Complex Hermite polynomials: Their combinatorics and integral operators

We consider two types of Hermite polynomials of a complex variable. For each type we obtain combinatorial interpretations for the linearization coefficients of products of these polynomials. We use the combinatorial interpretations to give new proofs of several orthogonality relations satisfied by these polynomials with respect to positive exponential weights in the complex plane. We also construct four integral operators of which the first type of complex Hermite polynomials are eigenfunctions and we identify the corresponding eigenvalues. We prove that the products of these complex Hermite polynomials are complete in certain L₂-spaces.