Combinatorial interpretation of integrals of products of hermite, Laguerre and Tchebycheff polynomials

Abstract

Certain integrals of products of Laguerre polynomials have been interpreted as numbers of generalized derangements by Kaplansky, Even, Gillis, Jackson, Askey, Ismail, and Rashed. The analog for the Hermite polynomials have been done by Azor, Gillis, Victor, Godsil in term of matchings. Here we give a simple combinatorial (i.e. with a bijection) proof of these results. An analogous bijection is constructed for the case of Tchebycheff polynomials and leads to an interpretation with Dyck words.