A generalization of a formula of Ramanujan
type: Item
instance of: scholarly article
title: A generalization of a formula of Ramanujan
zbMATH Open document ID: 0727.05004
DOI: 10.1016/0022-247X(90)90381-O
author: Pavel G. Todorov
published in: Journal of Mathematical Analysis and Applications
publication date: 1990-01-01T00:00:00Z
review text: Let \(k_{n+1}=\sum^{n+1}_{k=1}\frac{1}{2k-1}\), \(n=0,1,2,...,then\) for \(\frac{1}{2}<x<1\), Ramanujan's formula \[ \sum^{\infty}_{k=0}\frac{1}{(2k+1)^ 2}(\frac{x}{1+x})^{k+1}=\sum^{\infty}_{n=0}\frac{(-1)^ n2^{2n}(n!)^ 2h_{n+1}x^{n+1}}{(2n+1)!} \] has been proved. In this paper this formula is generalized and proved as follows: Let \(h^{(0)}_{n+1}=1\), \(n=0,1,2,...\), and \(h^{(p)}_{n+1}=\sum^{n+1}_{k=1}\frac{h_ k^{(p-1)}}{2k-1}\), \(n=0,1,2,...\), then for each \(p=0,1,2,...\), in the set \(\{\) \(z| Re z>-\frac{1}{2}\}\cap \{z| | z| <1\}\) \[ \sum^{\infty}_{k=0}\frac{1}{(2k+1)^{p+1}}(\frac{z}{z+1})^{k+1}=\sum^{\infty} _{k=0}\frac{(-1)^ n2^{2n}(n!)^ 2h^{(p)}_{n+1}z^{n+1}}{(2n+1)!}. \] Two explicit representations of the numbers \(h^{(p)}_{n+1}\) are also given.
Mathematics Subject Classification ID: 05A19
Mathematics Subject Classification ID: 33C99
zbMATH DE Number: 4200208
zbMATH Keywords: Ramanujan's formula
MaRDI profile type: Publication
cites work: Chapter 9 of Ramanujan’s Second Notebook: Infinite Series Identities, Transformations, and Evaluations
full work available at URL: 0022-247x(90)90381-o
OpenAlex ID: W2070291629
description: scientific article; zbMATH DE number 4200208