On the Rogers-Ramanujan continued fraction
type: Item
instance of: scholarly article
title: On the Rogers-Ramanujan continued fraction
zbMATH Open document ID: 0565.10009
DOI: 10.1007/BF02840651
author: K. G. Ramanathan
published in: Proceedings of the Indian Academy of Sciences. Mathematical Sciences
publication date: 1984-01-01T00:00:00Z
review text: \textit{L. J. Rogers} [Proc. Lond. Math. Soc. 25, 318--343 (1894; JFM 25.0432.01)] and S. Ramanujan [see \textit{G. H. Hardy}, ``Ramanujan'' (1940; Zbl 0025.10505)] independently found that the continued fraction \(\frac{1}{1+}\frac{x}{1+}\frac{x^ 2}{1+}\cdots\) is equal to the ratio of theta functions \[ \prod (1-x^{5n-1})(1-x^{5n-4})/(1-x^{5n-2})(1- x^{5n-3}). \] In unpublished work, Ramanujan used this equality to explicitly evaluate the continued fraction for certain values of \(x\). The author generalizes Ramanujan's results in finding a large class of \(x\)'s for which he can explicitly evaluate the continued fraction, namely \(x=-\exp(-\pi /\sqrt{5n})\) or \(-\exp(-\pi\sqrt{n/5})\) where \(n\) is a positive odd integer and \({\mathbb Q}(\sqrt{-5n})\) has discriminant \(-5n\) and class number two.
Mathematics Subject Classification ID: 11A55
zbMATH DE Number: 3900847
zbMATH Keywords: evaluation of continued fractions
zbMATH Keywords: theta functions
Wikidata QID: Q56458175
reviewed by: David M. Bressoud
MaRDI profile type: Publication
cites work: Q5521595
cites work: Ramanujan and the congruence properties of partitions
cites work: Q3934463
full work available at URL: https://doi.org/10.1007/bf02840651
OpenAlex ID: W2022582561
description: scientific article; zbMATH DE number 3900847