New series representations for Apéry's and other classical constants
type: Item
instance of: scholarly article
title: New series representations for Apéry's and other classical constants
zbMATH Open document ID: 1413.11017
author: Horst Alzer
author: Anthony Sofo
published in: Analysis Mathematica
publication date: 2019-02-01T00:00:00Z
review text: The authors establish an identity involving the sums \(A_k(a,p,N) =\sum_{j=0}^{k-1} \left((-1)^{jN}/(a+j)^p\right)\) and use it to produce new series representations for \(\log(2)\), \(\zeta(3)\), \(\pi^2\) and the constant of Catalan. For example they obtain \[ \sum_{k=1}^\infty \frac{1}{k^3} = \zeta(3) = \frac{149}{144} + \frac{1}{8} \sum_{k=2}^\infty \frac{(2k+1)(k^4+2k^3+3k^2+2k-2)}{((k-1)k(k+1)(k+2))^2} \left(2H_k^{(2)}-H_{[k/2]}^{(2)}\right), \] where \(H_k^{(2)} = \sum_{j=1}^k1/j^2\).
reviewed by: Władysław Narkiewicz
Mathematics Subject Classification ID: 11Y60
Mathematics Subject Classification ID: 41A58
zbMATH DE Number: 7011045
zbMATH Keywords: series representation
zbMATH Keywords: log(2)
zbMATH Keywords: zeta(3)
zbMATH Keywords: \(\pi^2\)
zbMATH Keywords: constant of Catalan
MaRDI profile type: Publication
full work available at URL: s10476-018-0502-8
OpenAlex ID: W2890149188
DOI: 10.1007/S10476-018-0502-8
description: scientific article; zbMATH DE number 7011045