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 approach to generating functions for the Hermite polynomials to obtain a triple lacunary generating function. We define renormalized Hermite polynomials hn(u) by ∞ ∑ n=0 hn(u) zn n! = e 2/2. and give a combinatorial proof of the following generating function: ∞ ∑ n=0 h3n(u) zn n! = e(w−u)(3u−w)/6 √ 1 − 6wz ∞ ∑ n=0 (6n)! 23n(3n)!(1 − 6wz)3n z2n (2n)! , where w = (1 − √1 − 12uz)/6z = uC(3uz) and C(x) = (1 − √1 − 4x)/(2x) is the Catalan generating function. We also give an umbral proof of this generating function.