Hidden \(q\)-analogues of Ramanujan-like \(\pi\)-series
type: Item
instance of: scholarly article
title: Hidden \(q\)-analogues of Ramanujan-like \(\pi\)-series
zbMATH Open document ID: 1476.33011
author: Xiaojing Chen
author: Wenchang Chu
published in: The Ramanujan Journal
publication date: 2021-05-06T00:00:00Z
review text: The authors take the famous series representations for \(1/\pi\) by Ramanujan, and show that the \(q\)-analogue of some of these are actually easily deducible by known identities scattered in the literature. For instance, the classical \(q\)-series equality \[ \begin{aligned} &\sum_{k=0}^{\infty} \frac{1-q^{2 k} a}{1-a}\left[\begin{array}{c} a, b, c, d \\ q, q a / b, q a / c, q a / d \end{array} \mid q\right]_{k}\left(\frac{q a}{b c d}\right)^{k} \\ &=\left[\begin{array}{c} q a, q a / b c, q a / b d, q a / c d \\ q a / b, q a / c, q a / d, q a / b c d \end{array} \mid q\right]_{\infty} \end{aligned} \] implies that \[ \sum_{k=0}^{\infty}(-1)^{k} \frac{1-q^{4 k+1}}{1-q} \frac{\left(q ; q^{2}\right)_{k}^{3}}{\left(q^{2} ; q^{2}\right)_{k}^{3}} q^{k^{2}}=\Gamma_{q^{2}}\left[\begin{array}{cc} 1,1 \\ \frac{1}{2}, \frac{3}{2} \end{array}\right]. \] This, in turn, is the \(q\)-analogue of \[ \sum_{k=0}^{\infty}(-1)^{k}\left\{\frac{\left(\frac{1}{2}\right)_{k}}{k !}\right\}^{3}(1+4 k)=\frac{2}{\pi}. \] Many similar identities and analogues, based on quadratic, cubic, quartic, and quintic series, are mentioned in the paper.
Mathematics Subject Classification ID: 33D15
Mathematics Subject Classification ID: 33C20
zbMATH DE Number: 7344949
zbMATH Keywords: infinite series for \(\pi\)
zbMATH Keywords: \(q\)-analogue
zbMATH Keywords: well-poised series
zbMATH Keywords: quadratic series
zbMATH Keywords: cubic series
zbMATH Keywords: quartic series
zbMATH Keywords: quintic series
zbMATH Keywords: Ramanujan's series for \(1/\pi\)
reviewed by: István Mező
MaRDI profile type: Publication
full work available at URL: s11139-019-00192-7
OpenAlex ID: W2998997387
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DOI: 10.1007/S11139-019-00192-7
description: scientific article; zbMATH DE number 7344949