Monomiality and a New Family of Hermite Polynomials

TL;DR

This paper explores the monomiality principle to analyze a new family of Hermite polynomials, deriving their properties, differential equations, and orthogonality, connecting them to the Kampe de Feriet family.

Contribution

It introduces a novel approach using the monomiality principle to study Hermite polynomials and their generalizations, expanding understanding of their properties and relationships.

Findings

Derived differential equations for the new polynomial family

Established orthogonality and generalized forms

Connected the polynomials to the Kampe de Feriet family

Abstract

In this article we go deeply into the formulation and meaning of the monomiality principle and employ it to study the properties of a set of polynomials, which, asymptotically, reduce to the ordinary two variable Kampe de Feriet family. We derive the relevant differential equations and discuss the associated orthogonality properties, along with the relevant generalized forms.