Analytic properties of complex Hermite polynomials

We study the complex Hermite polynomials { H m , n ( z , z ¯ ) } \{H_{m,n}(z, \bar z)\} 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 , n ( z , z ¯ ) } \{H_{m,n}(z,\bar z)\} . Some inverse relations are also given. We give a two dimensional moment representation for H m , n ( z , z ¯ ) H_{m,n}(z,\bar z) and evaluate several related integrals. We also introduce bivariate Appell polynomials and prove that { H m , n ( z , z ¯ ) } \{H_{m,n}(z, \bar z)\} are the only bivariate orthogonal polynomials of Appell type.