A few theta-function identities and some of Ramanujan's modular equations
type: Item
instance of: scholarly article
title: A few theta-function identities and some of Ramanujan's modular equations
zbMATH Open document ID: 0973.11050
DOI: 10.1023/A:1009835301693
author: Nayandeep Deka Baruah
published in: The Ramanujan Journal
publication date: 2001-12-02T00:00:00Z
review text: Let \(\varphi(q)=\sum_{k=-\infty}^\infty q^{k^2}\), \(\psi(q)=\sum_{k=0}^\infty q^{k(k+1)/2}\) and \[ \alpha(q)=16q\frac{\psi^4(q^2)}{\varphi^4(q)}. \] A modular equation of degree \(n\) is defined as an identity satisfied by \(\alpha(q)\) and \(\alpha(q^n)\). A modular equation of composite degrees \(r,s,rs\) is an identity satisfied by \(\alpha(q^j)\) with \(j=1,r,s,\) and \(rs\). In his notebooks, S. Ramanujan recorded many modular equations without proofs. The proofs of these modular equations are now available, thanks to \textit{B. C. Berndt} and his collaborators [see Ramanujan's notebooks. III, Springer, New York (1991; Zbl 0733.11001); IV (1994; Zbl 0785.11001); V (1998; Zbl 0886.11001)]. Unfortunately, some of the proofs provided by Berndt and others are verifications which require the knowledge of the modular equations in advance. As such, new proofs are needed in order to shed some light on the methods used by Ramanujan in his discovery of these identities. In this paper, the author succeeds in using elementary methods and theta-functions identities to derive elegant modular equations such as \[ Q^4 +\frac{1}{Q^4}-2\left(P^2+\frac{1}{P^2}\right)+3=0, \] where \[ P=\{256 \alpha\beta\gamma\delta(1-\alpha)(1-\beta)(1-\gamma)(1-\delta)\}^{1/48} \] and \[ Q=\left(\frac{\beta\delta(1-\beta)(1-\delta)}{\alpha\gamma(1-\alpha)(1-\gamma)}\right)^{1/48}, \] with \(\alpha=\alpha(q),\beta=\alpha(q^3),\gamma=\alpha(q^5),\) and \(\delta=\alpha(q^{15}).\) The reviewer wishes to highlight a paper written by \textit{M. L. Lang} and himself [Isr. J. Math. 103, 1-16 (1998; Zbl 0922.11040)] which indicates that the theory of modular forms can be used to derive (instead of just verifying) many new modular equations analogous to those considered by S. Ramanujan.
reviewed by: Heng Huat Chan
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 11F27
Mathematics Subject Classification ID: 33E05
zbMATH DE Number: 1563049
zbMATH Keywords: modular equations
zbMATH Keywords: theta function
MaRDI profile type: Publication
full work available at URL: a:1009835301693
OpenAlex ID: W148714787
description: scientific article; zbMATH DE number 1563049