On the transition of Charlier polynomials to the Hermite function

TL;DR

This paper introduces a new, more general asymptotic transition from Charlier polynomials to the Hermite function valid for all real parameters, with uniform convergence, rate bounds, and zero convergence results.

Contribution

It establishes a powerful, general transition from Charlier polynomials to the Hermite function applicable for any real parameter, extending classical results.

Findings

Uniform convergence of Charlier polynomials to Hermite function

Sharp bounds on the rate of convergence

Zeros of Charlier polynomials converge to zeros of Hermite function

Abstract

It has been known for over 70 years that there is an asymptotic transition of Charlier polynomials to Hermite polynomials. This transition, which is still presented in its classical form in modern reference works, is valid if and only if a certain parameter is integer. In this light, it is surprising that a much more powerful transition exists from Charlier polynomials to the Hermite function, valid for any real value of the parameter. This greatly strengthens the asymptotic connections between Charlier polynomials and special functions, with applications for instance in queueing theory. It is shown in this paper that the convergence is uniform in bounded intervals, and a sharp rate bound is proved. It is also shown that there is a transition of derivatives of Charlier polynomials to the derivative of the Hermite function, again with a sharp rate bound. Finally, it is proved that zeros of Charlier polynomials converge to zeros of the Hermite function. While rigorous, the proofs use only elementary techniques.