Orthogonal structure and orthogonal series in and on a double cone or a hyperboloid

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