Series Associated With the Zeta and Related FunctionsIn 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 ((s), the Hurwitz Zeta function ((s,a), and their such extensions and generalizations as (for example) Lerch's transcendent (or the Hurwitz-Lerch Zeta function) iI>(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 rediscoveries of a fairly rapidly convergent series representation for ((3), which is actually contained in a 1772 paper by Leonhard Euler (1707-1783), and from another known series representation for ((3), which was used by Roger Apery (1916-1994) in 1978 in his celebrated proof of the irrationality of ((3). This book is motivated essentially by the fact that the theories and applications of the various methods and techniques used in dealing with many different families of series associated with the Riemann Zeta function and its aforementioned relatives are to be found so far only"in widely scattered journal articles. Thus our systematic (and unified) presentation of these results on the evaluation and representation of the Zeta and related functions is expected to fill a conspicuous gap in the existing books dealing exclusively with these Zeta functions. |
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Series Associated with the Zeta and Related Functions Hari M. Srivastava,Junesang Choi No preview available - 2013 |
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1)² log Adamchik Amer analytic continuation applying arctan asymptotic Bernoulli numbers Bernoulli polynomials Bn(x C₁ Choi and Srivastava Comput convergent series Cvijović and Klinowski deduce denotes derived dt R(z Euler polynomials Euler-Maclaurin summation formula Euler-Mascheroni constant evaluation Gauss's given Hurwitz Zeta function hypergeometric series integral formula integral representation Laplacians Li₂ log G log G(t logarithms Math multiple Gamma functions numbers obtain the following Polylogarithm positive integer Problem proof Prove the following representations for 2n+1 resulting equation Riemann Zeta function s-plane Section sequence series involving series representations setting simple poles Srivastava 1998 Srivastava 2000a Srivastava and Tsumura Stirling numbers summation formula Taking the limit V5 log y+log yields Zhang and Williams πν πνο ΣΣ Зп
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