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

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.