Zeta and q-Zeta Functions and Associated Series and Integrals

Preface

Acknowledgements

1. Introduction and Preliminaries

1.1 Gamma and Beta Functions

1.2 The Euler-Mascheroni Constant γ

1.3 Polygamma Functions

1.4 The Multiple Gamma Functions

1.5 The Gaussian Hypergeometric Function and its Generalization

1.6 Stirling Numbers of the First and Second Kind

1.7 Bernoulli, Euler and Genocchi Polynomials and Numbers

1.8 Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi Polynomials and Numbers

1.9 Inequalities for the Gamma Function and the Double Gamma Function

Problems

2. The Zeta and Related Functions

2.1 Multiple Hurwitz Zeta Functions

2.2 The Hurwitz (or Generalized) Zeta Function

2.3 The Riemann Zeta Function

2.4 Polylogarithm Functions

2.5 Hurwitz–Lerch Zeta Functions

2.6 Generalizations of the Hurwitz–Lerch Zeta Function

2.7 Analytic Continuations of Multiple Zeta Functions

Problems

3. Series Involving Zeta Functions

3.1 Historical Introduction

3.2 Use of the Binomial Theorem

3.3 Use of Generating Functions

3.4 Use of Multiple Gamma Functions

Problems

4. Evaluations and Series Representations

4.1 Evaluation of

4.2 Rapidly Convergent Series for

4.3 Further Series Representations

4.4 Computational Results

Problems

5. Determinants of the Laplacians

5.1 The n-Dimensional Problem

5.2 Computations Using the Simple and Multiple Gamma Functions

5.3 Computations Using Series of Zeta Functions

5.4 Computations using Zeta Regularized Products

5.5 Remarks and Observations

Problems

6. q-Extensions of Some Special Functions and Polynomials

6.1 q-Shifted Factorials and q-Binomial Coefficients

6.2 q-Derivative, q-Antiderivative and Jackson q-Integral

6.3 q-Binomial Theorem

6.4 q-Gamma Function and q-Beta Function

6.5 A q-Extension of the Multiple Gamma Functions

6.6 q-Bernoulli Numbers and q-Bernoulli Polynomials

6.7 q-Euler Numbers and q-Euler Polynomials

6.8 The q-Apostol-Bernoulli Polynomials of Order

6.9 The q-Apostol-Euler Polynomials of Order

6.10 A Generalized q-Zeta Function

6.11 Multiple q-Zeta Functions

Problems

7. Miscellaneous Results

7.1 A Set of Useful Mathematical Constants

7.2 Log-Sine Integrals Involving Series Associated with the Zeta Function and Polylogarithms

7.3 Applications of the Gamma and Polygamma Functions Involving Convolutions of the Rayleigh Functions

7.4 Bernoulli and Euler Polynomials at Rational Arguments

7.5 Closed-Form Summation of Trigonometric Series

Problems

Bibliography