Asymptotic behaviours of q-orthogonal polynomials from a q-Riemann Hilbert Problem

TL;DR

This paper develops a Riemann-Hilbert framework to analyze the asymptotic behavior of q-orthogonal polynomials, revealing universal properties near their zeros and norms as the degree grows large.

Contribution

It introduces a Riemann-Hilbert problem approach for q-orthogonal polynomials and uncovers universal asymptotic behaviors independent of weight functions.

Findings

Asymptotic behavior near smallest zeros is universal.

Norms of polynomials become independent of weight functions as degree increases.

The approach applies to a family of q-orthogonal polynomials.

Abstract

We describe a Riemann-Hilbert problem for a family of $q$-orthogonal polynomials, $\{ P_n(x) \}_{n=0}^\infty$, and use it to deduce their asymptotic behaviours in the limit as the degree, $n$, approaches infinity. We find that the $q$-orthogonal polynomials studied in this paper share certain universal behaviours in the limit $n\to\infty$. In particular, we observe that the asymptotic behaviour near the location of their smallest zeros, $x \sim q^{n/2}$, and norm, $\|P_n\|_2$, are independent of the weight function as $n\to\infty$.