On certain identities of Ramanujan
type: Item
instance of: scholarly article
title: On certain identities of Ramanujan
zbMATH Open document ID: 0981.11016
author: Zhi-Guo Liu
published in: Journal of Number Theory
publication date: 2000-08-09T00:00:00Z
review text: Starting with a Ramanujan identity involving Lambert series related to modular equations of degree 7, the author gives new proofs of various theta function identities previously proved by \textit{B. Berndt} and \textit{L.-C. Zhang} [J. Number Theory 48, 224--242 (1994; Zbl 0812.11028)] using modular forms. The paper also gives a simple proof of one of Ramanujan's identities that implies the congruence \(p(7n+5)\equiv 0\pmod 7\) for the partition function. The author uses theta functions to construct an elliptic function whose poles are known and then notes that the sum of the residues at the poles inside a period parallelogram is zero.
Mathematics Subject Classification ID: 11B65
Mathematics Subject Classification ID: 11F27
Mathematics Subject Classification ID: 33E05
Mathematics Subject Classification ID: 11P83
zbMATH DE Number: 1486245
zbMATH Keywords: theta functions
zbMATH Keywords: elliptic functions
zbMATH Keywords: Lambert series
zbMATH Keywords: modular equation
zbMATH Keywords: partition
reviewed by: Tom M. Apostol
MaRDI profile type: Publication
full work available at URL: jnth.2000.2513
OpenAlex ID: W2028896193
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DOI: 10.1006/JNTH.2000.2513
description: scientific article; zbMATH DE number 1486245