$q$-Hermite polynomials, biorthogonal rational functions, and $q$-beta integrals
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- by M. E. H. Ismail and D. R. Masson PDF
- Trans. Amer. Math. Soc. 346 (1994), 63-116 Request permission
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.References
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Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 346 (1994), 63-116
- MSC: Primary 33D15; Secondary 33D05, 33D45
- DOI: https://doi.org/10.1090/S0002-9947-1994-1264148-6
- MathSciNet review: 1264148