Global Asymptotics of the Meixner Polynomials

TL;DR

This paper derives asymptotic formulas for Meixner polynomials in different complex plane regions using Riemann-Hilbert steepest descent methods, confirming earlier results on the positive real line.

Contribution

It applies the Deift-Zhou steepest descent method to obtain global asymptotics of Meixner polynomials in complex regions, extending previous real-line results.

Findings

Asymptotic formulas valid in two complex regions

Boundary asymptotics obtained by limiting procedures

Results consistent with earlier real-line asymptotic analyses

Abstract

Using the steepest descent method for oscillatory Riemann-Hilbert problems introduced by Deift and Zhou [Ann. Math. {\bf 137}(1993), 295-368], we derive asymptotic formulas for the Meixner polynomials in two regions of the complex plane separated by the boundary of a rectangle. The asymptotic formula on the boundary of the rectangle is obtained by taking limits from either inside or outside. Our results agree with the ones obtained earlier for $z$ on the positive real line by using the steepest descent method for integrals [Constr. Approx. {\bf 14}(1998), 113-150].