On Ramanujan's cubic continued fraction as a modular function
type: Item
instance of: scholarly article
title: On Ramanujan's cubic continued fraction as a modular function
zbMATH Open document ID: 1248.11110
author: Bumkyu Cho
author: Ja Kyung Koo
author: Yoon Kyung Park
published in: Tôhoku Mathematical Journal. Second Series
publication date: 2011-02-23T00:00:00Z
review text: The authors study Ramanujan's cubic continued fraction as a modular function. For a complex number \(\tau \in \mathbb{U}\), the Rogers-Ramanujan continued fraction is defined by \[ r(\tau )=\frac{q^{1/5}}{1+\frac{q}{1+\frac{q^{2}}{1+\frac{q^{3}}{1+\cdots}}}} =q^{1/5}\prod\limits_{n=1}^{\infty }(1-q^{n})^{(\frac{n}{5})}, \] where \(q=e^{2\pi i\tau }\) and \((\frac{n}{5})\) is the Legendre function. Ramanujan proved that \[ r(i)=\sqrt{\frac{5+\sqrt{5}}{2}}-\frac{\sqrt{5}+1}{2}\;\;\;\text{and} \;\;r(\frac{5+i}{2})=\sqrt{\frac{5-\sqrt{5}}{2}}-\frac{\sqrt{5}-1}{2}. \] Later in 2005 W. Duke mentioned that Ramanujan's cubic continued fraction \(C(\tau )\) defined as \[ C(\tau )=\frac{q^{1/3}}{1+\frac{q+q^{2}}{1+\frac{q^{2}+q^{4}}{1+\frac{q^{3}+q^{6}}{1+\cdots}}}}=q^{1/3}\prod\limits_{n=1}^{\infty }\frac{(1-q^{6n-1})(1-q^{6n-5})}{(1-q^{6n-3})^{3}} \] with modularity for \(\Gamma (6)\). They prove some nice results on Ramanujan's cubic continued fraction.
reviewed by: Ahmet Tekcan
Mathematics Subject Classification ID: 11Y65
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 11R37
Mathematics Subject Classification ID: 11R04
Mathematics Subject Classification ID: 14H55
zbMATH DE Number: 5856432
zbMATH Keywords: Ramanujan cubic continued fraction
zbMATH Keywords: modular form
zbMATH Keywords: class field theory
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1294170348
OpenAlex ID: W1980575252
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DOI: 10.2748/TMJ/1294170348
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description: scientific article; zbMATH DE number 5856432