Asymptotics for the ratio and the zeros of multiple Charlier polynomials

TL;DR

This paper studies the asymptotic behavior and zero distribution of multiple Charlier polynomials, revealing how their ratios and zeros behave under various parameter regimes using recurrence relations.

Contribution

It introduces new asymptotic results for the ratio and zeros of multiple Charlier polynomials, including cases with parameter dependence on polynomial degree.

Findings

Asymptotic ratio behavior derived from recurrence relations

Zeros are asymptotically uniformly distributed on an interval

Different zero distributions occur when parameters depend on degree

Abstract

We investigate multiple Charlier polynomials and in particular we will use the (nearest neighbor) recurrence relation to find the asymptotic behavior of the ratio of two multiple Charlier polynomials. This result is then used to obtain the asymptotic distribution of the zeros, which is uniform on an interval. We also deal with the case where one of the parameters of the various Poisson distributions depend on the degree of the polynomial, in which case we obtain another asymptotic distribution of the zeros.