  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.
