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

Abstract

For any partition λ let ω ( λ ) denote the four parameter weight ω ( λ ) = a ∑ i ≥ 1 ⌈ λ 2 i − 1 / 2 ⌉ b ∑ i ≥ 1 ⌊ λ 2 i − 1 / 2 ⌋ c ∑ i ≥ 1 ⌈ λ 2 i / 2 ⌉ d ∑ i ≥ 1 ⌊ λ 2 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.