Uniform asymptotics for discrete orthogonal polynomials on infinite nodes with an accumulation point

TL;DR

This paper develops a Riemann-Hilbert approach to derive uniform asymptotic formulas for discrete orthogonal polynomials on infinite nodes with an accumulation point, exemplified by Tricomi-Carlitz polynomials.

Contribution

It introduces a novel Riemann-Hilbert method to analyze asymptotics of discrete orthogonal polynomials with accumulation points, extending previous results.

Findings

Derived uniform Plancherel-Rotach asymptotics in the complex plane.

Results agree with earlier asymptotic formulas for Tricomi-Carlitz polynomials.

Established a new analytical framework for discrete orthogonal polynomials with accumulation points.

Abstract

In this paper, we develop the Riemann-Hilbert method to study the asymptotics of discrete orthogonal polynomials on infinite nodes with an accumulation point. To illustrate our method, we consider the Tricomi-Carlitz polynomials $f_n^{(\alpha)}(z)$ where $\alpha$ is a positive parameter. Uniform Plancherel-Rotach type asymptotic formulas are obtained in the entire complex plane including a neighborhood of the origin, and our results agree with the ones obtained earlier in [{\it SIAM J.\;Math.\;Anal} {\bf 25} (1994)] and [{{\it Proc.\;Amer.\;Math.\;Soc.\,}{\bf138} (2010)}].