Uniform asymptotics for the discrete Laguerre polynomials

TL;DR

This paper derives uniform asymptotic expansions for discrete Laguerre polynomials in the band-saturated region using Riemann-Hilbert techniques, providing detailed behavior near endpoints and the origin.

Contribution

It introduces uniform asymptotic formulas for discrete Laguerre polynomials in a specific parameter regime, expanding the understanding of their complex-plane behavior.

Findings

Uniform asymptotics for $P_{n,N}(z)$ in different regions

Explicit Airy and Gamma-function expansions near endpoints and origin

Asymptotics for normalization and recurrence coefficients

Abstract

In this paper, we consider the discrete Laguerre polynomials $P_{n, N}(z)$ orthogonal with respect to the weight function $w(x) = x^{\alpha} e^{-N cx}$ 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 $\mathscr{B}_{n, N}$ and $\mathscr{A}_{n, N}^2$, are also obtained. Our method is based on the Deift-Zhou steepest descent method for Riemann-Hilbert problems.