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A $q$-beta integral on the unit circle and some biorthogonal rational functions. (English) Zbl 0835.33011
The authors start with a pair of polynomial sets, due to P. I. Pastro [J. Math. Anal. Appl. 112, 517-540 (1985; Zbl 0582.33010)], biorthogonal on the unit circle with respect to a complex weight function. Using generating functions, they turn the biorthogonality relation into a certain $q$-beta integral, which in turn leads to a pair of sets of rational functions biorthogonal on the unit circle with respect to another complex weight function. The asymptotics of these biorthogonal pairs are exhibited. Remarkably, in both cases the weight function can be read off from this asymptotics, in a way reminiscent of the Szegö theory for orthogonal polynomials. To be more precise, if $\left\{{P}_{n}\left(z\right)\right\}$ and $\left\{{Q}_{n}\left(z\right)\right\}$ is one of these pairs and $K\left(z\right)$ is the weight function of the corresponding biorthogonality measure, then (under suitable conditions) ${P}_{n}\left(z\right)\overline{\left({Q}_{n}1/z}\right)\sim 1/K\left(z\right)$ as $n$ tends to $\infty$.

MSC:
 33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.) 33D05 $q$-gamma functions, $q$-beta functions and integrals 42A65 Completeness of sets of functions