A moment problem and a family of integral evaluations

Abstract

We study the Al-Salam–Chihara polynomials when $q>1$. Several solutions of the associated moment problem are found, and the orthogonality relations lead to explicit evaluations of several integrals. The polynomials are shown to have raising and lowering operators and a second order operator equation of Sturm-Liouville type whose eigenvalues are found explicitly. We also derive new measures with respect to which the Ismail-Masson system of rational functions is biorthogonal. An integral representation of the right inverse of a divided difference operator is also obtained.