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

We show that the sum of the four parameter weights over all ordinary or strict partitions with parts each less than or equal to a given integer $N$ is expressed by the Al-Salam Chihara polynomials. This weight is a generalization of the Andrews-Stanley partition function. As a corollary we prove C. Boulet’s results when the sum runs over all ordinary or strict partitions. In the last section we study the weighted sum of Schur’s $P$-functions and $Q$-functions, where the sum runs over all strict partitions.