Uniform asymptotics for the discrete Laguerre polynomials

In this paper, we consider the discrete Laguerre polynomials [Formula: see text] orthogonal with respect to the weight function [Formula: see text] supported on the infinite nodes [Formula: see text]. We focus on the “band-saturated region” situation when the parameter [Formula: see text]. As [Formula: see text], uniform expansions for [Formula: see text] are achieved for [Formula: see text] in different regions in the complex plane. Typically, the Airy-function expansions and Gamma-function expansions are derived for [Formula: see text] near the endpoints of the band and the origin, respectively. The asymptotics for the normalizing coefficient [Formula: see text], recurrence coefficients [Formula: see text] and [Formula: see text], are also obtained. Our method is based on the Deift–Zhou steepest descent method for Riemann–Hilbert problems.