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

摘要

For any partition λ let ω(λ) denote the four parameter weight ω(λ)=a∑i≥1λ2i?1/2b∑i≥1λ2i?1/2c∑i≥1λ2i/2d∑i≥1λ2i/2,and let ?(λ) be the length of λ. We show that the generating function ∑ω(λ)z?(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?(λ)Pλ(x) where Pλ(x) is Schur’s P-function and the sum runs over all strict partitions.