A combinatorial approach to the 2D-Hermite and 2D-Laguerre polynomials

The first author has recently proved a Kibble–Slepian type formula for the 2D-Hermite polynomials {Hm,n(z,z¯)} which extends the Poisson kernel for these polynomials. We provide a combinatorial proof of a closely related formula. The combinatorial structures involved are the so-called m-involutionary ℓ-graphs. They are enumerated in two different manners: first globally, then as the exponential of their connected components. We also give a combinatorial model for the 2D-Laguerre polynomials and study their linearization coefficients.