Septic theta function identities in Ramanujan's lost notebook
type: Item
instance of: scholarly article
title: Septic theta function identities in Ramanujan's lost notebook
zbMATH Open document ID: 0980.33015
author: Seung Hwan Son
published in: Acta Arithmetica
publication date: 2001-06-17T00:00:00Z
Mathematics Subject Classification ID: 33D99
Mathematics Subject Classification ID: 11F32
Mathematics Subject Classification ID: 11R29
zbMATH DE Number: 1605230
zbMATH Keywords: theta series
zbMATH Keywords: modular equation
zbMATH Keywords: Jacobi triple product identity
zbMATH Keywords: Lambert series
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4064/aa98-4-3
OpenAlex ID: W1994837615
DOI: 10.4064/AA98-4-3
review text: Let \(\varphi(q)\) be the theta series \(\sum _{k=-\infty} ^{\infty}q^{k^2}\). In his so-called ``Lost Notebook'', Ramanujan records an identity NEWLINE\[NEWLINE\frac {\varphi(q^{1/7})} {\varphi(q^{7})} =1+u+v+w,NEWLINE\]NEWLINE where \(u,v,w\) are implicitly given by certain algebraic equations. Explicit expressions for \(u,v,w\) can be extracted by comparison with an entry in Ramanujan's Second Notebook. The identity together with the algebraic equations are proven in this paper, by employing modular equations of degree seven, the Jacobi triple product identity, several Lambert series identities and a recent product formula for theta functions due to the author.
reviewed by: C. Krattenthaler
description: scientific article; zbMATH DE number 1605230