Lévy processes, polynomials and martingales

We study an unusual connection between orthogonal polynomials and martingales. We prove that all classical orthogonal polynomials from the Meixner class, when evaluated at a corresponding Levy process, are martingales. This result is well known for the case of Hermite polynomials evaluated in Brownian motion. Our results provide similar analogues for the Poisson process, for the Gamma process and for two less familiar processes related to Meixner polynomials