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

Masao Ishikawa, Jiang Zeng

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

For any partition λ let ω (λ) denote the four parameter weight ω (λ) = ai ≥ 1 ⌈ λ2 i - 1 / 2 ⌉ bi ≥ 1 ⌊ λ2 i - 1 / 2 ⌋ ci ≥ 1 ⌈ λ2 i / 2 ⌉ di ≥ 1 ⌊ λ2 i / 2 ⌋, and let ℓ (λ) be the length of λ. We show that the generating function ∑ ω (λ) zℓ (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) zℓ (λ) Pλ (x) where Pλ (x) is Schur's P-function and the sum runs over all strict partitions.

Original languageEnglish
Pages (from-to)151-175
Number of pages25
JournalDiscrete Mathematics
Volume309
Issue number1
DOIs
Publication statusPublished - Jan 6 2009
Externally publishedYes

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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