### Abstract

For any partition λ let ω (λ) denote the four parameter weight ω (λ) = a^{∑i ≥ 1 ⌈ λ2 i - 1 / 2 ⌉} b^{∑i ≥ 1 ⌊ λ2 i - 1 / 2 ⌋} c^{∑i ≥ 1 ⌈ λ2 i / 2 ⌉} d^{∑i ≥ 1 ⌊ λ2 i / 2 ⌋}, and let ℓ (λ) be the length of λ. We show that the generating function ∑ ω (λ) z^{ℓ (λ)}, where the sum runs over all ordinary (resp. strict) partitions with parts each ≤ N, can be expressed by the Al-Salam-Chihara polynomials. As a corollary we derive Andrews' result by specializing some parameters and Boulet's results by letting N → + ∞. In the last section we prove a Pfaffian formula for the weighted sum ∑ ω (λ) z^{ℓ (λ)} P_{λ} (x) where P_{λ} (x) is Schur's P-function and the sum runs over all strict partitions.

Original language | English |
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Pages (from-to) | 151-175 |

Number of pages | 25 |

Journal | Discrete Mathematics |

Volume | 309 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 6 2009 |

Externally published | Yes |

### Keywords

- Al-Salam-Chihara polynomials
- Andrews-Stanley partition function
- Basic hypergeometric series
- Minor summation formula of Pfaffians
- Schur's Q-functions

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

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## Cite this

*Discrete Mathematics*,

*309*(1), 151-175. https://doi.org/10.1016/j.disc.2007.12.064