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Zeta and \(q\)-zeta functions and associated series and integrals. (English) Zbl 1239.33002

Amsterdam: Elsevier (ISBN 978-0-12-385218-2/hbk). xvi, 657 p. (2012).
This book is a revised, enlarged and updated version of the authors’ previous work entitled “Series associated with the zeta and related functions” [Dordrecht: Kluwer Academic Publishers (2001; Zbl 1014.33001)]. It sets out a compendium of results on exact evaluations of infinite series involving the Riemann zeta function \(\zeta(s)\). The style of presentation is in between that of a text book, in which the theory is discussed in depth, and that of a handbook, in which formulas are listed sequentially.
The first two chapters set out the necessary analytical background to the main body of the book contained in Chapters 3 and 4. An account is given of the standard functions such as the Gamma and Beta functions, and the Polygamma and related functions. The basic properties of all these functions are collected together as a series of useful formulas. Then follows a discussion of some less common special functions, namely the double and multiple gamma functions. The chapter concludes with a summary of the Gauss and confluent hypergeometric functions, together with a brief mention of the generalized hypergeometric function \({}_pF_q(z)\), the Stirling numbers and the Bernoulli and Euler numbers and polynomials. Additional material in the revised text includes a discussion of the Genocchi, Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials and numbers. Chapter 2 deals with various types of zeta functions, including the Hurwitz zeta function \(\zeta(z,a)\) (which reduces to the Riemann zeta function when \(a=1\)) and the Hurwitz-Lerch zeta function. The most important properties and integral representations for these functions are listed. In addition, the Polylogarithm and related functions are also defined.
Chapter 3 is the main chapter of the book and discusses the most effective methods of evaluating series associated with \(\zeta(s)\) and related functions. There then follows a list of just under 750 evaluations (together with an additional 48 evaluations in the end-of-chapter exercises) involving the zeta function. These rational zeta series take the general forms \[ S_1=\sum_{n=2}^\infty a_n (\zeta(n)-1)\quad \text{or}\quad S_2=\sum_{n=2}^\infty b_n \zeta(n), \] where the coefficients \(a_n\) and \(b_n\) are rational functions of \(n\), with \(b_n=O(n^{-1})\) for large \(n\). The simplest such series have \(a_n=1\) and \(b_n=(-1)^n/n\) which yield the well-known results \(S_1=1\) and \(S_2=\gamma\). A few examples of the series listed in this chapter, chosen almost at random, are: \[ \sum_{n=2}^\infty\frac{(-1)^n}{n+1}\,\zeta(n)=1+\frac12\gamma-\frac12\log\,(2\pi), \]
\[ \sum_{n=1}^\infty (\zeta(2n)-1) (5/3)^{2n}=\frac{33}{16}+\frac{5}{18}\pi \surd 3, \]
\[ \sum_{n=2}^\infty\frac{(-1)^n}{n(n+1)2^{2n}} (\zeta(n)-1)=-\frac{17}{8}+\gamma/8+G/\pi+\log\,(2^{-23/2} 5^5 \pi^{1/2} A^{9/2}), \] where \(A\), \(G\) and \(\gamma\) denote the Glaisher, Catalan and Euler constants, respectively. The chapter concludes with the high-order derivatives of \(\Gamma(x)\) evaluated at \(x=\frac12\) and \(x=1\), and a discussion of Mathieu series. Chapter 4 is a shorter chapter dealing with series representations for \(\zeta(n)\) for \(n=2, 3, 4, \dots\;\). In particular, numerous rapidly convergent series for \(\zeta(2n+1)\) are discussed.
Chapter 5 presents a short discussion illustrating the use of some of the results in the earlier chapters in the determination of the determinants of the Laplacians for the \(n\)-dimensional sphere \({\mathbb S}^n\). Chapter 6 deals with \(q\)-extensions of some special functions and polynomials and is a completely new addition to the book. The \(q\)-extensions discussed are those of the Gamma, Beta, multiple Gamma, and zeta functions together with various polynomials, such as the Bernoulli and Euler polynomials, and the \(q\)-extension of the Stirling numbers. The final chapter presents a variety of miscellaneous results related to the evaluation of Bernoulii and Euler polynomials at rational arguments, the closed-form summation of several classes of trigonometric series and the evaluation of certain integrals using the Euler-Maclaurin summation formula.
At the end of each chapter there is a useful set of exercises intended both as an expansion of results contained in the chapter as well as a source of information to recent publications, with references should further details be required. These exercises have been extended to include material published since the first edition of this book. There is an extensive (over 1260 references) and up-to-date bibliography on the literature concerned with the evaluation of series of the zeta function and the various special functions discussed in the various chapters. Overall this is a very valuable reference for those with an interest in the Riemann zeta function or who have occasion to evaluate series involving the zeta function. Apart from the extensive list of series evaluations, the first two chapters also provide a valuable reference on the standard functions of analysis together with the more recondite multiple gamma and zeta functions, which are not usually found in the standard texts on special functions.

MSC:

33-02 Research exposition (monographs, survey articles) pertaining to special functions
33E20 Other functions defined by series and integrals
11M41 Other Dirichlet series and zeta functions

Citations:

Zbl 1014.33001
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