New proofs of Ramanujan's partition identities for moduli 5 and 7
type: Item
instance of: scholarly article
title: New proofs of Ramanujan's partition identities for moduli 5 and 7
zbMATH Open document ID: 0844.11064
author: Heng Huat Chan
published in: Journal of Number Theory
publication date: 1996-09-01T00:00:00Z
review text: Let \(|q|< 1\) and \((a; q)_\infty= \prod^\infty_{n=0} (1- aq^n)\). Ramanujan found identities for \(q(q^5; q^5 )^5_\infty/ (q;q)_\infty\) and \(q(q; q)^3_\infty (q^7; q^7)^3_\infty+ 8q^2 (q^7; q^7)^7_\infty/ (q; q)_\infty\). These identities lead to Ramanujan's celebrated congruences \(p(5n+ 4)\equiv 0\bmod 5\) and \(p(7n+ 5)\equiv 0\bmod 7\) where \(p(n)\) denotes the partition function. As to the proofs of the identities, see \textit{W. N. Bailey} [Q. J. Math., Oxf., II. Ser. 3, 29-31 (1952; Zbl 0046.04203) and 158-160 (1952; Zbl 0046.27202)], \textit{N. J. Fine} [Tôhoku Math. J., II. Ser. 8, 149-164 (1956; Zbl 0073.02904)]\ and \textit{S. Raghavan} [Q. J. Math., Oxf., II. Ser. 37, 221-229 (1986; Zbl 0585.10033)]. In the paper under review, new proofs of the identities are given. The common feature of these two proofs is that they begin with the same trigonometric identity. The proof of the first identity is elementary. As to the second identity, also a short and elegant proof by F. Garvan is included.
Mathematics Subject Classification ID: 11P83
Mathematics Subject Classification ID: 05A17
Mathematics Subject Classification ID: 11P82
Mathematics Subject Classification ID: 11F03
zbMATH DE Number: 792473
zbMATH Keywords: Ramanujan's partition identities
zbMATH Keywords: modular identities
zbMATH Keywords: congruences for the partition function
reviewed by: Mihály Szalay
MaRDI profile type: Publication
full work available at URL: jnth.1995.1083
OpenAlex ID: W2083645009
DOI: 10.1006/JNTH.1995.1083
description: scientific article; zbMATH DE number 792473