On a class of q-orthogonal polynomials and the q-Riemann Hilbert Problem

TL;DR

This paper solves a specific q-Riemann Hilbert problem related to q-orthogonal polynomials, establishing its uniqueness and analyzing the asymptotic behavior of polynomial zeros as degree increases.

Contribution

It provides an explicit solution and uniqueness proof for a q-Riemann Hilbert problem, along with new insights into zero distribution asymptotics.

Findings

Explicit solution to the q-Riemann Hilbert problem

Proof of uniqueness of the solution

Asymptotic behavior of polynomial zeros as degree approaches infinity

Abstract

We give an explicit solution of a q-Riemann Hilbert problem which arises in the theory of orthogonal polynomials, prove that it is unique, and deduce several properties. Our new results include the asymptotic behaviour of zeroes in the limit as the degree of the polynomial approaches infinity.