On the expansion of a continued fraction of Gordon
type: Item
instance of: scholarly article
title: On the expansion of a continued fraction of Gordon
zbMATH Open document ID: 0993.30003
DOI: 10.1023/A:1013939720688
author: Michael D. Hirschhorn
published in: The Ramanujan Journal
publication date: 2002-05-12T00:00:00Z
review text: The infinite product \[ P=\frac{(q^3;q^8)_\infty (q^5;q^8)_\infty}{(q;q^8)_\infty (q^7;q^8)_\infty} \] has a very nice and simple continued fraction expansion. In this paper Hirschhorn proves that the power series expansions of both \(P\) and \(1/P\) have beautiful regularities. Indeed, if \(P=\sum a_nq^n\) and \(P^{-1}=\sum b_nq^n\), then \[ a_{8n}>0,\quad a_{8n+1}>0,\quad a_{8n+2}>0,\quad a_{8n+3}=0, \] \[ a_{8n+12}>0,\quad a_{8n+5}<0,\quad a_{8n+6}<0,\quad a_{8n+7}=0, \] \[ b_{8n}>0,\quad b_{8n+1}<0,\quad b_{8n+2}=0,\quad b_{8n+3}>0, \] \[ b_{8n+4}<0,\quad b_{8n+5}>0,\quad b_{8n+6}=0,\quad b_{8n+7}<0 . \]
Mathematics Subject Classification ID: 11A55
Mathematics Subject Classification ID: 11B65
zbMATH DE Number: 1738884
zbMATH Keywords: Gordon's continued fraction
zbMATH Keywords: power series expansion
zbMATH Keywords: periodicity of sign of coefficients
zbMATH Keywords: Jacobi's triple product
reviewed by: Lisa Lorentzen
MaRDI profile type: Publication
full work available at URL: a:1013939720688
OpenAlex ID: W122926215
description: scientific article; zbMATH DE number 1738884