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

For any partition @l let @w(@l) denote the four parameter weight @w(@l)=a^@?^"^i^"^>=^"^1^@?^@l^"^2^"^i^"^-^"^1^/^2^@?b^@?^"^i^"^>=^"^1^@?^@l^"^2^"^i^"^-^"^1^/^2^@?c^@?^"^i^"^>=^"^1^@?^@l^"^2^"^i^/^2^@?d^@?^"^i^"^>=^"^1^@?^@l^"^2^"^i^/^2^@?, and let @?(@l) be the length of @l. We show that the generating function @[email protected](@l)z^@?^(^@l^), 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 @[email protected](@l)z^@?^(^@l^)P"@l(x) where P"@l(x) is Schur's P-function and the sum runs over all strict partitions.