Asymptotic analysis of the Askey-scheme II: from Charlier to Hermite

TL;DR

This paper provides an asymptotic analysis of Hermite polynomials and their zeros by leveraging the limit relation with Charlier polynomials, resulting in accurate approximation formulas.

Contribution

It introduces new asymptotic formulas for Hermite polynomials based on the connection with Charlier polynomials, enhancing approximation accuracy.

Findings

Derived precise asymptotic formulas for Hermite polynomials

Provided accurate zero approximations for large n

Utilized special functions to improve approximation quality

Abstract

We analyze the Hermite polynomials $H_{n}(\xi)$ and their zeros asymptotically as $n\to\infty,$ using the limit relation between the Charlier and Hermite polynomials. Our formulas involve some special functions and they yield very accurate approximations.