Ramanujan series upside-down
type: Item
instance of: scholarly article
zbMATH Open document ID: 1305.33014
DOI: 10.1017/S1446788714000147
author: Jesús Guillera
author: Mathew Rogers
published in: Journal of the Australian Mathematical Society
publication date: 2014-08-25T00:00:00Z
full work available at URL: https://arxiv.org/abs/1206.3981
Mathematics Subject Classification ID: 33C20
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 11F03
Mathematics Subject Classification ID: 11Y60
Mathematics Subject Classification ID: 33C75
Mathematics Subject Classification ID: 33E05
zbMATH DE Number: 6334050
zbMATH Keywords: Dirichlet \(L\)-values
zbMATH Keywords: Ramanujan type formulas for \(1/\pi\)
zbMATH Keywords: hypergeometric series
MaRDI profile type: Publication
arXiv ID: 1206.3981
cites work: Dirichlet L -series with real and complex characters and their application to solving double sums
cites work: Apéry limits and special values of \(L\)-functions
cites work: Some entries in Ramanujan's notebooks
cites work: Q3579593
cites work: Rational analogues of Ramanujan's series for 1/π
cites work: WZ-Proofs of “Divergent” Ramanujan-Type Series
cites work: Hypergeometric identities for 10 extended Ramanujan-type series
cites work: Ramanujan's series for \(1/\pi \) arising from his cubic and quartic theories of elliptic functions
OpenAlex ID: W2041457691
describes a project that uses: Digital Library of Mathematical Functions
title: Ramanujan series upside-down
review text: Based on the known example NEWLINE\[NEWLINE\frac{1}{\pi}=\sum_{n=0}^{\infty}\frac{(-1)^n}{2^{6n}}\begin{pmatrix} 2n,\\ 2\end{pmatrix}{}^3\left(\frac{1}{2}+2n\right)NEWLINE\]NEWLINE and the so-called \textit{companion series identity} NEWLINE\[NEWLINE8\,L_{-4}(2)=\sum_{n=1}^{\infty}\frac{(-1)^n2^{6n}}{n^3\begin{pmatrix} 2n,\\ n\end{pmatrix}{}^3}\left(\frac{1}{2}-2n\right),NEWLINE\]NEWLINE where \(L_{-4}(2)=1-\frac{1}{3^2}+\frac{1}{5^2}+\ldots\) is Catalan's constant and \(L_k(s):=\sum_{n=1}^{\infty}\chi_k(n)/n^s\) denotes the general Dirichlet \(L\)-series with \(\chi_k(x)=\left(\frac{k}{n}\right)\) being the Jacobi symbol, the authors study seventeen formulas identified by Ramanujan of the form NEWLINE\[NEWLINE\frac{1}{\pi}=\sum_{n=0}^{\infty}\frac{(s)_n(\frac{1}{2})_n(1-s)_n}{(1)_n^3}(a+bn)z^n,NEWLINE\]NEWLINE where \((x)_n=\Gamma(x+n)/\Gamma(x)\) is the Pochhammer symbol. In each case one has \(s\in\{\frac{1}{2},\frac{1}{3},\frac{1}{4}\,\frac{1}{6}\}\) and \((a,b,z)\) parametrized by modular functions. Preserving these modular parametrizations for \((a,b,z)\), the general \textit{companion series} is given by NEWLINE\[NEWLINE\sum_{n=1}^{\infty}\frac{(1)_n^3}{(s)_n(\frac{1}{2})_n(1-s)_n}\frac{(a-bn)}{n^3}z^{-n}.NEWLINE\]NEWLINE For \(s\in\{\frac{1}{2},\frac{1}{3},\frac{1}{4}\}\) many values of this companion series reduce to linear combinations of two Epstein zeta functions and elementary constants. If this linear combination of Epstein zeta functions reduces to Dirichlet \(L\)-values, then the companion series also reduces to Dirichlet \(L\)-values. Proofs are based upon a new idea called \textit{completing the hypergeometric function}. This approach fails completely when \(s=\frac{1}{6}\), which is explained. The methods used by the authors lead to several explicit formulas involving rapidly converging hypergeometric functions.
reviewed by: Roelof Koekoek
description: scientific article; zbMATH DE number 6334050