A Probabilistic Approach of Sum Rules for Heat Polynomials

In this paper, we show that the sum rules for generalized Hermite polynomials derived by Daboul and Mizrahi (2005 J. Phys. A: Math. Gen. 38 427-48) and by Graczyk and Nowak (2004 C. R. Acad. Sci., Ser. 1 338 849) can be interpreted and easily recovered using a probabilistic moment representation of these polynomials. The covariance property of the raising operator of the harmonic oscillator, which is at the origin of the identities proved in Daboul and Mizrahi and the dimension reduction effect expressed in the main result of Graczyk and Nowak are both interpreted in terms of the rotational invariance of the Gaussian distributions. As an application of these results, we uncover a probabilistic moment interpretation of two classical integrals of the Wigner function that involve the associated Laguerre polynomials.