On the expansion of Ramanujan's continued fraction
type: Item
instance of: scholarly article
title: On the expansion of Ramanujan's continued fraction
zbMATH Open document ID: 0924.11005
DOI: 10.1023/A:1009789012006
published in: The Ramanujan Journal
publication date: 1999-11-15T00:00:00Z
review text: Ramanujan's continued fraction \(R(q)= 1+ \frac{q}{1+ \frac{q^2}{1+ \frac{q^3}{1+\dots}}}\) and its reciprocal \(R(q)^{-1}= \frac{1}{1+ \frac{q}{1+ \frac{q^2}{1+ \frac{q^3}{1+\dots}}}}\) can be expanded as series \[ \begin{aligned} R(q)&= 1+q-q^3+ q^5+\dots= \sum_{n=0}^\infty c(n)q^n \\ \text{and} R(q)^{-1}&= 1-q+q^2-q^4+\dots= \sum_{n=0}^\infty d(n) q^n.\end{aligned} \] M. Hirschhorn and G. Szekeres noticed that the sign of \(c(n)\) and \(d(n)\) for large \(n\) is periodic with period 5. \textit{B. Richmond} and \textit{G. Szekeres} [Acta Sci. Math. 40, 347-369 (1978; Zbl 0397.10046)] found an explicit expression in asymptotic form for \(c(n)\) and \(d(n)\). \textit{G. Andrews} [Adv. Math. 41, 186-208 (1981; Zbl 0477.33009)] proved a theorem with a combinatorial interpretation of \(c(n)\) and \(d(n)\), from which the periodicity follows easily. The author of the present paper proves the periodicity of \(c(n)\) and \(d(n)\) by finding another formulae for \(R(q)\) and \(R(q)^{-1}\), and he gives combinatorial interpretations of \(c(n)\) and \(d(n)\).
reviewed by: Tonko Tonkov
Mathematics Subject Classification ID: 11A55
Mathematics Subject Classification ID: 33E99
zbMATH DE Number: 1251336
zbMATH Keywords: series expansion
zbMATH Keywords: sign of coefficients
zbMATH Keywords: Ramanujan's continued fraction
zbMATH Keywords: periodicity
author: Michael D. Hirschhorn
MaRDI profile type: Publication
full work available at URL: a:1009789012006
OpenAlex ID: W57725924
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description: scientific article; zbMATH DE number 1251336