A triple lacunary generating function for Hermite polynomials

Some of the classical orthogonal polynomials such as Hermite, Laguerre, Charlier, etc. have been shown to be the generating polynomials for certain combinatorial objects. These combinatorial interpretations are used to prove new identities and generating functions involving these polynomials. In this paper we apply Foata's combinatorial interpretation of the Hermite polynomials as counting matchings of a set to obtain a triple lacunary generating function for the Hermite polynomials. We also give an umbral proof of this generating function.