## 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 - q^{n}) . 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 + q^{n}) (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 | |

State | Published - Jan 2009 |

## Keywords

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