A deformation of Hermite polynomials

TL;DR

This paper introduces a new family of polynomials derived from Hermite polynomials through a deformation process, exploring their properties and suggesting the deformation's broad applicability beyond Hermite polynomials.

Contribution

It proposes and studies a novel set of deformed Hermite-related polynomials, expanding the understanding of polynomial deformations and their potential generality.

Findings

Defined new deformed Hermite polynomials $M_{neta, H}^{s}(z)$, $C_{neta, H}^{s}(z)$, $W_{neta, H}^{s}(z)$

Analyzed structural properties of the deformed polynomials

Indicated the deformation's general applicability beyond Hermite polynomials

Abstract

We propose and study the properties of a set of polynomials $M_{n\alpha, H\ }^{s}(z)$, $C_{n\alpha, H}^{s}(z)$ $W_{n\alpha, H}^{s}(z)$ with $n,s\in N$ $;\alpha =\pm 1;$and where $H$ stands for Hermite ; the ''root '' polynomial >.These polynomials are obtained from a deformation of Hermite polynomials $H_{n}(z).$The structure underlying the deformation seems quite general and not only restricted to Hermite polynomials.