New proofs of identities of Lebesgue and Göllnitz via tilings

David P. Little, James A. Sellers

Research output: Contribution to journalArticlepeer-review

7 Scopus citations

Abstract

In 1840, V.A. Lebesgue proved the following two series-product identities:under(∑, n ≥ 0) frac((- 1 ; q)n, (q)n) q((n + 1; 2)) = under(∏, n ≥ 1) frac(1 + q2 n - 1, 1 - q2 n - 1),under(∑, n ≥ 0) frac((- q ; q)n, (q)n) q((n + 1; 2)) = under(∏, n ≥ 1) frac(1 - q4 n, 1 - qn) . These can be viewed as specializations of the following more general result:under(∑, n ≥ 0) frac((- z ; q)n, (q)n) q((n + 1; 2)) = under(∏, n ≥ 1) (1 + qn) (1 + z q2 n - 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 Göllnitz identities.

Original languageEnglish (US)
Pages (from-to)223-231
Number of pages9
JournalJournal of Combinatorial Theory. Series A
Volume116
Issue number1
DOIs
StatePublished - Jan 2009

Keywords

  • Göllnitz identities
  • Lebesgue identities
  • Pell numbers
  • Rogers-Ramanujan identities
  • Tilings

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