ANALYTIC PROPERTIES OF COMPLEX HERMITE POLYNOMIALS

We study the complex Hermite polynomials {H-m,H- n(z, (z) over bar)} in some detail, establish operational formulas for them and prove a Kibble-Slepian type formula, which extends the Poisson kernel for these polynomials. Positivity of the associated kernels is discussed. We also give an infinite family of integral operators whose eigenfunctions are {H-m,H- n(z, (z) over bar)}. Some inverse relations are also given. We give a two dimensional moment representation for H-m,H- n(z, (z) over bar) and evaluate several related integrals. We also introduce bivariate Appell polynomials and prove that {H-m,H- n(z, (z) over bar)} are the only bivariate orthogonal polynomials of Appell type.