### 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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Title of host publication | FPSAC 2006 - Proceedings: 18th Annual International Conference on Formal Power Series and Algebraic Combinatorics |

Pages | 490-499 |

Number of pages | 10 |

Publication status | Published - 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 |

### Fingerprint

### 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

### Cite this

*FPSAC 2006 - Proceedings: 18th Annual International Conference on Formal Power Series and Algebraic Combinatorics*(pp. 490-499)

**The partition function of Andrews and Stanley and Al-Salam-Chihara polynomials.** / Ishikawa, Masao; Zeng, Jiang.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*FPSAC 2006 - Proceedings: 18th Annual International Conference on Formal Power Series and Algebraic Combinatorics.*pp. 490-499, 18th Annual International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2006, San Diego, CA, United States, 6/19/06.

}

TY - GEN

T1 - The partition function of Andrews and Stanley and Al-Salam-Chihara polynomials

AU - Ishikawa, Masao

AU - Zeng, Jiang

PY - 2006

Y1 - 2006

N2 - 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.

AB - 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.

KW - Al-Salam-Chihara polynomials

KW - Basic hypergeometric functions

KW - Minor summation formula of Pfaffians

KW - Partitions

KW - Pfaffians

KW - Schur's Q-functions

KW - Symmetric functions

UR - http://www.scopus.com/inward/record.url?scp=84860633562&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84860633562&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84860633562

SP - 490

EP - 499

BT - FPSAC 2006 - Proceedings: 18th Annual International Conference on Formal Power Series and Algebraic Combinatorics

ER -