New proofs of identities of Lebesgue and Göllnitz via tilings

David P. Little; James A. Sellers

doi:10.1016/j.jcta.2008.05.004

New proofs of identities of Lebesgue and Göllnitz via tilings

David P. Little
, James A. Sellers

Mathematics

Research output: Contribution to journal › Article › peer-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 + q^{2 n - 1}, 1 - q^{2 n - 1}),under(∑, n ≥ 0) frac((- q ; q)_n, (q)_n) q^{((n + 1; 2))} = under(∏, n ≥ 1) frac(1 - q^{4 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 q^{2 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 language	English (US)
Pages (from-to)	223-231
Number of pages	9
Journal	Journal of Combinatorial Theory. Series A
Volume	116
Issue number	1
DOIs	https://doi.org/10.1016/j.jcta.2008.05.004
State	Published - Jan 2009

All Science Journal Classification (ASJC) codes

Theoretical Computer Science
Discrete Mathematics and Combinatorics
Computational Theory and Mathematics

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10.1016/j.jcta.2008.05.004

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@article{74e658bec4e54984a8465b93645c80f0,

title = "New proofs of identities of Lebesgue and G{\"o}llnitz via tilings",

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 G{\"o}llnitz identities.",

author = "Little, \{David P.\} and Sellers, \{James A.\}",

year = "2009",

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doi = "10.1016/j.jcta.2008.05.004",

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AB - 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.

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New proofs of identities of Lebesgue and Göllnitz via tilings

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