Complex Hermite functions as Fourier-Wigner transform

TL;DR

This paper demonstrates that complex Hermite polynomials can be expressed as the Fourier-Wigner transform of real Hermite functions, simplifying previous proofs and deriving new generating functions and integral identities.

Contribution

It establishes a direct link between complex Hermite polynomials and Fourier-Wigner transforms of real Hermite functions, providing simpler proofs and new identities.

Findings

Expressed complex Hermite polynomials as Fourier-Wigner transforms of real Hermite functions

Derived a new generating function for complex Hermite polynomials

Established new integral identities involving these polynomials

Abstract

We prove that the complex Hermite polynomials H_{m,n} on the complex plane $\mathbb{C}$ can be realized as the Fourier-Wigner transform $\mathcal{V}$ of the well-known real Hermite functions $h_n$ on real line $\mathbb{R}$. This reduces considerably the Wong's proof giving the explicit expression of $\mathcal{V}(h_m,h_n)$ in terms of the Laguerre polynomials. Moreover, we derive a new generating function for the H_{m,n} as well as some new integral identities.