Orthogonal polynomials on the unit circle, $q$-Gamma weights, and discrete Painlev\'e equations

TL;DR

This paper studies orthogonal polynomials on the unit circle with $q$-gamma weights, revealing that their Verblunsky coefficients satisfy discrete Painlevé equations linked to Sakai's $A_3^{(1)}$ surface, thus connecting special functions, integrable systems, and algebraic geometry.

Contribution

It establishes a novel connection between $q$-gamma weighted orthogonal polynomials and discrete Painlevé equations, classified by Sakai's geometric framework.

Findings

Verblunsky coefficients satisfy discrete Painlevé equations

Connection to Sakai's $A_3^{(1)}$ surface

Identification of integrable structure in $q$-gamma weights

Abstract

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.