Representations of degenerate Hermite polynomials

TL;DR

This paper introduces degenerate Hermite polynomials, explores their relationships with other degenerate polynomials, and provides representations connecting these polynomial families.

Contribution

It presents the first systematic study of degenerate Hermite polynomials and their connections to higher-order degenerate Bernoulli, Euler, and Frobenius-Euler polynomials.

Findings

Representation of degenerate Hermite polynomials in terms of other degenerate polynomials

Representation of other degenerate polynomials in terms of degenerate Hermite polynomials

New formulas linking different families of degenerate polynomials

Abstract

We introduce degenerate Hermite polynomials as a degenerate version of the ordinary Hermite polynomials. Then, among other things, by using the formula about representing one lambda-Sheffer polynomial in terms of other lambda-Sheffer polynomials we represent the degenerate Hermite polynomials in terms of the higher-order degenerate Bernoulli, Euler, and Frobenius-Euler polynomials and vice versa.