𝑞-Hermite polynomials, biorthogonal rational functions, and 𝑞-beta integrals

Abstract

We characterize the solutions of the indeterminate moment problem associated with the continuous q q -Hermite polynomials when q > 1 q > 1 in terms of their Stieltjes transforms. The extremal measures are found explicitly. An analog of the Askey-Wilson integral is evaluated. It involves integrating a kernel, similar to the Askey-Wilson kernel, against any solution of the q q -Hermite moment problem, provided that certain integrability conditions hold. This led to direct evaluation of several q q -beta integrals and their discrete analogs as well as a generalization of Bailey’s 6 ψ 6 {}_6{\psi _6} , sum containing infinitely many parameters. A system of biorthogonal rational functions is also introduced.