New identities about operator Hermite polynomials and their related integration formulas

TL;DR

This paper introduces new identities and a binomial-like theorem for operator Hermite polynomials using the IWOP technique and entangled state representation, leading to novel integration formulas without traditional integration.

Contribution

It presents novel identities and a binomial-like theorem for operator Hermite polynomials, expanding their theoretical framework and applications.

Findings

Derived new identities for single- and two-variable operator Hermite polynomials

Established a binomial-like theorem linking single- and two-variable Hermite polynomials

Developed new integration formulas using operator techniques without conventional integration

Abstract

By virtue of the technique of integration within an ordered product (IWOP) of operators and the bipartite entangled state representation we derive some new identities about operator Hermite polynomials in both single- and two-variable, we also find a binomial-like theorem between the single-variable Hermite polynomials and the two-variable Hermite polynomials. Application of these identities in deriving new integration formulas, but without really doing the integration in the usual sense, is demonstrated.