We introduce holomorphic Hermite polynomials in n complex variables that generalize the Hermite polynomials in n real variables introduced by Hermite in the late 19th century. We discuss cases in which these polynomials are orthogonal and construct a reproducing kernel Hilbert space related to one such orthogonal family. We also introduce a multivariate analog of the Itô polynomials. We show how these multivariate polynomials generalize the univariate complex Hermite and Itô polynomials. Generating functions, orthogonality relations, Rodrigues formulas, recurrence and linearization relations, and operator formulas are also derived for these multivariate holomorphic Hermite and Itô polynomials. A Kibble–Slepian formula and a Mehler-type formula for the multivariate Itô polynomials are established.

我们介绍的全纯埃尔米特多项式ñ是概括埃尔米特多项式在复杂的变量ñHermite在19世纪后期引入的实际变量。我们讨论了这些多项式是正交的情况，并构造了与一个此类正交族有关的再生内核希尔伯特空间。我们还介绍了Itô多项式的多元类似物。我们展示了这些多元多项式如何推广单变量复数Hermite和Itô多项式。还为这些多元全纯Hermite和Itô多项式推导了生成函数，正交关系，Rodrigues公式，递归和线性化关系以及算子公式。建立了多元Itô多项式的Kibble-Slepian公式和Mehler型公式。