New proofs of identities of Lebesgue and Gollnitz via tilings

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:116

Article

Little, David R.¹  Sellers, James A.¹ 

[1] Penn State Univ, Dept Math, University Pk, PA 16802 USA

关键词: Pell numbers;    Lebesgue identities;    Gollnitz identities;    Rogers-Ramanujan identities;    Tilings;   

DOI  :  10.1016/j.jcta.2008.05.004

来源: Elsevier

【 摘 要 】

In 1840, V.A. Lebesgue proved the following two series-product identities: Sigma(n >= 0) (-1: q)(n)/(q)(n) q((n+1 2)) = Pi(n >= 1) 1+q(2n-1)/1-q(2n-1), Sigma(n >= 0) (-q: q)(n)/(q)(n) q((n+1 2)) = Pi(n >= 1) 1-q(4n)/1-q(n) , These can be viewed as specializations of the following more general result: Sigma(n >= 0) (-z: q)(n)/(q)(n) q((n+1 2)) = Pi(n >= 1) (1+q(n))(1+zq(2n-1)). There are numerous combinatorial proofs of this identity, all of which describe a bijection between different types of integer partitions. Our goal is to provide a new, novel combinatorial proof that demonstrates how both sides of the above identity enumerate the same collection of weighted Pell tilings. In the process, we also provide a new proof of the Gollnitz identities. (C) 2008 Elsevier Inc. All rights reserved.

【 授权许可】

Free