New proofs for two infinite product identities
type: Item
instance of: scholarly article
title: New proofs for two infinite product identities
zbMATH Open document ID: 0767.11048
author: Richard Blecksmith
author: John Brillhart
author: Irving Gerst
published in: Rocky Mountain Journal of Mathematics
publication date: 1993-05-16T00:00:00Z
review text: The identities in question are \[ \begin{aligned} \Pi_ 1(1-x^ n)\;\Pi_ 2(1+x^ n)&=\sum_{n=0}^ \infty (x^{2n(n+1)}+x^{6n(n+1)+1})\qquad\text{ and}\\ \Pi_ 3(1-x^ n)\;\Pi_ 4(1+x^ n)&=\sum_{n=0}^ \infty (x^{n(n+1)}+x^{5n(n+1)+1}),\end{aligned} \] where the products are extended over all positive integers in the following residue classes: \(\Pi_ 1\): \(n\equiv 0,\pm5,\pm7\pmod{24}\); \(\Pi_ 2\): \(n\equiv\pm1,\pm4,6\pmod{24}\); \(\Pi_ 3\): \(n\equiv 0,\pm3\pmod{10}\); \(\Pi_ 4\): \(n\equiv\pm1,\pm2,\pm4\pmod{10}\).
reviewed by: Tom M. Apostol
Mathematics Subject Classification ID: 11P99
Mathematics Subject Classification ID: 05A19
zbMATH DE Number: 168029
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181072697
OpenAlex ID: W1978960833
cites work: Chapter 16 of Ramanujan’s second notebook: theta-functions and 𝑞-series
cites work: Q3202197
cites work: Parity Results for Certain Partition Functions and Identities Similar to Theta Function Identities
cites work: Some Infinite Product Identities
DOI: 10.1216/RMJM/1181072697
description: scientific article; zbMATH DE number 168029