Zeta Functions and Basic Analogues

Zeta Functions and Basic Analogues by Peter John Anderson B.Sc. (Hon.), Simon Fraser University, 1985 M.Sc., Simon Fraser University, 1991 A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY in the Department of Mathematics and Statistics c Peter John Anderson, 2009 University of Victoria All rights reserved. This dissertation may not be reproduced in whole or in part, by photocopying or other means, without the permission of the author. 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The author retains copyright L’auteur conserve la propriété du droit d’auteur ownership and moral rights in this et des droits moraux qui protège cette thèse. Ni thesis. Neither the thesis nor la thèse ni des extraits substantiels de celle-ci substantial extracts from it may be ne doivent être imprimés ou autrement printed or otherwise reproduced reproduits sans son autorisation. without the author’s permission. In compliance with the Canadian Conformément à la loi canadienne sur la Privacy Act some supporting forms protection de la vie privée, quelques may have been removed from this formulaires secondaires ont été enlevés de thesis. cette thèse. While these forms may be included Bien que ces formulaires aient inclus dans in the document page count, their la pagination, il n’y aura aucun contenu removal does not represent any loss manquant. of content from the thesis. ii Zeta Functions and Basic Analogues by Peter John Anderson B.Sc. (Hon.), Simon Fraser University, 1985 M.Sc., Simon Fraser University, 1991 Supervisory Committee Dr. H.M. Srivastava, Senior Supervisor (Department of Mathematics and Statistics) Dr. Jing Huang, Departmental Member (Department of Mathematics and Statistics) Dr. A.R. Sourour, Departmental Member (Department of Mathematics and Statistics) Dr. Frank Ruskey, Outside Member (Department of Computer Science) iii Supervisory Committee Dr. H.M. Srivastava, Senior Supervisor (Department of Mathematics and Statistics) Dr. Jing Huang, Departmental Member (Department of Mathematics and Statistics) Dr. A.R. Sourour, Departmental Member (Department of Mathematics and Statistics) Dr. Frank Ruskey, Outside Member (Department of Computer Science) ABSTRACT We present results evolving from established connections between zeta functions and different systems of polynomials, particularly the Riemann and Hurwitz zeta functions and the Bernoulli and Euler polynomials. In particular, we develop certain results related to Apostol’s deformation of the Bernoulli polynomials and obtain iden- tities of Carlitz by a novel approach using generating functions instead of difference equations. In the last two chapters, we work out new rapidly convergent series expansions of the Riemann zeta function, find coefficient symmetries of a polynomial sequence obtained from the cyclotomic polynomials by a linear fractional transformation of argument, and obtain an expression for the constant term in an identity involving the gamma function. iv Contents Supervisory Committee ii Abstract iii Table of Contents iv Acknowledgements vi Dedication viii 1 Introduction 1 2 Classical and Basic Zeta Functions 3 2.1 Zeta functions in general ....................... 3 2.2 Particular classical zeta functions ................. 4 2.3 Basic analogues (q-analogues) in general ............. 16 2.4 Basic analogues of zeta functions (q-zeta functions) . 18 3 Basic Analogues of Bernoulli and Euler Numbers and Polynomials, and of the Hurwitz Zeta Function 23 3.1 Introduction and Preliminaries ................... 23 3.2 q-Eta Numbers and Polynomials .................. 26 3.3 q-Stirling Numbers of the Second Kind .............. 33 3.4 The Polynomial βn;q(x) ........................ 38 3.5 q-Euler Numbers and Polynomials ................. 43 3.6 A q-Hurwitz Zeta Function ..................... 48 4 Apostol-Bernoulli Polynomials and the Multiple Hurwitz-Lerch Zeta Function 63 4.1 Introduction and Preliminaries ................... 63 v 4.2 Further Properties of the Multiple Hurwitz-Lerch Zeta Func- tion .................................... 71 4.3 Basic Analogues of the Apostol-Bernoulli and Apostol-Euler Polynomials ............................... 81 4.4 A Basic Analogue of the Multiple Hurwitz Lerch-Zeta Function 90 4.5 An Identity Relating the Bernoulli Polynomials of the First and Second Kinds ........................... 92 5 Cyclotomy and Basic Gamma Functions 97 5.1 A Consequence of Gauss’s Multiplication Formula . 97 5.2 Coefficient Sequences of Transformed Cyclotomic Polynomials 102 6 Rapidly Convergent Series Expansions for ζ(2n + 1) 108 6.1 A Series Obtained from a Twelvefold Division of the Unit Interval .................................. 109 6.2 A Series Obtained from an Eightfold Division of the Unit Interval .................................. 121 7 Conclusions 127 Bibliography 128 vi ACKNOWLEDGEMENTS It gives me great pleasure to acknowledge the different forms of help provided to me in the course of my doctoral studies. Many who assisted in crucial ways cannot be singled out for specific mention here, but it will please me no less to acknowledge their contributions separately by appropriate means. I would like to thank my senior supervisor Professor Hari Srivastava for guiding my first steps into a new field, for uprooting naive misconceptions about the nature of mathematical research, and for providing unfailing encouragement. His personal example is also of inestimable value: I live in the hope of inheriting his all-embracing self-discipline. I am proud to be his student. I gratefully thank Professor Junesang Choi for many, many hours of collegial dialogue. I have greatly enjoyed his hospitality, generosity, and buoyant good cheer. I thank Professor Victor Adamchik for participating as external examiner. I thank Professor Reinhard Illner for demolishing an obstruction in LaTeX; this greatly fortified my morale at a decisive point. I would like to extend thanks to the members of my supervisory committee as well. I thank Professor Frank Ruskey for astute advice concerning software and certain general aspects of preparing a dissertation. I thank Professor Jing Huang, the longest-serving member of my committee, for financial aid early in my program, for general encouragement, and for illustrating by practice with what great harmony research and teaching activity can support one another. To Professor Ahmed Sourour I extend thanks for similar illustration, including (direct and indirect) discourse on a number of algebraic topics years ago. The occasions of this discourse created an abiding impression of lucidity. I am thankful to (and for) friends old and new, especially those who are, or have been, students. Universities are good places to discover that bonds of friendship can be strengthened by sharing not only suffering or enjoyment, but curiosity as well. Thanks go to Allison for deftly illustrating the power of words, especially first words and kind ones. Blessed be. I wholeheartedly thank my family for interest and support that, quite naturally, extend far beyond the realm of education. My mother, father, and step-mother, in particular, have been inspiration, safe haven, cheering section, and chamber of sober second thought for as long as I have known them. Like most other sets of parents, they are the world’s very best. vii My partner Lisa has played a central role in my life for more than a decade. It delights me now to honour her for deep understanding — and bottomless patience. viii DEDICATION To my teachers and my students. To the helpful, for helping; to the unhelpful, for not exercising a decisive influence. Chapter 1 Introduction The five chapters following this Introduction contain results pertaining to the Rie- mann zeta function and a number of its generalizations. Chapter 2, ‘Classical and Basic Zeta Functions,’ sets the stage by introducing the functions under consideration and describing some of the operations that apply to them. The chapter concludes with a variety of examples extracted from the current literature, including some peripheral discussion of the many interesting properties they satisfy. The first new results and arguments of this thesis appear in Chapter 3, ‘Basic Analogues of Bernoulli and Euler Numbers and Polynomials.’ This chapter, like the one that follows it, is largely the outcome of collaborative work with Professors Junesang Choi and Hari Srivastava carried out during Professor Choi’s sabbatical at the University of Victoria during the year 2007. Although I was involved in this work from the beginning,

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