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A q-Beta Integral on the Unit Circle and Some Biorthogonal Rational Functions

Waleed A. Al-Salam and Mourad E. H. Ismail
Proceedings of the American Mathematical Society
Vol. 121, No. 2 (Jun., 1994), pp. 553-561
DOI: 10.2307/2160434
Stable URL: http://www.jstor.org/stable/2160434
Page Count: 9
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A q-Beta Integral on the Unit Circle and Some Biorthogonal Rational Functions
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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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