A new class of bivariate poly-analytic Hermite polynomials is considered. We show that they are realizable as the Fourier-Wigner transform of the univariate complex Hermite functions and form a nontrivial orthogonal basis of the classical Hilbert space on the two-complex space with respect to the Gaussian measure. Their basic properties are discussed, such as their three term recurrence relations, operational realizations and differential equations (Bochner's property) they obey. Different generating functions of exponential type are obtained. Integral and exponential operational representations are also derived. Some applications in the context of integral transforms and the concrete spectral theory of specific magnetic Laplacians are discussed.

中文翻译：

---

二元多元解析 Hermite 多项式

考虑了一类新的双变量多解析 Hermite 多项式。我们证明它们可以作为单变量复 Hermite 函数的傅立叶-维格纳变换来实现，并在关于高斯测度的二复空间上形成经典希尔伯特空间的非平凡正交基。讨论了它们的基本性质，例如它们的三项递推关系、运算实现和它们服从的微分方程（Bochner 性质）。得到不同的指数型生成函数。还导出了积分和指数运算表示。讨论了积分变换和特定磁拉普拉斯算子的具体谱理论中的一些应用。