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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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

Authors: M. E. H. Ismail and D. R. Masson
Journal: Trans. Amer. Math. Soc. 346 (1994), 63-116
MSC: Primary 33D15; Secondary 33D05, 33D45
MathSciNet review: 1264148
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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 $ {}_6{\psi _6}$, sum containing infinitely many parameters. A system of biorthogonal rational functions is also introduced.

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Keywords: Hamburger moment problem, indeterminate moment problem, Bailey's sum, Poisson kernel, biorthogonal rational functions, $ q$-beta integrals, asymptotics, $ q$-Hermite polynomials, theta functions, generating functions, Askey-Wilson integral
Article copyright: © Copyright 1994 American Mathematical Society