On some slowly convergent series involving the Hurwitz zeta-function.
type: Item
instance of: scholarly article
title: On some slowly convergent series involving the Hurwitz zeta-function.
zbMATH Open document ID: 1038.11055
DOI: 10.1016/S0377-0427(03)00617-4
published in: Journal of Computational and Applied Mathematics
publication date: 2003-11-20T00:00:00Z
review text: Let, as usual, \(\zeta(s,\alpha)\) be the Hurwitz zeta-function. Let \(\lambda\) and \(1\leq m<q\), \(q\geq2\) be non-negative integers, \(\Re\alpha>0\) and \(| z| <| \alpha| \). In this paper [see also \textit{S. Kanemitsu, H. Kumagai}, and \textit{M. Yoshimoto}, Ramanujan J. 5, 5--19 (2001; Zbl 0989.11043)] the authors consider series involving the Hurwitz zeta-function, namely: \(\sum_{n=2}^\infty{(n-1)!\over (n+\lambda)!}(-1)^n\zeta(n,\alpha)z^{n+\lambda}\), \(\sum_{n=1}^\infty{z^{nq}\over n}(-1)^m\zeta(nq,\alpha)\), \(\sum_{n=0}^\infty{\zeta(nq+m, \alpha)\over nq+m}\), \(\sum_{n=2}^\infty n^\lambda z^n\zeta(n, \alpha)\). They express these series by finite sums involving Bernoulli numbers, Stirling numbers of the second kind, the Euler gamma function and the Hurwitz zeta-function. The authors also discuss related results from the book of \textit{H. M. Srivastava} and \textit{J. Choi} [Series associated with the zeta and related functions. Dordrecht: Kluwer (2001; Zbl 1014.33001)].
Mathematics Subject Classification ID: 11M35
Mathematics Subject Classification ID: 33B15
zbMATH DE Number: 2006420
zbMATH Keywords: Hurwitz zeta-function
zbMATH Keywords: gamma function
zbMATH Keywords: Bernoulli numbers
zbMATH Keywords: Stirling numbers of the second kind
author: Masahiro Hashimoto
author: Shigeru Kanemitsu
author: Yoshio Tanigawa
author: Masami Yoshimoto
author: Wenpeng Zhang
reviewed by: Ramūnas Garunkštis
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2006420