Asymptotics of the Charlier polynomials via difference equation methods

TL;DR

This paper develops uniform and non-uniform asymptotic formulas for Charlier polynomials using difference equation techniques, addressing their unique position outside traditional turning point theory.

Contribution

It introduces novel asymptotic methods for Charlier polynomials that do not rely on turning point theory, including uniform approximations at coalescing points and near the origin.

Findings

Derived uniform asymptotics at coalescing turning points

Obtained asymptotics in outside, intermediate, and near-turning point regions

Provided uniform approximation at the origin using dominant balance and matching

Abstract

We derive uniform and non-uniform asymptotics of the Charlier polynomials by using difference equation methods alone. The Charlier polynomials are special in that they do not fit into the framework of the turning point theory, despite the fact that they are crucial in the Askey scheme. In this paper, asymptotic approximations are obtained respectively in the outside region, an intermediate region, and near the turning points. In particular, we obtain uniform asymptotic approximation at a pair of coalescing turning points with the aid of a local transformation. We also give a uniform approximation at the origin by applying the method of dominant balance and several matching techniques.