Q-HERMITE POLYNOMIALS, BIORTHOGONAL RATIONAL FUNCTIONS, AND Q-BETA-INTEGRALS

We characterize the solutions of the indeterminate moment problem asso ciated with the continuous q-Hermite polynomials when q > 1 in terms o f their Stieltjes transforms. The extremal measures are found explicit ly. An analog of the Askey-Wilson integral is evaluated. It involves i ntegrating a kernel, similar to the Askey-Wilson kernel, against any s olution of the q-Hermite moment problem, provided that certain integra bility conditions hold. This led to direct evaluation of several q-bet a integrals and their discrete analogs as well as a generalization of Bailey's 6psi6 sum containing infinitely many parameters. A system of biorthogonal rational functions is also introduced.