$(p,q)$-complex It\^o--Hermite polynomials

TL;DR

This paper introduces two new classes of $(p,q)$-Itô--Hermite polynomials, extending the $q$-analogues with post-quantum parameters, and explores their fundamental properties and differential equations.

Contribution

It presents the first study of $(p,q)$-Itô--Hermite polynomials, including their operational formulas, generating functions, and differential equations, expanding the theory of $q$-Hermite polynomials.

Findings

Defined two classes of $(p,q)$-Itô--Hermite polynomials.

Derived Rodrigues-type formulas and generating functions.

Established differential equations for these polynomials.

Abstract

We introduce two classes of $(p,q)$-It\^o--Hermite polynomials, the post-quantum analogs of the $q$-It\^o--Hermite polynomials introduced recently by Ismail and Zhang. We study their basic properties such as their operational formulas of Rodrigues type, the corresponding raising and lowering operators as well as their generating functions and the $(p,p$-differential equations they obey.