Uniform asymptotics for the discrete Laguerre polynomials

Abstract

In this paper, we consider the discrete Laguerre polynomials $P_{n,N}(z)$ orthogonal with respect to the weight function $w(x)=x^{\alpha }{e}^{-Ncx}$ supported on the infinite nodes $L_N=\{x_{k,N}=\frac{k^2}{N^2},k\in \mathbb{N}\}$. We focus on the "band-saturated region" situation when the parameter $c>\frac{{\pi }^2}{4}$. As $n\to \infty$, uniform expansions for $P_{n,n}(z)$ are achieved for $z$ in different regions in the complex plane. Typically, the Airy-function expansions and Gamma-function expansions are derived for $z$ near the endpoints of the band and the origin, respectively. The asymptotics for the normalizing coefficient $h_{n,N}$, recurrence coefficients ${\mathcal{B}}_{n,N}$ and ${\mathcal{A}}_{n,N}^2$, are also obtained. Our method is based on the Deift-Zhou steepest descent method for Riemann-Hilbert problems.

37 pages, 6 figures