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q-Hermite Polynomials, Biorthogonal Rational Functions, and q-Beta Integrals

M. E. H. Ismail and D. R. Masson
Transactions of the American Mathematical Society
Vol. 346, No. 1 (Nov., 1994), pp. 63-116
DOI: 10.2307/2154943
Stable URL: http://www.jstor.org/stable/2154943
Page Count: 54
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q-Hermite Polynomials, Biorthogonal Rational Functions, and q-Beta Integrals
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Abstract

We characterize the solutions of the indeterminate moment problem associated with the continuous q-Hermite polynomials when $q > 1$ 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-Hermite moment problem, provided that certain integrability conditions hold. This led to direct evaluation of several q-beta integrals and their discrete analogs as well as a generalization of Bailey's 6ψ6 sum containing infinitely many parameters. A system of biorthogonal rational functions is also introduced.

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