### Abstract

For any partition λ let ω(λ) denote the four parameter weight (equation presented) and let ℓ(λ) be the length of λ. We show that the generating function Σω(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 prove G.E. Andrews' result by specializing some parameters and C. Boulet's results when N → +∞. In the last section we study the weighted sum Σω(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 | 490-499 |

Number of pages | 10 |

Publication status | Published - Dec 1 2006 |

Externally published | Yes |

Event | 18th Annual International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2006 - San Diego, CA, United States Duration: Jun 19 2006 → Jun 23 2006 |

### Other

Other | 18th Annual International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2006 |
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Country | United States |

City | San Diego, CA |

Period | 6/19/06 → 6/23/06 |

### Keywords

- Al-Salam-Chihara polynomials
- Basic hypergeometric functions
- Minor summation formula of Pfaffians
- Partitions
- Pfaffians
- Schur's Q-functions
- Symmetric functions

### ASJC Scopus subject areas

- Algebra and Number Theory

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## Cite this

Ishikawa, M., & Zeng, J. (2006).

*The partition function of Andrews and Stanley and Al-Salam-Chihara polynomials*. 490-499. Paper presented at 18th Annual International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2006, San Diego, CA, United States.