On some slowly convergent series involving the Hurwitz zeta-function.
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)].
scientific article; zbMATH DE number 2006420