Theta function identities
type: Item
instance of: scholarly article
title: Theta function identities
zbMATH Open document ID: 0707.11033
DOI: 10.1016/0022-247X(90)90387-U
author: Ron Evans
published in: Journal of Mathematical Analysis and Applications
publication date: 1990-01-01T00:00:00Z
review text: All but one of the identities in the 21 chapters of Ramanujan's second Notebook had been proved by 1986. This paper proves the remaining identity, which involves a ratio of Dedekind eta functions \(\eta^ 2(z/p)/\eta^ 2(z)\) with \(p=13\). The author's method is quite general and constructs, for each odd integer \(p>1\), a class of modular functions on a subgroup \(\Gamma^ 0(p)\) of the modular group with no poles in the upper half plane or at the cusp 0. This yields a number of interesting identities involving the ratio \(\eta^ 2(z/p)/\eta^ 2(z)\) for any odd \(p>1\). These include the case with \(p=13\) mentioned above and several other special cases with \(p=5,7,9\), and 17 stated by Ramanujan.
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 11F20
Mathematics Subject Classification ID: 11F03
zbMATH DE Number: 4160842
zbMATH Keywords: Ramanujan's second Notebook
zbMATH Keywords: Dedekind eta functions
zbMATH Keywords: modular functions
zbMATH Keywords: modular group
MaRDI profile type: Publication
cites work: Chapter 16 of Ramanujan’s second notebook: theta-functions and 𝑞-series
cites work: Q3218192
cites work: Q3995284
cites work: Ramanujan's Notebooks
cites work: Q3476890
cites work: Q3803117
cites work: Q5675772
cites work: Q5666618
cites work: Generalisations of some theorems of Ramanujan
cites work: Q5513003
cites work: Q4040264
cites work: Q4154638
cites work: Q4771024
cites work: \(L\)-functions at \(s=1\). IV: First derivatives at \(s=0\)
full work available at URL: 0022-247x(90)90387-u
OpenAlex ID: W2021864007
Recommended article: Some theta-function identities
Recommended article: Some identities involving theta functions
Recommended article: On a theta function identity
Recommended article: Q3012448
Recommended article: A theta function identity and its implications
Recommended article: A theta function identity and its applications
Recommended article: Q3584314
Recommended article: Q3338381
Recommended article: Q3520665
Recommended article: Q3534546
description: scientific article; zbMATH DE number 4160842