Theta-function identities and the explicit formulas for theta-function and their applications
type: Item
instance of: scholarly article
title: Theta-function identities and the explicit formulas for theta-function and their applications
zbMATH Open document ID: 1060.11027
author: Jinhee Yi
published in: Journal of Mathematical Analysis and Applications
publication date: 2004-08-06T00:00:00Z
review text: Let \(h_{k,n}= \frac{\varphi(e^{-\pi\sqrt{n/k}})} {k^{1/4}\varphi(e^{-\pi\sqrt{nk}})}\) and \(h_{k,n}'= \frac{\varphi(-e^{-2\pi\sqrt{n/k}})} {k^{1/4}\varphi(-e^{-2\pi\sqrt{nk}})},\) where \(\varphi(q)=\sum_{j=-\infty}^\infty q^{j^2}\). Properties of \(h_{k,n}\) and \(h_{k,n}'\) are studied, and \(h_{k,n}\) and \(h_{k,n}'\) are explicitly evaluated for several values of \(k\) and \(n\). These evaluations come from so-called P-Q modular equations, some of which were given by Ramanujan and others are established using results in his notebooks. The paper ends with explicit values for \(\varphi(e^{-k\pi})\) and \(\varphi(-e^{-2k\pi})\) for \(1\leq k \leq 6\).
reviewed by: Shaun Cooper
Mathematics Subject Classification ID: 11F27
Mathematics Subject Classification ID: 11F20
Mathematics Subject Classification ID: 33E05
zbMATH DE Number: 2082180
zbMATH Keywords: modular equation
zbMATH Keywords: theta function
zbMATH Keywords: Ramanujan-Weber class invariant
MaRDI profile type: Publication
full work available at URL: j.jmaa.2003.12.009
OpenAlex ID: W1966874439
Wikidata QID: Q56270731
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DOI: 10.1016/J.JMAA.2003.12.009
description: scientific article; zbMATH DE number 2082180