Orthogonal polynomials on the unit circle, $q$-Gamma weights, and discrete Painlevé equations

Details Orthogonal polynomials on the unit circle, $q$-Gamma weights, and discrete Painlevé equations Orthogonal polynomials on the unit circle, $q$-Gamma weights, and discrete Painlevé equations

2009-01-07

arxiv.org/abs/0901.0947v2

Mathematics

Classical Analysis and ODEs

26 pages 2 figures

Scientific paper

We consider orthogonal polynomials on the unit circle with respect to a
weight which is a quotient of $q$-gamma functions. We show that the Verblunsky
coefficients of these polynomials satisfy discrete Painlev\'e equations, in a
Lax form, which correspond to an $A_3^{(1)}$ surface in Sakai's classification.

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