Some Lambert series expansions of products of theta functions
type: Item
instance of: scholarly article
title: Some Lambert series expansions of products of theta functions
zbMATH Open document ID: 0938.11017
DOI: 10.1023/A:1009853106329
author: Kenneth S. Williams
published in: The Ramanujan Journal
publication date: 2000-06-21T00:00:00Z
review text: Let \(q\) be a complex number satisfying \(|q|=1\) and define the theta function \(\varphi(q)\) by \(\varphi(q)= \sum_{k=-\infty}^\infty q^{x^2}\). Ramanujan has given a number of Lambert series expansions such as \[ \varphi(q) \varphi(q^2)= 1-2 \sum_{n=1}^\infty \frac{(-1)^{\frac{n(n+1)}{2}} q^{2n-1}} {1-q^{2n-1}}. \] A formula is proved which includes this and other expansions as special cases, in fact the theorem proved works for all 101 discriminants \(d\) for which every binary quadratic genus has only one class. The above example is the case \(d=-8\).
Mathematics Subject Classification ID: 11E16
Mathematics Subject Classification ID: 33D15
zbMATH DE Number: 1406508
zbMATH Keywords: theta functions
zbMATH Keywords: Lambert series
zbMATH Keywords: binary quadratic forms
reviewed by: Meinhard Peters
MaRDI profile type: Publication
full work available at URL: a:1009853106329
OpenAlex ID: W1496320939
description: scientific article; zbMATH DE number 1406508