Analytic properties of complex Hermite polynomials

Ismail, Mourad

doi:10.1090/tran/6358

Analytic properties of complex Hermite polynomials
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by Mourad E. H. Ismail PDF

Trans. Amer. Math. Soc. 368 (2016), 1189-1210 Request permission

Abstract:

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

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