New Differential Formulae Related to Hermite Polynomials and their Applications in Quantum Optics

TL;DR

This paper introduces new differential formulae related to Hermite polynomials derived from quantum operator methods, with applications in analyzing nonclassical properties of quantum optical fields.

Contribution

It presents novel differential identities involving Hermite polynomials using quantum operator techniques and Weyl's mapping, expanding tools for quantum optics analysis.

Findings

Derived a generating function for two-variable Hermite polynomials.

Established generalized differential expressions invariant under Weyl ordering.

Applied identities to study nonclassical properties of quantum optical fields.

Abstract

In this work, based on quantum operator Hermite polynomials and Weyl's mapping rule, we find a generation function of the two-variable Hermite polynomials. And then, noting that the Weyl ordering is invariant under the similar transformations, we obtain another generalized differential expression related to the Hermite polynomials. Those identites can be applied to investigate the nonclasscial properties of quantum optical fields.