3.1 and 3.2
We first note that 8(z,s,a) is an entire function of s when |z|< 1, converges for
<(s) > 0 when |z| = 1 and z 6= 1 and reduces to ζ(s,a) when z= 1. So, when |z|5 1
and z 6= 1, we have
L1(z,a)= 0 and L2(z,a)=8(z,1,a)= a
−1
2F1(1, a ; 1+ a ; z). (59)
It follows from 2.5(26) that
L1(1,a)= lim
s→−n
(s+ n)ζ(s+ n+ 1,a)= 1, (60)
since ζ(s,a) has a simple pole at s= 1 with residue 1. From (34), we get
L2(1,a)=−ψ(a). (61)
Upon setting z= 1 in (36) and (47) and considering 2.5(26), (60) and (61), we have
certain families of series associated with the generalized (or Hurwitz) Zeta function
ζ(s,a):
Corollary 3.3 For every nonnegative integer n,
∞∑
k=2
ζ(k, a)
(k)n+1
tn+k =
(−1)n
n!
[ζ ′(−n, a− t)− ζ ′(−n, a)]
+
n∑
k=1
(−1)n+k
n!
(
n
k
)[
(Hn−Hn−k) ζ(k− n, a)− ζ
′(k− n, a)
]
tk
+ [Hn+ψ(a)]
tn+1
(n+ 1)!
(|t|< |a|; n ∈ N0),
(62)
where Hn denotes the harmonic numbers defined by (37), it being understood that
ζ ′(s,a)=
∂
∂s
ζ(s,a).
Corollary 3.4 For every nonnegative integer n,
∞∑
k=2
ζ(k,a)
k+ n
tk+n =
n∑
k=0
(
n
k
)
ζ ′(−k,a− t) tn−k−
n−1∑
k=0
ζ(−k,a)
n− k
tn−k
+ [ψ(a)−Hn]
tn+1
n+ 1
− ζ ′(−n,a) (|t|< |a|; n ∈ N0),
(63)
where Hn denotes the harmonic numbers defined by (37), it being understood that
ζ ′(s,a)=
∂
∂s
ζ(s,a).
258 Zeta and q-Zeta Functions and Associated Series and Integrals
Infinite sums of the type occurring in (62) can also be evaluated, in a markedly dif-
ferent manner, in terms of such higher transcendental functions as the multiple Gamma
functions (see, for details, [628, p. 10, Theorem 3.1]).
The series identities (62) and (63) can include a number of formulas involving
various classes of earlier closed-form evaluations of series associated with the Zeta
function as its special cases. For example, we can get the following (easily derivable)
consequence of the summation formula (63):
∞∑
k=1
ζ(2k,a)
k+ n
t2k+2n =
2n∑
k=0
(
2n
k
)[
ζ ′(−k,a− t)+ (−1)k ζ ′(−k,a+ t)
]
t2n−k
(64)
−
n−1∑
`=0
ζ(−2`,a)
n− `
t2n−2`− 2ζ ′(−2n,a) (n ∈ N0; |t|< |a|);
∞∑
k=1
ζ(2k,a)
2k+ 2n+ 1
t2k+2n+1
=
1
2
2n+1∑
k=0
(
2n+ 1
k
)[
ζ ′(−k,a− t)+ (−1)kζ ′(−k,a+ t)
]
t2n+1−k (65)
−
n∑`=0
ζ(−2`,a)
2n+ 1− 2`
t2n+1−2` (n ∈ N0; |t|< |a|);
∞∑
k=1
ζ(2k+ 1,a)
2k+ 2n+ 1
t2k+2n+1
=
1
2
2n∑
k=0
(
2n
k
) [
ζ ′(−k,a− t)− (−1)k ζ ′(−k,a+ t)
]
t2n−k
(66)
−
n∑
`=1
ζ(1− 2`,a)
2n− 2`+ 1
t2n−2`+1−
t2n+1
2n+ 1
[ψ(2n+ 1)−ψ(a)+ γ ]
(n ∈ N0; |t|< |a|);
∞∑
k=1
ζ(2k+ 1,a)
k+ n+ 1
t2k+2n+2
=
2n+1∑
k=0
(
2n+ 1
k
) [
ζ ′(−k,a− t)− (−1)kζ ′(−k,a+ t)
]
t2n+1−k
(67)
−
n∑
`=1
ζ(1− 2`,a)
n− `+ 1
t2n+2−2`−
t2n+2
n+ 1
[ψ(2n+ 2)−ψ(a)+ γ ]
− 2ζ ′(−2n− 1,a) (n ∈ N0; |t|< |a|).
Series Involving Zeta Functions 259
Setting a= 1 and t =−1 in (63) and using various identities given in the previous
chapters, for example, 2.2(4) and 1.3(7), we obtain
∞∑
k=2
(−1)k
k+ n
ζ(k)=
n−1∑
`=0
(−1)`
(
n
`
)
ζ ′(−`)−
n−1∑
`=0
(−1)`
rn− `
ζ(−`)
+
1
n+ 1
(γ +Hn) (n ∈ N0).
(68)
Likewise, setting a= 1 and t = 1/2 in (63), we get
∞∑
k=2
ζ(k)
(k+ n)2k
=
n−1∑
`=0
[(
n
`
)
log2−
2`
n− `
]
ζ(−`)+
n−1∑
`=0
(
n
`
) (
1− 2`
)
ζ ′(−`)
(69)
−
γ +Hn
2(n+ 1)
+ ζ(−n) log2+
(
1− 2n+1
)
ζ ′(−n) (n ∈ N0).
The special cases of (64) and (65) when a= 1 and t = 1/2 are written here:
∞∑
k=1
ζ(2k)
(k+ n)22k
=
1
2n
− log2+ (−1)n (2n)!
1− 22n+1
(2π)2n
ζ(2n+ 1)
+
n−1∑
k=1
(−1)k
(
2n
2k
)
(2k)!
(2π)2k
(
1− 22k
)
ζ(2k+ 1) (n ∈ N).
(70)
∞∑
k=1
ζ(2k)
(2k+ 2n− 1)22k
=
1
2(2n− 1)
−
1
2
log2
+
1
2
n−1∑
k=1
(−1)k
(
2n− 1
2k
)
(2k)!
(2π)2k
(
1− 22k
)
ζ(2k+ 1) (n ∈ N),
(71)
the special case n= 1 of which yields 3.4(519).
Setting a= 1 and t = 1/2 in (66), we get
∞∑
k=1
ζ(2k+ 1)
(2k+ 2n+ 1)22k
=
n∑
`=1
22`−1 B2`
`(2n− 2`+ 1)
− log2
n∑
`=1
(
2n
2`− 1
)
B2`
`
+ 2
n∑
`=1
(
2n
2`− 1
)(
1− 22`−1
)
ζ ′(1− 2`)
−
γ +H2n
2n+ 1
+ log2
2n∑
`=0
(−1)`
(
2n
`
)
(n ∈ N0),
(72)
260 Zeta and q-Zeta Functions and Associated Series and Integrals
where we have assumed that
2n∑
`=0
(−1)`
(
2n
`
)
:= 1 (n= 0).
Of course, it is true that
2n∑
`=0
(−1)`
(
2n
`
)
= 0 (n ∈ N). (73)
Setting a= 1 and t = 1/2 in (67), we obtain
∞∑
k=1
ζ(2k+ 1)
(k+ n+ 1)22k
=−2log2
n∑
`=0
(
2n+ 1
2`+ 1
)
B2`+2
`+ 1
−
γ +H2n+1
n+ 1
+
n∑
`=1
22`−1
`(n− `+ 1)
B2`+ 4
n−1∑
`=0
(
1− 22`+1
) (2n+ 1
2`+ 1
)
ζ ′(−2`− 1)
+ 4
(
1− 22n+2
)
ζ ′(−2n− 1) (n ∈ N0).
(74)
If we set n= 1 in (74), we get
∞∑
k=1
ζ(2k+ 1)
(k+ 2) · 22k
=−
2
3
−
γ
2
−
29
30
log2+ 12logA+ 30 logC. (75)
Upon setting a= 1 in (65) and then t = 12 in the resulting equation, if we use the
various identities for ζ(s) and ζ(s,a) given in Chapter 2, in particular, together with
the known identity 2.3(22), we obtain (see [256, p. 262, Eq. (14)])
∞∑
k=1
ζ(2k)
(2k+ 2`− 1)22k
=
1
2(2`− 1)
−
1
2
log2
+
1
2
`−1∑
k=1
(−1)k
(
2`− 1
2k
)
(2k)!
(2π)2k
(
1− 22k
)
ζ(2k+ 1) (` ∈ N),
(76)
whose special case `= 1 is also recorded in 3.4(519).
As a consequence, the following integral can be evaluated as a finite series involv-
ing the Riemann Zeta function (see [256, Eq. (18)]):
π
2∫
0
θ2` cotθ dθ =
(π
2
)2` [
log2+ (−1)` (2`)!
22`+1− 1
(2π)2`
ζ(2`+ 1)
+
`−1∑
k=1
(−1)k+1
(
2`
2k
)
(2k)!
(2π)2k
(
1− 22k
)
ζ(2k+ 1)
]
(` ∈ N).
(77)
Series Involving Zeta Functions 261
We note that the integral in (77) was evaluated earlier as an infinite series involving
the Riemann Zeta function (see [505, p. 428, Entry 3.748.2]; also see [256]).
We deduce another interesting identity by suitably combining the special cases of
(62) when a= 1 and a= 2. By applying 2.2(4), 1.3(4) and 1.3(7), we, thus, find that
∞∑
k=1
tn+k
(k)n+1
=
(−1)n+1
n!
(1− t)n log(1− t)
+
n∑
k=1
(−1)n+k
n!
(
n
k
)
(Hn−Hn−k) t
k (|t|< 1; n ∈ N0),
(78)
which, in the special case when n= 2, is a known result recorded (for example) by
Hansen [531, p. 37, Entry (5.7.40); p. 74, Entry (5.16.26)].
We shall deal with other interesting applications of Corollary 3.3 in Section 3.4.
3.3 Use of Generating Functions
Adamchik and Srivastava [11] proposed and developed a novel method of evalua-
tion of the sums of series involving Zeta functions by using generating functions. The
key ingredients in their approach happen to include the familiar integral representa-
tions 2.2(21) and 2.3(30). This method has already been implemented in Mathematica
(Version 3.0).
Consider the sum:
�(a)=
∞∑
k=1
f (k)ζ(k+ 1,a) (<(a) > 0), (1)
where the sequence { f (n)}∞n=1 is assumed to possess a generating function:
F(t)=
∞∑
k=1
f (k)
tk
k!
(2)
and
f (n)= O
(
1
n
)
(n→∞). (3)
Upon replacing the Hurwitz (or generalized) Zeta function in (1) by its integral rep-
resentation given by 2.2(21) with s= k+ 1, if we invert the order of summation and
integration, we obtain
�(a)=
∞∫
0
F(t)
e−(a−1)t
et− 1
dt, (4)
where we have also made use of the generating function (2).
262 Zeta and q-Zeta Functions and Associated Series and Integrals
Thus, the problem of summation of series of the type (1) has been reduced for-
mally to that of integration in (4). Although the integral in (4) appears to be fairly
involved for symbolic integration, it may be possible to reduce it to 2.2(21) or 2.3(30)
(or another known integral), especially when F(t) is a power, exponential, trigonome-
tric or hyperbolic function.
The first example (Proposition 3.5 below) would illustrate how this technique actu-
ally works (see Adamchik and Srivastava [11, p. 135, Proposition 1]):
Proposition 3.5 Let n be a positive integer. Then,
∞∑
k=1
(−1)k
k
{ζ(nk)− 1} = log
n−1∏
j=0
0
(
2− (−1)(2j+1)/n
). (5)
Remark 1 In Proposition 3.5, as also in Equations (7), (8) and (10) below, it is tacitly
assumed that
−1= elog(−1),
where logz denotes the principal branch of the logarithmic function in the complex
z-plane for which
−π < arg(z)5 π (z 6= 0).
Proof. Denote, for convenience, the left-hand side of the summation formula (5) by
2(n). Then, in view of the special case of 2.3(9) when n= 1, we can apply the integral
representation 2.2(21) with s= nk and a= 2. Upon inverting the order of summation
and integration, which can be justified by the absolute convergence of the series and
the integral involved, we, thus, find that
2(n)= n
∞∫
0
dt
tet(et− 1)
∞∑
k=1
(−tn)k
0(nk+ 1)
. (6)
By recognizing the series in (6) as a trigonometric function of order n (see, for details,
Erdélyi et al. [422, Section 18.2]) or, alternatively, by using an easily derivable special
case of a known result in Hansen [531, p. 207, Entry (10.49.1)], we have
∞∑
k=1
(−tn)k
0(nk+ 1)
=−1+
1
n
n∑
j=1
exp
(
t(−1)(2j+1)/n
)
. (7)
Substituting from (7) into (6) and inverting the order of summation and integration
once again, we obtain
2(n)= lim
r→1−
{
−n
∞∫
0
dt
tr et(et− 1)
+
n∑
j=1
∞∫
0
exp
(
−
[
1− (−1)(2j+1)/n
]
t
)
tr(et− 1)
dt
}
,
(8)
Series Involving Zeta Functions 263
where the parameter r < 1 is inserted with a view to providing convergence of the
integrals at their lower terminal t = 0.
Finally, we evaluate each integral in (8) by means of 2.2(21) and proceed to the
limit as r→ 1−. Indeed, by making use of the known behavior of ζ(1− r,a) near
r = 1 for fixed a (cf., e.g., Erdélyi et al. [421, p. 26]):
ζ(1− r,a)∼
1
2
− a+ (1− r) log
(
0(a)
√
2π
)
(r→ 1−; a fixed), (9)
we arrive at the right-hand side of the assertion (5).
In precisely the same manner, we can prove a mild generalization of Propo-
sition 3.5, which we state here as (see Adamchik and Srivastava [11, p. 136,
Proposition 2]) �
Proposition 3.6 Let n be a positive integer. Then,
∞∑
k=1
(−1)k
k
ζ(nk,a)=−n log0(a)
+ log
n−1∏
j=0
0
(
a− (−1)(2j+1)/n
) (<(a)= 1). (10)
Next we prove (see Adamchik and Srivastava [11, p. 136, Proposition 3])
Proposition 3.7 Let n be a positive integer. Then, in terms of the Bernoulli numbers
Bn, defined by 1.7(2), and the Stirling numbers S(n,k), defined by 1.6(14),
∞∑
k=2
(−1)k {ζ(k)− 1} kn =−1+
1− 2n+1
n+ 1
Bn+1
−
n∑
k=1
(−1)k k! ζ(k+ 1)S(n+ 1,k+ 1).
(11)
Proof. Making use of the integral representation 2.2(21) with a= 2, if we invert the
order of summation and integration and then evaluate the resulting integral and sum,we find that
3(n) :=
∞∑
k=2
(−1)k {ζ(k)− 1} kn
= lim
r→1
(
r
d
dr
)n
{r [ψ(r+ 2)−ψ(2)]} .
(12)
Since(
r
d
dr
)r
{r f (r)} =
n∑
k=0
S(n+ 1,k+ 1) f (k)(r) rk+1 (13)
264 Zeta and q-Zeta Functions and Associated Series and Integrals
and (cf. Equation 1.2(53))
ψ (k)(3)= (−1)k k!
{
1+ 2−k−1− ζ(k+ 1)
}
(k ∈ N), (14)
we find from (12) that
3(n)=−1−
n∑
k=1
(−1)k k! ζ(k+ 1)S(n+ 1,k+ 1)
+
n∑
k=0
(−1)k k! 2−k−1S(n+ 1,k+ 1).
(15)
To complete the proof of Proposition 3.7, we, thus, need to show that
n∑
k=0
(−1)k k! 2−k−1 S(n+ 1,k+ 1)=
1− 2n+1
n+ 1
Bn+1 (16)
in terms of the Bernoulli numbers Bn, defined by 1.7(2). In fact, it is known that
(Hansen [531, p. 351, Entry (52.2.36)])
n∑
k=1
(−1)k k! 2−k S(n,k)=
2
n+ 1
(
1− 2n+1
)
Bn+1, (17)
which, in view of the recurrence relation 1.6(18) with k replaced by k+ 1, yields the
desired identity (16).
Similarly, we can prove (see Adamchik and Srivastava [11, p. 137, Proposition 4])
�
Proposition 3.8 Let n be a positive integer. Then,
∞∑
k=2
{ζ(k)− 1} kn = 1+
n∑
k=1
k! ζ(k+ 1)S(n+ 1,k+ 1). (18)
The following list provides further summation formulas, involving series of Zeta
functions, which can be derived by applying the foregoing technique (see Adamchik
and Srivastava [11, pp. 138–139]):
∞∑
k=2
k2
k+ 1
{ζ(k)− 1} =
3
2
−
γ
2
+
π2
6
−
1
2
log(2π); (19)
∞∑
k=2
k2
k+ 1
{ζ(2k+ 1)− 1} =
9
16
− γ + log2−
1
2
ζ(3); (20)
Series Involving Zeta Functions 265
∞∑
k=1
{ζ(4k)− 1} =
7
8
−
π
4
cothπ; (21)
∞∑
k=1
(−1)k {ζ(4k)− 1} = 1+
π
2
√
2
sin
(
π
√
2
)
+ sinh
(
π
√
2
)
cos
(
π
√
2
)
− cosh
(
π
√
2
) ; (22)
∞∑
k=1
{ζ(4k)− 1} z4k =
3z4− 1
2(z4− 1)
−
πz
4
{cot(πz)+ coth(πz)} (|z|< 2), (23)
which contains both (21) and (22) as limiting cases;
∞∑
k=1
{ζ(2k)− 1}sink =−
1
2
cot
(
1
2
)
+
π
2
sin
(
1
2
)
sin
(
2π cos
(
1
2
))
− cos
(
1
2
)
sinh
(
2πsin
(
1
2
))
cos
(
2πcos
(
1
2
))
− cosh
(
2πsin
(
1
2
)) ;
(24)
∞∑
k=1
(
p+ k
k
)
ζ(p+ k+ 1,a)zk =
(−1)p
p!
{
ψ (p)(a)−ψ (p)(a− z)
}
(p ∈ N; <(a) > 0; |z|< |a|);
(25)
∞∑
k=2
k
2k
ζ
(
k+ 1,
3
4
)
= 8G (26)
and
∞∑
k=2
k
2k
ζ
(
k+ 1,
5
4
)
= 8(1−G), (27)
where G denotes Catalan’s constant, defined by 1.3(16).
Remark 2 Since the right-hand side of the relationship 1.3(53) (with n! replaced by
0(n+ 1)) is well-defined for n ∈ C \ {−1}, the summation formula (25) may be put
in a slightly more general form (cf. Wilton [1233]; see also Srivastava [1072, p. 137,
Equation (6.6)]):
∞∑
k=0
(
s+ k− 1
k
)
ζ(s+ k,a)zk = ζ(s,a− z)
(
s ∈ C \ {1}; a 6= Z−0 ; |z|< |a|
)
, (28)
which, in view of 1.1(20), is equivalent to 3.2(7) for λ= s and t = z.
266 Zeta and q-Zeta Functions and Associated Series and Integrals
The foregoing method has been implemented in Mathematica (Version 3.0), as we
remarked at the beginning of this section.
Series Involving Polygamma Functions
The generalized harmonic numbers H(s)n , defined by (cf. Graham et al. [507])
H(s)n :=
n∑
k=1
1
ks
(n ∈ N, s ∈ C), (29)
which, by virtue of 2.3(9), can be written in the form:
H(s)n = ζ(s)− ζ(s,n+ 1) (<(s) > 1; n ∈ N). (30)
In view of the relationships 1.2(53) and (30), the foregoing techniques can be applied
also to series involving Polygamma functions and generalized harmonic numbers. We
first state (see Adamchik and Srivastava [11, p. 139, Proposition 5])
Proposition 3.9 Let p be a positive integer. Then, in terms of the function 8(z,s,a),
defined by 2.5(1),
∞∑
k=1
ψ (p)(a+ k)zk =
(−1)p+1 p!
ap+1
+
ψ (p)(a+ 1)
1− z
+
(−1)p p! z2
1− z
8(z,p+ 1,a+ 1)
(
|z|< 1; a 6= Z−0 \ {0}
)
.
(31)
Proof. Denoting, for convenience, the left-hand side of the summation formula (31)
by 4(z), it is not difficult to find from 1.3(53) and 2.2(21) that
4(z)= (−1)p+1
∞∫
0
tp e−(a−1)t
et− 1
(
∞∑
k=1
zk e−kt
)
dt
=−
1∫
0
τ a−1(logτ)p
(1− τ)(1− zτ)
dτ (<(a) > 0).
Upon evaluating this last integral and waiving the restriction on the parameter a by
appealing to the principle of analytic continuation, we complete the proof of Proposi-
tion 3.9.
Next, we turn to a family of linear harmonic sums:
Sp,q :=
∞∑
n=1
H(p)n
nq
, (32)
Series Involving Zeta Functions 267
which were discussed extensively by Flajolet and Salvy [454]. By applying (30),
2.2(21) and 2.3(30), it is not difficult to prove (see Adamchik and Srivastava [11,
p. 140, Proposition 6]) �
Proposition 3.10 Let Sp,q be defined by (32). Then,
Sp,q = ζ(p)ζ(q)+
(−1)p
(p− 1)!
1∫
0
(log t)p−1 Liq(t)
dt
1− t
(33)
and
Sp,q = ζ(p+ q)−
(−1)q
(q− 1)!
1∫
0
(log t)q−1 Lip(t)
dt
1− t
. (34)
Remark 3 In view of the following symmetry relation in Flajolet and Salvy [454]:
Sp,q+Sq,p = ζ(p)ζ(q)+ ζ(p+ q), (35)
the integral representations (33) and (34) are essentially the same. Although, in gen-
eral, the integrals occurring in (33) and (34) cannot be evaluated in closed forms,
many interesting particular cases of linear harmonic sums would follow from Propo-
sition 3.10.
Series Involving Polylogarithm Functions
We shall derive, among other results, an integral representation for the Khintchine
constant K0, which arises in the measure theory of continued fractions. Every positive
irrational number µ can, indeed, be written uniquely as a simple continued fraction as
follows:
µ= a0+
a1
a2+
a3
a4+
·· ·
an
an+1+
·· · , (36)
that is, with a0 a non-negative integer and with all other aj (j ∈ N) positive integers.
The Gauss-Kuz’min distribution (cf., e.g., Khintchine [647]) predicts that the density
of occurrence of some chosen positive integer k in the continued fraction (36) of a
random real number is given by
Prob{an = k} = −log2
(
1−
1
(k+ 1)2
)
. (37)
And, making use of the Gauss-Kuz’min distribution involving (37), Khintchine [647]
showed that, for almost all irrational numbers, the limiting geometric mean of the
268 Zeta and q-Zeta Functions and Associated Series and Integrals
positive integer elements aj (j ∈ N) of the relevant continued fraction exists and equals
K0 :=
∞∏
k=1
{
1+
1
k(k+ 2)
}log2 k
=
∞∏
k=1
{
k
log2
(
1+ 1k(k+2)
)}
.
(38)
An interesting explicit representation of the Khintchine constant K0 in terms of Poly-
logarithm functions was proven recently by Bailey et al. [85, p. 422]:
log(K0) log2= (log2)
2
+Li2
(
−
1
2
)
+
1
2
∞∑
n=2
(−1)n Li2
(
4
n2
)
. (39)
If we set
L(n) :=
∞∑
n=2
(−1)n Li2
(
4
n2
)
, (40)
replace the Dilogarithm function by its series representation given by 2.4(3), change
the order of summation and evaluate the inner sum, we shall obtain
L(n)= 2
∞∑
k=1
ζ(2k)
k2
−
∞∑
k=1
{ζ(2k)− 1}
4k
k2
. (41)
It seems very unlikely that the sums occurring in (41) can be evaluated in terms of
well-known functions. Nevertheless, by noting that
∞∑
k=1
tk
k2
ζ(2k)=
1∫
0
dt
t
∞∑
k=1
tk
k
ζ(2k) (42)
and evaluating the inner sum by the method illustrated in the preceding sections, we
find that
∞∑
k=1
tk
k2
ζ(2k)= log
(
π
√
t csc
(
π
√
t
))
. (43)
Combining (41), (42) and (43), we have
L(n) :=
∞∑
n=2
(−1)n Li2
(
4
n2
)
=
1∫
0
dt
t
log
(
π
√
t cot
(
π
√
t
)
1− 4t
)
, (44)
Series Involving Zeta Functions 269
which leads us immediately to the following integral representation for the Khintchine
constant K0 (see Adamchik and Srivastava [11, p. 141, Eq. (4.10)]):
log(K0) log2=
π2
12
+
(log2)2
2
+
π∫
0
log(t|cot t|)
dt
t
. (45)
Other sums involving the Polylogarithm function, which were also evaluated by
Adamchik and Srivastava [11], are given below.
∞∑
k=1
(
−
1
2
)k
Lik
(
z2
)
= 1− z−1 arctanhz; (46)
∞∑
k=1
(
1
2
)k
Lik
(
z2
)
=
1
2
z log
(
1+ z
1− z
)
. (47)
Since arctanh
(
1
2
)
= log3, both (46) and (47) can be expressed in terms of log3 when
z= 12 .
3.4 Use of Multiple Gamma Functions
We provide a rather extensive list of evaluations of series involving the Zeta functions,
by making use of the Gamma and multiple Gamma functions. Although our list in this
section includes some of the series identities considered in previous sections or earlier
works, we chooseto present, here, all of the identities evaluated by the use of multiple
Gamma functions for the sake of completeness.
Evaluation by Using the Gamma Function
Upon setting n= 0 in 3.2(35) and applying 2.2(17), we obtain the familiar result (cf.
Whittaker and Watson [1225, p. 276], Hansen [531, p. 358, Entry (54.11.1)], and
Srivastava [1072, p. 18]):
∞∑
k=2
ζ(k,a)
tk
k
= log0(a− t)− log0(a)+ tψ(a) (|t|< |a|), (1)
which, upon replacing t by −t, yields
∞∑
k=2
(−1)kζ(k,a)
tk
k
= log0(a+ t)− log0(a)− tψ(a) (|t|< |a|). (2)
270 Zeta and q-Zeta Functions and Associated Series and Integrals
By adding and subtracting, we find from (1) and (2) that
∞∑
k=1
ζ(2k,a)
t2k
k
= log0(a+ t)+ log0(a− t)− 2log0(a) (|t|< |a|); (3)
∞∑
k=1
ζ(2k+ 1,a)
t2k+1
2k+ 1
=
1
2
{log0(a− t)− log0(a+ t)}+ tψ(a) (|t|< |a|).
(4)
Differentiating both sides of (1), (2), (3) and (4) with respect to t, we have
∞∑
k=2
ζ(k,a)tk−1 =−ψ(a− t)+ψ(a) (|t|< |a|); (5)
∞∑
k=2
(−1)kζ(k,a)tk−1 = ψ(a+ t)−ψ(a) (|t|< |a|); (6)
∞∑
k=1
ζ(2k,a)t2k−1 =
1
2
{ψ(a+ t)−ψ(a− t)} (|t|< |a|); (7)
∞∑
k=1
ζ(2k+ 1,a)t2k =−
1
2
{ψ(a+ t)+ψ(a− t)}+ψ(a) (|t|< |a|). (8)
Setting a= 1 in (1) through (8) and using 1.3(4) and 2.3(2), we obtain
∞∑
k=2
ζ(k)
tk
k
= log0(1− t)− γ t (|t|< 1); (9)
∞∑
k=2
(−1)kζ(k)
tk
k
= log0(1+ t)+ γ t (|t|< 1); (10)
∞∑
k=1
ζ(2k)
t2k
k
= log0(1+ t)+ log0(1− t) (|t|< 1); (11)
∞∑
k=1
ζ(2k+ 1)
t2k+1
2k+ 1
=
1
2
{log0(1− t)− log0(1+ t)}− γ t (|t|< 1); (12)
∞∑
k=2
ζ(k)tk−1 =−ψ(1− t)− γ (|t|< 1); (13)
Series Involving Zeta Functions 271
∞∑
k=2
(−1)kζ(k)tk−1 = ψ(1+ t)+ γ (|t|< 1); (14)
∞∑
k=1
ζ(2k)t2k−1 =
1
2
{ψ(1+ t)−ψ(1− t)} (|t|< 1); (15)
∞∑
k=1
ζ(2k+ 1)t2k =−
1
2
{ψ(1+ t)+ψ(1− t)}− γ (|t|< 1). (16)
Equations (11) and (15) can, in view of 1.1(12) and 1.3(9), be written in their
respective equivalent forms:
∞∑
k=1
ζ(2k)
t2k
k
= log
(
π t
sinπ t
)
(|t|< 1); (17)
∞∑
k=1
ζ(2k)t2k−1 =−
π
2
cot(π t)+
1
2t
(|t|< 1). (18)
If we use some elementary trigonometric identities, we obtain the following
trigonometric evaluations:
sin
(π
8
)
= sin
(
7π
8
)
=
1
2
√
2−
√
2;
(19)
sin
(
3π
8
)
= sin
(
5π
8
)
=
1
2
√
2+
√
2;
sin
(π
5
)
= sin
(
4π
5
)
= cos
(
3π
10
)
=
√
5−
√
5
2
√
2
;
(20)
sin
(
2π
5
)
= sin
(
3π
5
)
= cos
( π
10
)
=
√
5+
√
5
2
√
2
;
cos
(π
5
)
=−cos
(
4π
5
)
= sin
(
3π
10
)
=
√
5+ 1
4
;
(21)
cos
(
2π
5
)
=−cos
(
3π
5
)
= sin
( π
10
)
=
√
5− 1
4
;
cot
(π
5
)
=−cot
(
4π
5
)
=
√
1+
2
5
√
5; cot
(
2π
5
)
=−cot
(
3π
5
)
=
√
1−
2
5
√
5;
(22)
cot
( π
10
)
=−cot
(
9π
10
)
=
√
5+ 2
√
5; cot
(
3π
10
)
=−cot
(
7π
10
)
=
√
5− 2
√
5.
272 Zeta and q-Zeta Functions and Associated Series and Integrals
Taking the limit as t→ 1 on both sides of (10), we have the well-known result:
∞∑
k=2
(−1)k
ζ(k)
k
= γ. (23)
For suitable special values of the argument t, we can deduce the following series
identities from (9) through (18), by making use of such evaluations as (19) through
(22), as well as the ψ-function values listed in Section 1.3.
∞∑
k=2
ζ(k)
k ·2k
=
1
2
logπ −
γ
2
; (24)
∞∑
k=2
(−1)k
ζ(k)
k ·2k
=
γ
2
+
1
2
logπ − log2; (25)
∞∑
k=1
ζ(2k)
k ·22k
= log
(π
2
)
; (26)
∞∑
k=1
ζ(2k+ 1)
(2k+ 1)22k
= log2− γ ; (27)
∞∑
k=1
ζ(2k)
k ·32k
= log
(
2π
3
√
3
)
; (28)
∞∑
k=1
ζ(2k)
k
(
2
3
)2k
= log
(
4π
3
√
3
)
; (29)
∞∑
k=1
ζ(2k)
k ·24k
= log
(
π
2
√
2
)
; (30)
∞∑
k=1
ζ(2k)
k
(
3
4
)2k
= log
(
3π
2
√
2
)
; (31)
∞∑
k=1
ζ(2k)
k ·52k
= log
(
2
√
2π
5
√
5−
√
5
)
; (32)
∞∑
k=1
ζ(2k)
k
(
2
5
)2k
= log
(
4
√
2π
5
√
5+
√
5
)
; (33)
∞∑
k=1
ζ(2k)
k
(
3
5
)2k
= log
(
6
√
2π
5
√
5+
√
5
)
; (34)
∞∑
k=1
ζ(2k)
k
(
4
5
)2k
= log
(
8
√
2π
5
√
5+
√
5
)
; (35)
∞∑
k=1
ζ(2k)
k ·102k
= log

(√
5+ 1
)
π
10
 ; (36)
Series Involving Zeta Functions 273
∞∑
k=1
ζ(2k)
k
(
3
10
)2k
= log
3
(√
5− 1
)
π
10
 ; (37)
∞∑
k=1
ζ(2k)
k
(
7
10
)2k
= log
7
(√
5− 1
)
π
10
 ; (38)
∞∑
k=1
ζ(2k)
k
(
9
10
)2k
= log
9
(√
5+ 1
)
π
10
 ; (39)
∞∑
k=1
ζ(2k)
k ·62k
= log
(π
3
)
; (40)
∞∑
k=1
ζ(2k)
k
(
5
6
)2k
= log
(
5π
3
)
; (41)
∞∑
k=1
ζ(2k)
k ·26k
= log
(
π
4
√
2−
√
2
)
; (42)
∞∑
k=1
ζ(2k)
k
(
3
8
)2k
= log
(
3π
4
√
2+
√
2
)
; (43)
∞∑
k=1
ζ(2k)
k
(
5
8
)2k
= log
(
5π
4
√
2+
√
2
)
; (44)
∞∑
k=1
ζ(2k)
k
(
7
8
)2k
= log
(
7π
8
√
4+ 2
√
2
)
; (45)
∞∑
k=1
ζ(2k)
k ·122k
= log
π
(√
3+ 1
)
6
√
2
; (46)
∞∑
k=1
ζ(2k)
k
(
5
12
)2k
= log
5π
(√
3− 1
)
6
√
2
; (47)
∞∑
k=1
ζ(2k)
k
(
7
12
)2k
= log
7π
(√
3− 1
)
6
√
2
; (48)
∞∑
k=1
ζ(2k)
k
(
11
12
)2k
= log
11π
(√
3+ 1
)
6
√
2
; (49)
∞∑
k=2
(−1)k
ζ(k)
2k
= 1− log2; (50)
274 Zeta and q-Zeta Functions and Associated Series and Integrals
∞∑
k=2
ζ(k)
2k
= log2; (51)
∞∑
k=1
ζ(2k)
22k−1
= 1; (52)
∞∑
k=1
ζ(2k+ 1)
22k
= 2log2− 1; (53)
∞∑
k=2
(−1)k
ζ(k)
3k
= 1−
1
18
π
√
3−
1
2
log3; (54)
∞∑
k=2
ζ(k)
3k
=
1
2
log3−
1
18
π
√
3; (55)
∞∑
k=1
ζ(2k)
32k
=
1
2
−
1
18
π
√
3; (56)
∞∑
k=1
ζ(2k+ 1)
32k
=
3
2
(log3− 1); (57)
∞∑
k=2
(−1)kζ(k)
(
2
3
)k
= 1+
1
9
π
√
3− log3; (58)
∞∑
k=2
ζ(k)
(
2
3
)k
=
1
9
π
√
3+ log3; (59)
∞∑
k=1
ζ(2k)
(
2
3
)2k
=
1
2
+
1
9
π
√
3; (60)
∞∑
k=1
ζ(2k+ 1)
(
2
3
)2k
=
3
2
log3−
3
4
; (61)
∞∑
k=2
(−1)k
ζ(k)
22k
= 1−
1
8
π −
3
4
log2; (62)
∞∑
k=2
ζ(k)
22k
=
3
4
log2−
1
8
π; (63)
∞∑
k=1
ζ(2k)
24k
=
1
2
−
1
8
π; (64)
∞∑
k=1
ζ(2k+ 1)
24k
= 3log2− 2; (65)
∞∑
k=2
(−1)kζ(k)
(
3
4
)k
= 1+
3
8
π −
9
4
log2; (66)
Series Involving Zeta Functions 275
∞∑
k=2
ζ(k)
(
3
4
)k
=
3
8
π +
9
4
log2; (67)
∞∑
k=1
ζ(2k)
(
3
4
)2k
=
1
2
+
3
8
π; (68)
∞∑
k=1
ζ(2k+ 1)
(
3
4
)2k
= 3log2−
2
3
; (69)
∞∑
k=2
(−1)k
ζ(k)
5k
= 1−
π
10
√
1+
2
5
√
5−
1
4
log5−
√
5
10
log
1+
√
5
2
; (70)
∞∑
k=2
ζ(k)
5k
=−
π
10
√
1+
2
5
√
5+
1
4
log5+
√
5
10
log
1+
√
5
2
; (71)
∞∑
k=1
ζ(2k)
52k
=
1
2
−
π
10
√
1+
2
5
√
5; (72)
∞∑
k=1
ζ(2k+ 1)
52k
=−
5
2
+
5
4
log5+
√
5
2
log
1+
√
5
2
; (73)
∞∑
k=2
(−1)kζ(k)
(
2
5
)k
= 1−
π
5
√
1−
2
5
√
5−
1
2
log5+
√
5
5
log
1+
√
5
2
; (74)
∞∑
k=2
ζ(k)
(
2
5
)k
=−
π
5
√
1−
2
5
√
5+
1
2
log5−
√
5
5
log
1+
√
5
2
; (75)
∞∑
k=1
ζ(2k)
(
2
5
)2k
=
1
2
−
π
5
√
1−
2
5
√
5; (76)
∞∑
k=1
ζ(2k+ 1)
(
2
5
)2k
=−
5
4
+
5
4
log5−
√
5
2
log
1+
√
5
2
; (77)
∞∑
k=2
(−1)kζ(k)
(
3
5
)k
= 1+
3π
10
√
1−
2
5
√
5−
3
4
log5+
3
√
5
10
log
1+
√
5
2
; (78)
∞∑
k=2
ζ(k)
(
3
5
)k
=
3π
10
√
1−
2
5
√
5+
3
4
log5−
3
√
5
10
log
1+
√
5
2
; (79)
∞∑
k=1
ζ(2k)
(
3
5
)2k
=
1
2
+
3π
10
√
1−
2
5
√
5; (80)
∞∑
k=1
ζ(2k+ 1)
(
3
5
)2k
=−
5
6
+
5
4
log5−
√
5
2
log
1+
√
5
2
; (81)
∞∑
k=2
(−1)kζ(k)
(
4
5
)k
= 1+
2π
5
√
1+
2
5
√
5− log5−
2
√
5
5
log
1+
√
5
2
; (82)
276 Zeta and q-Zeta Functions and Associated Series and Integrals
∞∑
k=2
ζ(k)
(
4
5
)k
=
2π
5
√
1+
2
5
√
5+ log5+
2
√
5
5
log
1+
√
5
2
; (83)
∞∑
k=1
ζ(2k)
(
4
5
)2k
=
1
2
+
2π
5
√
1+
2
5
√
5; (84)
∞∑
k=1
ζ(2k+ 1)
(
4
5
)2k
=−
5
8
+
5
4
log5+
√
5
2
log
1+
√
5
2
; (85)
∞∑
k=2
(−1)k
ζ(k)
6k
= 1−
1
12
π
√
3−
1
4
log3−
1
3
log2; (86)
∞∑
k=2
ζ(k)
6k
=−
1
12
π
√
3+
1
4
log3+
1
3
log2; (87)
∞∑
k=1
ζ(2k)
62k
=
1
2
−
1
12
π
√
3; (88)
∞∑
k=1
ζ(2k+ 1)
62k
=−3+
3
2
log3+ 2log2; (89)
∞∑
k=2
(−1)kζ(k)
(
5
6
)k
= 1+
5
12
π
√
3−
5
4
log3−
5
3
log2; (90)
∞∑
k=2
ζ(k)
(
5
6
)k
=
5
12
π
√
3+
5
4
log3+
5
3
log2; (91)
∞∑
k=1
ζ(2k)
(
5
6
)2k
=
1
2
+
5
12
π
√
3; (92)
∞∑
k=1
ζ(2k+ 1)
(
5
6
)2k
=−
3
5
+
3
2
log3+ 2log2; (93)
∞∑
k=2
(−1)k
ζ(k)
23k
= 1−
π
16
(√
2+ 1
)
−
1
2
log2+
√
2
16
log
(
3− 2
√
2
)
; (94)
∞∑
k=2
ζ(k)
23k
=−
π
16
(√
2+ 1
)
+
1
2
log2−
√
2
16
log
(
3− 2
√
2
)
; (95)
∞∑
k=1
ζ(2k)26k
=
1
2
−
π
16
(√
2+ 1
)
; (96)
∞∑
k=1
ζ(2k+ 1)
26k
=−4+ 4log2−
√
2
2
log
(
3− 2
√
2
)
; (97)
∞∑
k=2
(−1)kζ(k)
(
3
8
)k
=1−
3π
16
(√
2−1
)
−
3
2
log2−
3
√
2
16
log
(
3−2
√
2
)
; (98)
Series Involving Zeta Functions 277
∞∑
k=2
ζ(k)
(
3
8
)k
=−
3π
16
(√
2− 1
)
+
3
2
log2+
3
√
2
16
log
(
3− 2
√
2
)
; (99)
∞∑
k=1
ζ(2k)
(
3
8
)2k
=
1
2
−
3π
16
(√
2− 1
)
; (100)
∞∑
k=1
ζ(2k+ 1)
(
3
8
)2k
=−
4
3
+ 4log2+
√
2
2
log
(
3− 2
√
2
)
; (101)
∞∑
k=2
(−1)kζ(k)
(
5
8
)k
= 1+
5π
16
(√
2− 1
)
−
5
2
log2−
5
√
2
16
log
(
3− 2
√
2
)
;
(102)
∞∑
k=2
ζ(k)
(
5
8
)k
=
5π
16
(√
2− 1
)
+
5
2
log2+
5
√
2
16
log
(
3− 2
√
2
)
; (103)
∞∑
k=1
ζ(2k)
(
5
8
)2k
=
1
2
+
5π
16
(√
2− 1
)
; (104)
∞∑
k=1
ζ(2k+ 1)
(
5
8
)2k
=−
4
5
+ 4log2+
√
2
2
log
(
3− 2
√
2
)
; (105)
∞∑
k=2
(−1)kζ(k)
(
7
8
)k
=1+
7π
16
(√
2+1
)
−
7
2
log2+
7
√
2
16
log
(
3−2
√
2
)
;
(106)
∞∑
k=2
ζ(k)
(
7
8
)k
=
7π
16
(√
2+ 1
)
+
7
2
log2−
7
√
2
16
log
(
3− 2
√
2
)
; (107)
∞∑
k=1
ζ(2k)
(
7
8
)2k
=
1
2
+
7π
16
(√
2+ 1
)
; (108)
∞∑
k=1
ζ(2k+ 1)
(
7
8
)2k
=−
4
7
+ 4log2−
√
2
2
log
(
3− 2
√
2
)
; (109)
∞∑
k=2
(−1)k
ζ(k)
10k
= 1−
π
20
√
5+ 2
√
5−
1
5
log2−
1
8
log5−
√
5
20
log
(
2+
√
5
)
;
(110)
∞∑
k=2
ζ(k)
10k
=−
π
20
√
5+ 2
√
5+
1
5
log2+
1
8
log5+
√
5
20
log
(
2+
√
5
)
; (111)
∞∑
k=1
ζ(2k)
102k
=
1
2
−
π
20
√
5+ 2
√
5; (112)
∞∑
k=1
ζ(2k+ 1)
102k
=−5+ 2log2+
5
4
log5+
√
5
2
log
(
2+
√
5
)
; (113)
278 Zeta and q-Zeta Functions and Associated Series and Integrals
∞∑
k=2
(−1)kζ(k)
(
3
10
)k
= 1−
3π
20
√
5− 2
√
5−
3
5
log2
−
3
8
log5+
3
√
5
20
log
(
2+
√
5
)
; (114)
∞∑
k=2
ζ(k)
(
3
10
)k
=−
3π
20
√
5− 2
√
5+
3
5
log2+
3
8
log5−
3
√
5
20
log
(
2+
√
5
)
;
(115)
∞∑
k=1
ζ(2k)
(
3
10
)2k
=
1
2
−
3π
20
√
5− 2
√
5; (116)
∞∑
k=1
ζ(2k+ 1)
(
3
10
)2k
=−
5
3
+ 2log2+
5
4
log5−
√
5
2
log
(
2+
√
5
)
; (117)
∞∑
k=2
(−1)kζ(k)
(
7
10
)k
= 1+
7π
20
√
5− 2
√
5−
7
5
log2
−
7
8
log5+
7
√
5
20
log
(
2+
√
5
)
; (118)
∞∑
k=2
ζ(k)
(
7
10
)k
=
7π
20
√
5− 2
√
5+
7
5
log2+
7
8
log5−
7
√
5
20
log
(
2+
√
5
)
;
(119)
∞∑
k=1
ζ(2k)
(
7
10
)2k
=
1
2
+
7π
20
√
5− 2
√
5; (120)
∞∑
k=1
ζ(2k+ 1)
(
7
10
)2k
=−
5
7
+ 2log2+
5
4
log5−
√
5
2
log
(
2+
√
5
)
; (121)
∞∑
k=2
(−1)kζ(k)
(
9
10
)k
= 1+
9π
20
√
5+ 2
√
5−
9
5
log2
−
9
8
log5−
9
√
5
20
log
(
2+
√
5
)
; (122)
∞∑
k=2
ζ(k)
(
9
10
)k
=
9π
20
√
5+ 2
√
5+
9
5
log2+
9
8
log5+
9
√
5
20
log
(
2+
√
5
)
;
(123)
∞∑
k=1
ζ(2k)
(
9
10
)2k
=
1
2
+
9π
20
√
5+ 2
√
5; (124)
∞∑
k=1
ζ(2k+ 1)
(
9
10
)2k
=−
5
9
+ 2log2+
5
4
log5+
√
5
2
log
(
2+
√
5
)
; (125)
Series Involving Zeta Functions 279
∞∑
k=2
(−1)k
ζ(k)
12k
= 1−
π
24
(
2+
√
3
)
+
√
3
24
log
(
7− 4
√
3
)
−
1
8
log3+
1
4
log2;
(126)
∞∑
k=2
ζ(k)
12k
=−
π
24
(
2+
√
3
)
−
√
3
24
log
(
7− 4
√
3
)
+
1
8
log3+
1
4
log2; (127)
∞∑
k=1
ζ(2k)
122k
=
1
2
−
π
24
(
2+
√
3
)
; (128)
∞∑
k=1
ζ(2k+ 1)
122k
=−6−
1
2
√
3log
(
7− 4
√
3
)
+
3
2
log3+ 3log2; (129)
∞∑
k=2
(−1)kζ(k)
(
5
12
)k
= 1−
5π
24
(
2−
√
3
)
−
5
√
3
24
log
(
7− 4
√
3
)
−
5
8
log3−
5
4
log2; (130)
∞∑
k=2
ζ(k)
(
5
12
)k
=−
5π
24
(
2−
√
3
)
+
5
√
3
24
log
(
7− 4
√
3
)
+
5
8
log3+
5
4
log2; (131)
∞∑
k=1
ζ(2k)
(
5
12
)2k
=
1
2
−
5π
24
(
2−
√
3
)
; (132)
∞∑
k=1
ζ(2k+ 1)
(
5
12
)2k
=−
6
5
+
√
3
2
log
(
7− 4
√
3
)
+
3
2
log3+ 3log2; (133)
∞∑
k=2
(−1)kζ(k)
(
7
12
)k
(134)
= 1+
7π
24
(
2−
√
3
)
−
7
√
3
24
log
(
7− 4
√
3
)
−
7
8
log3−
7
4
log2;
∞∑
k=2
ζ(k)
(
7
12
)k
(135)
=
7π
24
(
2−
√
3
)
+
7
√
3
24
log
(
7− 4
√
3
)
+
7
8
log3+
7
4
log2;
∞∑
k=1
ζ(2k)
(
7
12
)2k
=
1
2
+
7π
24
(
2−
√
3
)
; (136)
∞∑
k=1
ζ(2k+ 1)
(
7
12
)2k
=−
6
7
+
√
3
2
log
(
7− 4
√
3
)
+
3
2
log3+ 3log2; (137)
280 Zeta and q-Zeta Functions and Associated Series and Integrals
∞∑
k=2
(−1)kζ(k)
(
11
12
)k
(138)
= 1+
11π
24
(
2+
√
3
)
+
11
√
3
24
log
(
7− 4
√
3
)
−
11
8
log3−
11
4
log2;
∞∑
k=2
ζ(k)
(
11
12
)k
(139)
=
11π
24
(
2+
√
3
)
−
11
√
3
24
log
(
7− 4
√
3
)
+
11
8
log3+
11
4
log2;
∞∑
k=1
ζ(2k)
(
11
12
)2k
=
1
2
+
11π
24
(
2+
√
3
)
; (140)
∞∑
k=1
ζ(2k+ 1)
(
11
12
)2k
=−
6
11
−
√
3
2
log
(
7− 4
√
3
)
+
3
2
log3+ 3log2. (141)
In view of the various identities given in previous sections and the special case
of 2.3(9) when n= 1, by setting a= 2 in (1) through (8), we obtain the following
additional series identities:
∞∑
k=2
{ζ(k)− 1}
tk
k
= log0(2− t)+ (1− γ )t (|t|< 2); (142)
∞∑
k=2
(−1)k{ζ(k)− 1}
tk
k
= log0(2+ t)+ (γ − 1)t (|t|< 2); (143)
∞∑
k=1
{ζ(2k)− 1}
t2k
k
= log0(2+ t)+ log0(2− t) (|t|< 2); (144)
∞∑
k=1
{ζ(2k+ 1)− 1}
t2k+1
2k+ 1
(145)
=
1
2
{log0(2− t)− log0(2+ t)}+ (1− γ )t (|t|< 2);
∞∑
k=2
{ζ(k)− 1} tk−1 =−ψ(2− t)+ 1− γ (|t|< 2); (146)
∞∑
k=2
(−1)k(ζ(k)− 1)tk−1 = ψ(2+ t)+ γ − 1 (|t|< 2); (147)
∞∑
k=1
{ζ(2k)− 1} t2k−1 =
1
2
{ψ(2+ t)−ψ(2− t)} (|t|< 2); (148)
∞∑
k=1
{ζ(2k+ 1)− 1} t2k =−
1
2
{ψ(2+ t)+ψ(2− t)}+ 1− γ (|t|< 2); (149)
Series Involving Zeta Functions 281
∞∑
k=1
{ζ(2k)− 1}
t2k
k
= log
(
π t(1− t2)
sinπ t
)
(|t|< 2); (150)
∞∑
k=1
{ζ(2k)− 1} t2k−1 =−
π
2
cot(π t)+
3t2− 1
2t(t2− 1)
(|t|< 2). (151)
Similarly, as above, setting various suitable arguments in (142) through (151), we
readily obtain the following evaluations: for instance, by taking the limit in (143) as
t→ 2, we obtain
∞∑
k=2
(−1)k{ζ(k)− 1}
2k
k
= 2γ − 2+ log6; (152)
∞∑
k=2
(−1)k
ζ(k)− 1
k ·2k
=
γ
2
−
1
2
+ log
(
3
√
π
4
)
; (153)
∞∑
k=2
ζ(k)− 1
k ·2k
=
1
2
−
γ
2
+ log
(√
π
2
)
; (154)
∞∑
k=1
ζ(2k)− 1
k ·22k
= log
(
3π
8
)
; (155)
∞∑
k=1
ζ(2k+ 1)− 1
(2k+ 1)22k
= 1− γ + log
(
2
3
)
; (156)
∞∑
k=2
(−1)k
ζ(k)− 1
k
= γ − 1+ log2; (157)
∞∑
k=2
ζ(k)− 1
k
= 1− γ ; (158)
∞∑
k=1
ζ(2k)− 1
k
= log2; (159)
∞∑
k=1
ζ(2k+ 1)− 1
2k+ 1
= 1− γ −
1
2
log2; (160)
∞∑
k=2
(−1)k
ζ(k)− 1
k
(
3
2
)k
=
3γ
2
−
3
2
+ log
(
15
√
π
8
)
; (161)
∞∑
k=2
ζ(k)− 1
k
(
3
2
)k
=
3
2
−
3γ
2
+
1
2
logπ; (162)
∞∑
k=1
ζ(2k)− 1
k
(
3
2
)2k
= log
(
15π
8
)
; (163)
282 Zeta and q-Zeta Functions and Associated Series and Integrals
∞∑
k=1
ζ(2k+ 1)− 1
2k+ 1
(
3
2
)2k
= 1− γ +
1
3
log
(
8
15
)
; (164)
∞∑
k=1
ζ(2k)− 1
k ·32k
= log
(
16π
27
√
3
)
; (165)
∞∑
k=1
ζ(2k)− 1
k
(
2
3
)2k
= log
(
20π
27
√
3
)
; (166)
∞∑
k=1
ζ(2k)− 1
k
(
4
3
)2k
= log
(
56π
27
√
3
)
; (167)
∞∑
k=1
ζ(2k)− 1
k
(
5
3
)2k
= log
(
160π
27
√
3
)
; (168)
∞∑
k=1
ζ(2k)− 1
k ·24k
= log
(
15π
32
√
2
)
; (169)
∞∑
k=1
ζ(2k)− 1
k
(
3
4
)2k
= log
(
21π
32
√
2
)
; (170)
∞∑
k=1
ζ(2k)− 1
k
(
5
4
)2k
= log
(
45π
32
√
2
)
; (171)
∞∑
k=1
ζ(2k)− 1
k
(
7
4
)2k
= log
(
231π
32
√
2
)
; (172)
∞∑
k=1
ζ(2k)− 1
k ·52k
= log
(
48
√
2π
125
√
5−
√
5
)
; (173)
∞∑
k=1
ζ(2k)− 1
k
(
2
5
)2k
= log
(
84
√
2π
125
√
5+
√
5
)
; (174)
∞∑
k=1
ζ(2k)− 1
k
(
3
5
)2k
= log
(
96
√
2π
125
√
5+
√
5
)
; (175)
∞∑
k=1
ζ(2k)− 1
k
(
4
5
)2k
= log
(
72
√
2π
125
√
5−
√
5
)
; (176)
∞∑
k=1
ζ(2k)− 1
k
(
6
5
)2k
= log
(
132
√
2π
125
√
5−
√
5
)
; (177)
∞∑
k=1
ζ(2k)− 1
k
(
7
5
)2k
= log
(
336
√
2π
125
√
5+
√
5
)
; (178)
Series Involving Zeta Functions 283
∞∑
k=1
ζ(2k)− 1
k
(
8
5
)2k
= log
(
624
√
2π
125
√
5+
√
5
)
; (179)
∞∑
k=1
ζ(2k)− 1
k
(
9
5
)2k
= log
(
1008
√
2π
125
√
5−
√
5
)
; (180)
∞∑
k=1
ζ(2k)− 1
k ·102k
= log
99
(√
5+ 1
)
π
1000
 ; (181)
∞∑
k=1
ζ(2k)− 1
k
(
3
10
)2k
= log
273
(√
5− 1
)
π
1000
 ; (182)
∞∑
k=1
ζ(2k)− 1
k
(
7
10
)2k
= log
357
(√
5− 1
)
π
1000
 ; (183)
∞∑
k=1
ζ(2k)− 1
k
(
9
10
)2k
= log
171
(√
5+ 1
)
π
1000
 ; (184)
∞∑
k=1
ζ(2k)− 1
k
(
11
10
)2k
= log
231
(√
5+1
)
π
1000
 ; (185)
∞∑
k=1
ζ(2k)− 1
k
(
13
10
)2k
= log
897
(√
5− 1
)
π
1000
 ; (186)
∞∑
k=1
ζ(2k)− 1
k
(
17
10
)2k
= log
3213
(√
5− 1
)
π
1000
 ; (187)
∞∑
k=1
ζ(2k)− 1
k
(
19
10
)2k
= log
4959
(√
5+ 1
)
π
1000
 ; (188)
∞∑
k=1
ζ(2k)− 1
k ·62k
= log
(
35π
108
)
; (189)
∞∑
k=1
ζ(2k)− 1
k
(
5
6
)2k
= log
(
55π
108
)
; (190)
∞∑
k=1
ζ(2k)− 1
k
(
7
6
)2k
= log
(
91π
108
)
; (191)
∞∑
k=1
ζ(2k)− 1
k
(
11
6
)2k
= log
(
935π
108
)
; (192)
284 Zeta and q-Zeta Functions and Associated Series and Integrals
∞∑
k=1
ζ(2k)− 1
k ·26k
= log
(
63π
256
√
2−
√
2
)
; (193)
∞∑
k=1
ζ(2k)− 1
k
(
3
8
)2k
= log
(
165π
256
√
2+
√
2
)
; (194)
∞∑
k=1
ζ(2k)− 1
k
(
5
8
)2k
= log
(
195π
256
√
2+
√
2
)
; (195)
∞∑
k=1
ζ(2k)− 1
k
(
7
8
)2k
= log
(
105π
256
√
2−
√
2
)
; (196)
∞∑
k=1
ζ(2k)− 1
k
(
9
8
)2k
= log
(
153π
256
√
2−
√
2
)
; (197)
∞∑
k=1
ζ(2k)− 1
k
(
11
8
)2k
= log
(
627π
256
√
2+
√
2
)
; (198)
∞∑
k=1
ζ(2k)− 1
k
(
13
8
)2k
= log
(
1365π
256
√
2+
√
2
)
; (199)
∞∑
k=1
ζ(2k)− 1
k
(
15
8
)2k
= log
(
2415π
256
√
2−
√
2
)
; (200)
∞∑
k=1
ζ(2k)− 1
k ·122k
= log
(
143π(
√
3+ 1)
864
√
2
)
; (201)
∞∑
k=1
ζ(2k)− 1
k
(
5
12
)2k
= log
(
595π(
√
3− 1)
864
√
2
)
; (202)
∞∑
k=1
ζ(2k)− 1
k
(
7
12
)2k
= log
(
665π(
√
3− 1)
864
√
2
)
; (203)
∞∑
k=1
ζ(2k)− 1
k
(
11
12
)2k
= log
(
253π(
√
3+ 1)
864
√
2
)
; (204)
∞∑
k=1
ζ(2k)− 1
k
(
13
12
)2k
= log
(
325π(
√
3+ 1)
864
√
2
)
; (205)
∞∑
k=1
ζ(2k)− 1
k
(
17
12
)2k
= log
(
2465π(
√
3− 1)
864
√
2
)
; (206)
∞∑
k=1
ζ(2k)− 1
k
(
19
12
)2k
= log
(
4123π(
√
3− 1)
864
√
2
)
; (207)
Series Involving Zeta Functions 285
∞∑
k=1
ζ(2k)− 1
k
(
23
12
)2k
= log
(
8855π(
√
3+ 1)
864
√
2
)
; (208)
∞∑
k=2
(−1)k
ζ(k)− 1
2k
=
5
6
− log2; (209)
∞∑
k=2
ζ(k)− 1
2k
= log2−
1
2
; (210)
∞∑
k=1
ζ(2k)− 1
22k
=
1
6
; (211)
∞∑
k=1
ζ(2k+ 1)− 1
22k
=−
4
3
+ 2log2; (212)
∞∑
k=2
(−1)k{ζ(k)− 1} =
1
2
; (213)
∞∑
k=2
{ζ(k)− 1} = 1; (214)
∞∑
k=1
{ζ(2k)− 1} =
3
4
; (215)
∞∑
k=1
{ζ(2k+ 1)− 1} =
1
4
; (216)
∞∑
k=2
(−1)k{ζ(k)− 1}
(
3
2
)k
=
31
10
− 3log2; (217)
∞∑
k=2
{ζ(k)− 1}
(
3
2
)k
=
3
2
+ 3log2; (218)
∞∑
k=1
{ζ(2k)− 1}
(
3
2
)2k
=
23
10
; (219)
∞∑
k=1
{ζ(2k+ 1)− 1}
(
3
2
)2k
=−
8
15
+ 2log2; (220)
∞∑
k=2
(−1)k
ζ(k)− 1
3k
=
11
12
−
π
18
√
3−
1
2
log3; (221)
∞∑
k=2
ζ(k)− 1
3k
=−
1
6
−
π
18
√
3+
1
2
log3; (222)
286 Zeta and q-Zeta Functions and Associated Series and Integrals
∞∑
k=1
ζ(2k)− 1
32k
=
3
8
−
1
18
π
√
3; (223)
∞∑
k=1
ζ(2k+ 1)− 1
32k
=−
13
8
+
3
2
log3; (224)
∞∑
k=2
(−1)k(ζ(k)− 1)
(
2
3
)k
=
11
15
+
1
9
π
√
3− log3; (225)
∞∑
k=2
{ζ(k)− 1}
(
2
3
)k
=−
4
3
+
1
9
π
√
3+ log3; (226)
∞∑
k=1
{ζ(2k)− 1}
(
2
3
)2k
=−
3
10
+
1
9
π
√
3; (227)
∞∑
k=1
{ζ(2k+ 1)− 1}
(
2
3
)2k
=−
31
20
+
3
2
log3; (228)
∞∑
k=2
(−1)k{ζ(k)− 1}
(
4
3
)k
=
89
21
−
2
9
π
√
3− 2log3; (229)
∞∑
k=2
{ζ(k)− 1}
(
4
3
)k
=
4
3
−
2
9
π
√
3+ 2log3; (230)
∞∑
k=1
{ζ(2k)− 1}
(
4
3
)2k
=
39
14
−
2
9
π
√
3; (231)
∞∑
k=1
{ζ(2k+ 1)− 1}
(
4
3
)2k
=−
61
56
+
3
2
log3; (232)
∞∑
k=2
(−1)k{ζ(k)− 1}
(
5
3
)k
=
59
24
+
5
18
π
√
3−
5
2
log3; (233)
∞∑
k=2
{ζ(k)− 1}
(
5
3
)k
=
5
3
+
5
18
π
√
3+
5
2
log3; (234)
∞∑
k=1
{ζ(2k)− 1}
(
5
3
)2k
=
33
16
+
5
18
π
√
3; (235)
∞∑
k=1
{ζ(2k+ 1)− 1}
(
5
3
)2k
=−
19
80
+
3
2
log3; (236)
∞∑
k=2
(−1)k
ζ(k)− 1
22k
=
19
20
−
1
8
π −
3
4
log2; (237)
∞∑
k=2
ζ(k)− 1
22k
=−
1
12
−
1
8
π +
3
4
log2; (238)
Series Involving Zeta Functions 287
∞∑
k=1
ζ(2k)− 1
24k
=
13
30
−
1
8
π; (239)
∞∑
k=1
ζ(2k+ 1)− 1
24k
=−
31
15
+ 3log2; (240)
∞∑
k=2
(−1)k{ζ(k)− 1}
(
3
4
)k
=
19
28
+
3
8
π −
9
4
log2; (241)
∞∑
k=2
{ζ(k)− 1}
(
3
4
)k
=−
9
4
+
3
8
π +
9
4
log2; (242)
∞∑
k=1
{ζ(2k)− 1}
(
3
4
)2k
=−
11
14
+
3
8
π; (243)
∞∑
k=1
{ζ(2k+ 1)− 1}
(
3
4
)2k
=−
41
21
+ 3log2; (244)
∞∑
k=2
(−1)k{ζ(k)− 1}
(
5
4
)k
=
191
36
−
5
8
π −
15
4
log2; (245)
∞∑
k=2
{ζ(k)− 1}
(
5
4
)k
=
5
4
−
5
8
π +
15
4
log2; (246)
∞∑
k=1
{ζ(2k)− 1}
(
5
4
)2k
=
59
18
−
5
8
π; (247)
∞∑
k=1
{ζ(2k+ 1)− 1}
(
5
4
)2k
=−
73
45
+ 3log2; (248)
∞∑
k=2
(−1)k{ζ(k)− 1}
(
7
4
)k
=
293
132
+
7
8
π −
21
4
log2; (249)
∞∑
k=2
{ζ(k)− 1}
(
7
4
)k
=
7
4
+
7
8
π +
21
4
log2; (250)
∞∑
k=1
{ζ(2k)− 1}
(
7
4
)2k
=
131
66
+
7
8
π; (251)
∞∑
k=1
{ζ(2k+ 1)− 1}
(
7
4
)2k
=−
31
231
+ 3log2; (252)
∞∑
k=2
(−1)k
ζ(k)− 1
5k
=
29
30
−
π
10
√
1+
2
5
√
5−
1
4
log5−
√
5
10
log
1+
√
5
2
; (253)
∞∑
k=2
ζ(k)− 1
5k
=−
1
20
−
π
10
√
1+
2
5
√
5+
1
4
log5+
√
5
10
log
1+
√
5
2
; (254)
288 Zeta and q-Zeta Functions and Associated Series and Integrals
∞∑
k=1
ζ(2k)− 1
52k
=
11
24
−
π
10
√
1+
2
5
√
5; (255)
∞∑
k=1
ζ(2k+ 1)− 1
52k
=−
61
24
+
5
4
log5+
√
5
2
log
1+
√
5
2
; (256)
∞∑
k=2
(−1)k{ζ(k)− 1}
(
2
5
)k
(257)
=
31
35
−
π
5
√
1−
2
5
√
5−
1
2
log5+
√
5
5
log
1+
√
5
2
;
∞∑
k=2
{ζ(k)− 1}
(
2
5
)k
(258)
=−
4
15
−
π
5
√
1−
2
5
√
5+
1
2
log5−
√
5
5
log
1+
√
5
2
;
∞∑
k=1
{ζ(2k)− 1}
(
2
5
)2k
=
13
42
−
π
5
√
1−
2
5
√
5; (259)
∞∑
k=1
{ζ(2k+ 1)− 1}
(
2
5
)2k
=−
121
84
+
5
4
log5−
√
5
2
log
1+
√
5
2
; (260)
∞∑
k=2
(−1)k{ζ(k)− 1}
(
3
5
)k
(261)
=
31
40
+
3π
10
√
1−
2
5
√
5−
3
4
log5+
3
√
5
10
log
1+
√
5
2
;
∞∑
k=2
{ζ(k)− 1}
(
3
5
)k
(262)
=−
9
10
+
3π
10
√
1−
2
5
√
5+
3
4
log5−
3
√
5
10
log
1+
√
5
2
;
∞∑
k=1
{ζ(2k)− 1}
(
3
5
)2k
=−
1
16
+
3π
10
√
1−
2
5
√
5; (263)
∞∑
k=1
{ζ(2k+ 1)− 1}
(
3
5
)2k
=−
67
48
+
5
4
log5−
√
5
2
log
1+
√
5
2
; (264)
∞∑
k=2
(−1)k{ζ(k)− 1}
(
4
5
)k
(265)
=
29
45
+
2π
5
√
1+
2
5
√
5− log5−
2
√
5
5
log
1+
√
5
2
;
Series Involving Zeta Functions 289
∞∑
k=2
{ζ(k)− 1}
(
4
5
)k
(266)
=−
16
5
+
2π
5
√
1+
2
5
√
5+ log5+
2
√
5
5
log
1+
√
5
2
;
∞∑
k=1
{ζ(2k)− 1}
(
4
5
)2k
=−
23
18
+
2π
5
√
1+
2
5
√
5; (267)
∞∑
k=1
{ζ(2k+ 1)− 1}
(
4
5
)2k
=−
173
72
+
5
4
log5+
√
5
2
log
1+
√
5
2
; (268)
∞∑
k=2
(−1)k{ζ(k)− 1}
(
6
5
)k
(269)
=
349
55
−
3π
5
√
1+
2
5
√
5−
3
2
log5−
3
√
5
5
log
1+
√
5
2
;
∞∑
k=2
{ζ(k)− 1}
(
6
5
)k
(270)
=
6
5
−
3π
5
√
1+
2
5
√
5+
3
2
log5+
3
√
5
5
log
1+
√
5
2
;
∞∑
k=1
{ζ(2k)− 1}
(
6
5
)2k
=
83
22
−
3π
5
√
1+
2
5
√
5; (271)
∞∑
k=1
{ζ(2k+ 1)− 1}
(
6
5
)2k
=−
283
132
+
5
4
log5+
√
5
2
log
1+
√
5
2
; (272)
∞∑
k=2
(−1)k{ζ(k)− 1}
(
7
5
)k
(273)
=
221
60
−
7π
10
√
1−
2
5
√
5−
7
4
log5+
7
√
5
10
log
1+
√
5
2
;
∞∑
k=2
{ζ(k)− 1}
(
7
5
)k
(274)
=
7
5
−
7π
10
√
1−
2
5
√
5+
7
4
log5−
7
√
5
10
log
1+
√
5
2
;
∞∑
k=1
{ζ(2k)− 1}
(
7
5
)2k
=
61
24
−
7π
10
√
1−
2
5
√
5; (275)
∞∑
k=1
{ζ(2k+ 1)− 1}
(
7
5
)2k
=−
137
168
+
5
4
log5−
√
5
2
log
1+
√
5
2
; (276)
290 Zeta and q-Zeta Functions and Associated Series and Integrals
∞∑
k=2
(−1)k{ζ(k)− 1}
(
8
5
)k
(277)
=
523
195
+
4π
5
√
1−
2
5
√
5− 2log5+
4
√
5
5
log
1+
√
5
2
;
∞∑
k=2
{ζ(k)− 1}
(
8
5
)k
(278)
=
8
5
+
4π
5
√
1−
2
5
√
5+ 2log5−
4
√
5
5
log
1+
√
5
2
;
∞∑
k=1
{ζ(2k)− 1}
(
8
5
)2k
=
167
78
+
4π
5
√
1−
2
5
√
5; (279)
∞∑
k=1
{ζ(2k+ 1)− 1}
(
8
5
)2k
=−
211
624
+
5
4
log5−
√
5
2
log
1+
√
5
2
; (280)
∞∑
k=2
(−1)k{ζ(k)− 1}
(
9
5
)k
(281)
=
293
140
+
9π
10
√
1+
2
5
√
5−
9
4
log5−
9
√
5
10
log
1+
√
5
2
;
∞∑
k=2
{ζ(k)− 1}
(
9
5
)k
(282)
=
9
5
+
9π
10
√
1+
2
5
√
5+
9
4
log5+
9
√
5
10
log
1+
√
5
2
;
∞∑
k=1
{ζ(2k)− 1}
(
9
5
)2k
=
109
56
+
9π
10
√
1+
2
5
√
5; (283)
∞∑
k=1
{ζ(2k+ 1)− 1}
(
9
5
)2k
=−
41
504
+
5
4
log5+
√
5
2
log
1+
√
5
2
; (284)
∞∑
k=2
(−1)k
ζ(k)− 1
6k
=
41
42
−
π
12
√
3−