Cranks and dissections in Ramanujan's lost notebook
type: Item
instance of: scholarly article
title: Cranks and dissections in Ramanujan's lost notebook
zbMATH Open document ID: 1114.11082
author: Bruce C. Berndt
author: Heng Huat Chan
author: Song Heng Chan
author: Wen-Chin Liaw
published in: Journal of Combinatorial Theory. Series A
publication date: 2005-02-22T00:00:00Z
review text: Employing the standard \(q\)-series notation, let \(F_a(q)\) be the crank generating function \[ F_a(q) = \frac{(q)_{\infty}}{(aq,q/a)_{\infty}}. \] In Ramanujan's lost notebook there appear two congruences for \(F_a(q)\), one of which we display here: \[ F_a(\sqrt{q}) \equiv \frac{f(-q^3,-q^5)}{(-q^2;q^2)_{\infty}} + \left(a-1+\frac{1}{a}\right) \sqrt{q} \frac{f(-q,-q^7)}{(-q^2;q^2)_{\infty}} \pmod{a^2 + a^{-2}}. \] The function \(f(a,b)\) is Ramanujan's theta function, \[ f(a,b) = \sum_{n \in \mathbb{Z}} a^{n(n+1)/2}b^{n(n-1)/2}. \] In this paper the authors prove Ramanujan's two congruences for \(F_a(q)\) as well as three others with a similar flavor. In each case two different proofs are given. In the conclusion, the authors explain an elegant observation, communicated to them by Frank Garvan, which shows that the congruences for \(F_a(q)\) are actually equivalent to some of Garvan's own identities involving the crank generating function [\textit{F. G. Garvan}, Trans. Am. Math. Soc. 305, No. 1, 47--77 (1988; Zbl 0641.10009) and Trans. Am. Math. Soc. 322, No. 1, 79--94 (1990; Zbl 0717.11042)].
reviewed by: Jeremy Lovejoy
Mathematics Subject Classification ID: 11P82
Mathematics Subject Classification ID: 11P83
zbMATH DE Number: 2136714
zbMATH Keywords: cranks
zbMATH Keywords: dissections
zbMATH Keywords: Ramanujan's lost notebook
zbMATH Keywords: partitions
zbMATH Keywords: congruences
MaRDI profile type: Publication
full work available at URL: j.jcta.2004.06.013
OpenAlex ID: W1967604935
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DOI: 10.1016/J.JCTA.2004.06.013
description: scientific article; zbMATH DE number 2136714