We consider orthogonal polynomials on the surface of a double cone or a hyperboloid of revolution, either finite or infinite in axis direction, and on the solid domain bounded by such a surface and, when the surface is finite, by hyperplanes at the two ends. On each domain a family of orthogonal polynomials, related to the Gegebauer polynomials, is study and shown to share two characteristic properties of spherical harmonics: they are eigenfunctions of a second order linear differential operator with eigenvalues depending only on the polynomial degree, and they satisfy an addition formula that provides a closed form formula for the reproducing kernel of the orthogonal projection operator. The addition formula leads to a convolution structure, which provides a powerful tool for studying the Fourier orthogonal series on these domains. Furthermore, another family of orthogonal polynomials, related to the Hermite polynomials, is defined and shown to be the limit of the first family, and their properties are derived accordingly.

中文翻译：

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双锥或双曲面内和上的正交结构和正交级数

我们考虑在轴方向有限或无限的双锥或旋转双曲面表面上的正交多项式，以及在以这样的表面为界的实体域上，当表面有限时，在两端的超平面上。在每个域上，研究了与 Gegebauer 多项式相关的一系列正交多项式，并证明它们具有球谐函数的两个特征：它们是二阶线性微分算子的特征函数，特征值仅取决于多项式次数，并且它们满足一个加法公式，为正交投影算子的再现核提供封闭形式的公式。加法公式导致卷积结构，这为研究这些域上的傅立叶正交级数提供了强大的工具。此外，