Inequalities and infinite product formula for Ramanujan generalized modular equation function
type: Item
instance of: scholarly article
title: Inequalities and infinite product formula for Ramanujan generalized modular equation function
zbMATH Open document ID: 1444.11058
author: Miao-Kun Wang
author: Yongmin Li
author: Yu-Ming Chu
published in: The Ramanujan Journal
publication date: 2018-05-31T00:00:00Z
review text: Let \[ F(a, b ; c ; x)= {}_2F_{1}(a, b ; c ; x)=\sum_{n=0}^{\infty} \frac{(a, n)(b, n)}{(c, n)} \frac{x^{n}}{n !} \quad(|x|<1) \] be the Gaussian hypergeometric function. Let also \(a \in(0,1 / 2], r \in(0,1)\) and \(p>0 .\) Then the Ramanujan generalized modular equation with signature \(1 / a\) and degree \(p\) is given by \[ \frac{F\left(a, 1-a ; 1 ; 1-s^{2}\right)}{F\left(a, 1-a ; 1 ; s^{2}\right)}=p \frac{F\left(a, 1-a ; 1 ; 1-r^{2}\right)}{F\left(a, 1-a ; 1 ; r^{2}\right)} \] Making use of the decreasing homeomorphism \(\mu_{a}:(0,1) \rightarrow(0, \infty)\) defined by \[ \mu_{a}(r)=\frac{\pi}{2 \sin (\pi a)} \frac{F\left(a, 1-a ; 1 ; 1-r^{2}\right)}{F\left(a, 1-a ; 1 ; r^{2}\right)} \] we obtain the Ramanujan generalized modular equation function. The authors introduce several inequalities for \(\mu_{a}(r)\) and provide an infinite product formula for \(\mu_{1/4}(r)\) .
reviewed by: Pierluigi Vellucci
Mathematics Subject Classification ID: 11F03
Mathematics Subject Classification ID: 33C05
zbMATH DE Number: 6879666
zbMATH Keywords: Gaussian hypergeometric function
zbMATH Keywords: Ramanujan generalized modular equation
zbMATH Keywords: quadratic transformation
zbMATH Keywords: infinite product
MaRDI profile type: Publication
full work available at URL: s11139-017-9888-3
OpenAlex ID: W2603784699
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DOI: 10.1007/S11139-017-9888-3
describes a project that uses: Digital Library of Mathematical Functions
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description: scientific article; zbMATH DE number 6879666