Note on trigonometric expansions of theta functions
type: Item
instance of: scholarly article
title: Note on trigonometric expansions of theta functions
zbMATH Open document ID: 1026.33012
DOI: 10.1016/S0377-0427(02)00631-3
author: A. Raouf Chouikha
published in: Journal of Computational and Applied Mathematics
publication date: 2003-05-22T00:00:00Z
review text: Let \(\theta_4(\nu,\tau) = \sum_{n=-\infty}^\infty (-1)^nq^{n^2}e^{2\pi in\nu}\), where \(q=e^{i\pi\tau}\). Then it is well known that \[ \log\frac{\theta_4(\nu,\tau)}{\theta_4(0,\tau)} = 2\sum_{n=1}^\infty \frac{q^n}{n(1-q^{2n})}(1-\cos 2n\pi\nu). \] One of the main results of this article is an expansion in powers of \(\sin\pi\nu\), namely \[ \log\frac{\theta_4(\nu,\tau)}{\theta_4(0,\tau)} =\sum_{p=1}^\infty c_{2p}(\tau)(\sin\pi\nu)^{2p}. \] Explicit formulas for the coefficients \(c_{2p}(\tau)\) are obtained by expanding \(\cos 2n\pi\nu\) as a polynomial in \(\sin \pi \nu\). The other main result is a system of differential recurrence relations for the coefficients \(c_{2p}(\tau)\). These relations are obtained from the partial differential equation (i.e., the heat equation) satisfied by \(\theta_4(\nu,\tau).\)
reviewed by: Shaun Cooper
Mathematics Subject Classification ID: 33E05
Mathematics Subject Classification ID: 33E20
Mathematics Subject Classification ID: 35C10
zbMATH DE Number: 1915226
zbMATH Keywords: theta functions
zbMATH Keywords: elliptic functions
zbMATH Keywords: heat equation
zbMATH Keywords: differential equation
MaRDI profile type: Publication
cites work: Q3477147
cites work: Q4938571
cites work: Q5847356
cites work: Tata lectures on theta. II: Jacobian theta functions and differential equations. With the collaboration of C. Musili, M. Nori, E. Previato, M. Stillman, and H. Umemura
cites work: Q4716292
full work available at URL: s0377-0427(02)00631-3
OpenAlex ID: W2048639594
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description: scientific article; zbMATH DE number 1915226