Partition identities and a continued fraction of Ramanujan
type: Item
instance of: scholarly article
title: Partition identities and a continued fraction of Ramanujan
zbMATH Open document ID: 0781.05006
DOI: 10.1016/0097-3165(93)90061-C
author: Krishnaswami Alladi
author: Basil Gordon
published in: Journal of Combinatorial Theory. Series A
publication date: 1993-12-15T00:00:00Z
review text: The authors explore identities for the numerator and denominator of Ramanujan's continued fraction: \[ R(a,b)=1+{bq \over \displaystyle 1+aq+ {bq^ 2 \over \displaystyle 1+aq^ 2+ {bq^ 3 \over \displaystyle 1+\cdots}}}. \] In particular, they find partition identities that generalize results of Sylvester, Göllnitz, and this reviewer. Representative of their results is the identity that the number of partitions of \(n+j\) into \(i+j\) distinct red parts and \(j\) distinct blue parts, each blue part being \(\leq i+j\), is equal to the number of partitions of \(n\) into \(i\) red parts and \(j\) blue parts such that all parts have distinct values and no part is one greater than any blue part. They also generalize the Odlyzko and Wilf result on the number of \(n\)- coin fountains with \(k\) coins at the base to fountains of two-colored coins.
Mathematics Subject Classification ID: 05A17
Mathematics Subject Classification ID: 11A55
Mathematics Subject Classification ID: 11P81
zbMATH DE Number: 217949
zbMATH Keywords: numerator
zbMATH Keywords: denominator
zbMATH Keywords: Ramanujan's continued fraction
zbMATH Keywords: partition identities
reviewed by: David M. Bressoud
MaRDI profile type: Publication
cites work: Chapter 16 of Ramanujan’s second notebook: theta-functions and 𝑞-series
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cites work: The Editor's Corner: n Coins in a Fountain
cites work: Q4040264
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description: scientific article; zbMATH DE number 217949