Global Asymptotics of the Hahn Polynomials

TL;DR

This paper derives uniform asymptotic formulas for Hahn polynomials as their degree increases, using a modified Riemann-Hilbert method, covering the entire complex plane.

Contribution

It provides the first comprehensive uniform asymptotic analysis of Hahn polynomials for fixed parameters and varying degree ratios, employing a novel modification of the Riemann-Hilbert approach.

Findings

Derived uniform asymptotics in terms of Airy functions

Covered the entire complex plane with three overlapping regions

Extended the Riemann-Hilbert method for Hahn polynomials

Abstract

In this paper, we study the asymptotics of the Hahn polynomials Q_n(x; {\alpha}, {\beta}, N) as the degree n grows to infinity, when the parameters {\alpha} and {\beta} are fixed and the ratio of n/N = c is a constant in the interval (0, 1). Uniform asymptotic formulas in terms of Airy functions and elementary functions are obtained for z in three overlapping regions, which together cover the whole complex plane. Our method is based on a modified version of the Riemann-Hilbert approach introduced by Deift and Zhou.