Series involving zeta and related functions
type: Item
instance of: scholarly article
title: Series involving zeta and related functions
zbMATH Open document ID: 1231.11101
published in: Applied Mathematics and Computation
publication date: 2011-03-08T00:00:00Z
review text: The author proves: For \(\text{Re}(a)>0\), \(\text{Re}(a+p)>0\), \(m\in{\mathbb N}\), \(n\in{\mathbb N},N\in{\mathbb N}\), \[ \begin{multlined} (-1)^N N!\sum_{j=N+2}^{\max(m,n)}\frac{(\binom{m}{j}\zeta(j-N,a+p)-(-1)^j\binom{n}{j}\zeta(j-N,a))}{\binom{m+n}{j}(j-N)_{N+1}}p^j\\ =[\zeta'(-N,a)-\zeta'(-N,a+p)]\\ +\sum_{j=1}^N \frac{\binom{m}{j}}{\binom{m+n}{j}}\binom{N}{j}[(H_N-H_{N-j})\zeta(j-N,a+p)-\zeta'(j-N,a+p)](-p)^j\\ - \sum_{j=1}^N \frac{\binom{n}{j}}{\binom{m+n}{j}}\binom{N}{j}[(H_N-H_{N-j})\zeta(j-N,a)-\zeta'(j-N,a)]p^j \\ -\frac{\binom{m}{N+1}}{\binom{m+n}{N+1}}(H_N+H_{a+p-1}-\gamma)\frac{(-p)^{N+1}}{N+1}\\ +\frac{\binom{n}{N+1}}{\binom{m+n}{N+1}}(H_N+H_{a-1}-\gamma)\frac{p^{N+1}}{N+1} +\varepsilon_{n,m}^{(p)}(N,a),\end{multlined} \] where \(\zeta(s,a)\) is the Hurwitz zeta function, \(H_r:=\int_0^1\frac{1-x^r}{1-x}\,dx\), \(\text{Re}(r)> -1\), \((\lambda)_n:=\lambda(\lambda+1)\ldots (\lambda+n-1)\), \(n\geq 1\), \((\lambda)_0:=1\), \[ \begin{multlined} \varepsilon_{n,m}^{(p)}(N,a):=\\ \begin{cases} (-1)^{N+n+1}p^{m+n+1}\binom{m+n}{N}^{-1}\int_0^1 x^m(1-x)^n\zeta(m+n+1-N,a+px)\,dx,\, 0\leq N\leq m+n-1,\\ (-1)^mp^{m+n+1}\int_0^1 x^m(1-x)^n(H_{m+n}+H_{a+px-1}-\gamma)\,dx, \, N=m+n, \\(-1)^{N+m+1}p^{m+n+1}\binom{N}{m+n}\int_0^1 x^m(1-x)^n[(H_N-H_{N-m-n-1})\zeta(m+n+1-N,a+px)\\ -\zeta'(m+n+1-N,a+px)]\,dx,\, N\geq m+n+1. \end{cases}\end{multlined} \]
reviewed by: Florin Nicolae
Mathematics Subject Classification ID: 11M06
Mathematics Subject Classification ID: 11M35
zbMATH DE Number: 5862467
zbMATH Keywords: zeta function
zbMATH Keywords: values
author: M. Prevost
MaRDI profile type: Publication
full work available at URL: j.amc.2010.12.001
OpenAlex ID: W2044640114
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DOI: 10.1016/J.AMC.2010.12.001
description: scientific article; zbMATH DE number 5862467