Uniform Asymptotics of the Meixner Polynomials

TL;DR

This paper derives uniform asymptotic formulas for Meixner polynomials using the Deift-Zhou steepest descent method, including a novel local formula near the origin involving a special Riemann-Hilbert problem solution.

Contribution

It provides the first known asymptotic formula for Meixner polynomials near the origin, involving a special bounded solution to a scalar Riemann-Hilbert problem.

Findings

Derived uniform asymptotics for Meixner polynomials

Obtained a new local asymptotic formula near the origin

Validated formulas through numerical comparisons

Abstract

Using the steepest descent method of Deift-Zhou, we derive locally uniform asymptotic formulas for the Meixner polynomials. These include an asymptotic formula in a neighborhood of the origin, a result which as far as we are aware has not yet been obtained previously. This particular formula involves a special function, which is the uniformly bounded solution to a scalar Riemann-Hilbert problem, and which is asymptotically (as the polynomial degree $n$ tends to infinity) equal to the constant $"1"$ except at the origin. Numerical computation by using our formulas, and comparison with earlier results, are also given.