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A $ q$-beta integral on the unit circle and some biorthogonal rational functions


Authors: Waleed A. Al-Salam and Mourad E. H. Ismail
Journal: Proc. Amer. Math. Soc. 121 (1994), 553-561
MSC: Primary 33D45; Secondary 33D05
DOI: https://doi.org/10.1090/S0002-9939-1994-1215197-0
MathSciNet review: 1215197
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Abstract: In this paper we first consider a pair of polynomial sets which are biorthogonal on the unit circle with respect to a complex weight function. We then show how the biorthogonality of this pair of polynomial sets implies a q-beta integral which in turn leads to a pair of biorthogonal rational functions. Finally we show that the asymptotics for these pairs of rational functions exhibit qualitative properties reminiscent of the Szegö theory for orthogonal polynomials.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1994-1215197-0
Keywords: Biorthogonal rational functions, q-beta integrals, asymptotics, generating functions, Askey-Wilson integral
Article copyright: © Copyright 1994 American Mathematical Society

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