Asymptotics of Discrete $q$-Freud $\mathrm{II}$ orthogonal polynomials from the $q$-Riemann Hilbert Problem

TL;DR

This paper uses Riemann-Hilbert problem techniques to analyze the asymptotic behavior of discrete $q$-Freud II orthogonal polynomials, revealing new limits and properties as the polynomial degree grows large.

Contribution

It introduces a Riemann-Hilbert framework for studying $q$-orthogonal polynomials and derives novel asymptotic results, including effects of lattice translations and connections to $q$-Painlevé equations.

Findings

Asymptotic formulas for large-degree polynomials

Effects of lattice translation on polynomial behavior

Asymptotics of related $q$-Painlevé equations

Abstract

We investigate a Riemann-Hilbert problem (RHP), whose solution corresponds to a group of $q$-orthogonal polynomials studied earlier by Ismail et al. Using RHP theory we determine new asymptotic results in the limit as the degree of the polynomials approach infinity. The RHP formulation also enables us to obtain further properties. In particular, we consider how the class of polynomials and their asymptotic behaviours change under translations of the $q$-discrete lattice and determine the asymptotics of related $q$-Painlev\'e equations.