Complex Hermite functions as Fourier–Wigner transform

We prove that the complex Hermite polynomials Hm,n on the complex plane C can be realized as the Fourier–Wigner transform V of the well-known real Hermite functions hn on the real line R. This reduces considerably Wong's proof [Wong MW. Weyl transforms. New York: Universitext. Springer-Verlag; 1998. Chapter 21] giving the explicit expression of V(hm,hn) in terms of the Laguerre polynomials. Moreover, we derive some new integral identities for the classical real Hermite polynomials Hm.