On the recurrence coefficients for the $q$-Laguerre weight and discrete Painlev\'e equations

TL;DR

This paper investigates how recurrence coefficients for orthogonal polynomials with a deformed $q$-Laguerre weight depend on the degree, revealing they satisfy a novel discrete Painlevé equation linked to $A_{5}^{(1)}$ Sakai surfaces.

Contribution

It demonstrates that the recurrence coefficients follow a new composition of discrete Painlevé equations, illustrating the effectiveness of a geometric identification scheme.

Findings

Recurrence coefficients are governed by a new discrete Painlevé equation.

The equation is a composition of two standard discrete Painlevé equations.

The study showcases the geometric approach to identifying discrete Painlevé equations.

Abstract

We study the dependence of recurrence coefficients in the three-term recurrence relation for orthogonal polynomials with a certain deformation of the $q$-Laguerre weight on the degree parameter $n$. We show that this dependence is described by a discrete Painlev\'e equation on the family of $A_{5}^{(1)}$ Sakai surfaces, but this equation is different from the standard examples of discrete Painlev\'e equations of this type and instead is a composition of two such. This case study is a good illustration of the effectiveness of a recently proposed geometric identification scheme for discrete Painlev\'e equations.