Analytic properties of complex Hermite polynomials (Q2796518)

This contribution deals with the complex Hermite polynomials NEWLINE\[NEWLINE H_{m,n}(z,\overline{z}) = \sum_{k=0}^{m\wedge n} (-1)^k k!\binom{m}{k} \binom{n}{k} z^{m-k} \overline{z}^{n-k}. NEWLINE\]NEWLINE One of the main results of this remarkable work is establishing a multilinear generating function similar to the Kibble-Slepian formula, which extends the Poisson kernel for these polynomials. By using the Poisson kernel, the author introduce an integral operator and find its eigenvalues and eigenfunctions. A limiting case of this integral operator is the Fourier transform in two dimensions. Also, the author solved the linearization problem expanding the product of two complex Hermite polynomials in terms of polynomials of the same kind, where the linearization coefficients are given explicitly. Another main result of this work is to prove the positivity of the determinant whose entries are \((-i)^{m+n+2s} \pi H_{m+s,n+s}(i z,i\overline{z}).\) Finally, the author proved that the complex Hermite polynomials are the only orthogonal Appell polynomials in two variables.