Complex Hermite functions as Fourier–Wigner transform

Abstract

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