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We characterize the solutions of the indeterminate moment problem associated with the continuous      q q      -Hermite polynomials when      q &gt; 1 q &gt; 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.

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