Modular equations for the Rogers-Ramanujan continued fraction and the Dedekind eta-function
type: Item
instance of: scholarly article
title: Modular equations for the Rogers-Ramanujan continued fraction and the Dedekind eta-function
zbMATH Open document ID: 1043.11041
DOI: 10.1023/A:1013991704758
author: Jinhee Yi
published in: The Ramanujan Journal
publication date: 2002-05-12T00:00:00Z
review text: For \(| q|<1\), let \(R(q)\) denote the Ramanujan continued fraction, that is, \[ R(q)=\frac {q^{1/5}}{1+ \frac {q}{1+ \frac {q^2}{1+ \frac {q^3}{\ddots}}}} \] In his Lost Notebook, S. Ramanujan found five stunning polynomial identities relating \(R(q)\) and \(R(-q)\), \(R(q^2)\), \(R(q^3)\), \(R(q^4)\), and \(R(q^5)\), respectively. The author provides new proofs for these identities and, in addition, proves a new relation between \(u=R(q)\) and \(v=R(q^7)\), namely \[ \frac {1} {7}(u^7-v)(1+uv^7)=uv^3(1-u^5)+u^2v^5(1+u^5) -u^5v(1-v^5)-u^3v^2(1+v^5)-5u^4v^4. \] The proofs are based on certain modular equations, which are established by the use of the theory of modular forms.
reviewed by: C. Krattenthaler
Mathematics Subject Classification ID: 11F20
Mathematics Subject Classification ID: 33D90
Mathematics Subject Classification ID: 11A55
zbMATH DE Number: 1738885
zbMATH Keywords: modular equation
zbMATH Keywords: Rogers-Ramanujan continued fraction
zbMATH Keywords: Dedekind eta-function
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1738885