Asymptotics for a generalization of Hermite polynomials

TL;DR

This paper investigates the asymptotic behavior of a generalized class of Hermite polynomials involving derivatives, deriving formulas for their zeros' convergence and extending classical results.

Contribution

It provides the first asymptotic analysis of these generalized Hermite polynomials, including Mehler--Heine formulas and zero convergence acceleration.

Findings

Derived Mehler--Heine type formulas for the generalized polynomials.

Proved accelerated convergence of the smallest positive zeros.

Extended classical Hermite polynomial asymptotics to a broader class.

Abstract

We consider a generalization of the classical Hermite polynomials by the addition of terms involving derivatives in the inner product. This type of generalization has been studied in the literature from the point of view of the algebraic properties. Thus, our aim is to study the asymptotics of this sequence of nonstandard orthogonal polynomials. In fact, we obtain Mehler--Heine type formulas for these polynomials and, as a consequence, we prove that there exists an acceleration of the convergence of the smallest positive zeros of these generalized Hermite polynomials towards the origin.