Asymptotic root distribution of Charlier polynomials with large negative parameter

TL;DR

This paper investigates the asymptotic distribution of roots of Charlier polynomials with large negative parameters, revealing their clustering on complex curves and deriving the limiting density through advanced analytical methods.

Contribution

It provides the first detailed analysis of root distributions for Charlier polynomials with negative parameters, including explicit equations for the clustering curves and the limiting density.

Findings

Roots cluster on specific complex curves

Derived implicit equations for the clustering curves

Established the limiting density of roots on these curves

Abstract

We analyze the asymptotic distribution of roots of Charlier polynomials with negative parameter depending linearly on the index. The roots cluster on curves in the complex plane. We determine implicit equations for these curves and deduce the limiting density of the root distribution supported on these curves. The proof is based on a determination of the limiting Cauchy transform in a specific region and a careful application of the saddle point method. The obtained result represents a solvable example of a more general open problem.