New proofs of identities of Lebesgue and Göllnitz via tilings
type: Item
instance of: scholarly article
title: New proofs of identities of Lebesgue and Göllnitz via tilings
zbMATH Open document ID: 1189.05024
author: David P. Little
author: James A. Sellers
published in: Journal of Combinatorial Theory. Series A
publication date: 2009-03-04T00:00:00Z
review text: The authors propose a new method for proving the q-identity \[ \sum_{n\geq 0} \frac{(-z,q)_n}{(q)_n} q^{\binom{n+1}{2}} = \prod_{n\geq 1} (1+q^n)(1+z q^{2n-1}). \tag{\(*\)} \] This identity includes the classical identities of V.A. Lebesgue (1840), corresponding to the cases of \(z=1\) and \(z=q\). Moreover, from identities obtained in this paper two known Göllnitz identities are derived. The method proposed in this paper is in fact original, according to the authors:``that demonstrates how both sides of (\(*\)) enumerate the same collection of weighted Pell tilings''. The whole charm of the Little-Sellers method consists in introducting the generating function that counts all Pell tilings by a special weight. It allows, afterwards, to generate many useful identities for this function, from which the \(q\)-identities discussed here easily follow. Let us mention that \textit{J. P. O. Santos} and \textit{A. V. Sills} in their paper [``q-Pell sequences and two identities of V. A. Lebesgue'', Discrete Math. 257, No.\,1, 125--142 (2002; Zbl 1007.05017)] considered finite analogues of Lebesgue's \(q\)-identities. It was in that paper that the connections between Lebesgue's identities and \(q\)-Pell sequences were presented for the first time.
reviewed by: Roman Wituła
Mathematics Subject Classification ID: 05A19
Mathematics Subject Classification ID: 05A17
zbMATH DE Number: 5523128
zbMATH Keywords: Pell numbers
zbMATH Keywords: Lebesgue identities
zbMATH Keywords: Göllnitz identities
zbMATH Keywords: Rogers-Ramanujan identities
zbMATH Keywords: tilings
MaRDI profile type: Publication
full work available at URL: j.jcta.2008.05.004
OpenAlex ID: W2102814797
cites work: Partition identities and a continued fraction of Ramanujan
cites work: Q3801558
cites work: A bijection for Lebesgue's partition identity in the spirit of Sylvester
cites work: Arithmetic properties of overpartitions into odd parts
cites work: Partition bijections, a survey
cites work: A NEW EXPLORATION OF THE LEBESGUE IDENTITY
cites work: \(q\)-Pell sequences and two identities of V. A. Lebesgue
cites work: Further Identities of the Rogers-Ramanujan Type
DOI: 10.1016/J.JCTA.2008.05.004
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description: scientific article; zbMATH DE number 5523128