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Orthogonal polynomials on $n$-spheres: Gegenbauer, Jacobi and Heun. (English) Zbl 0867.33005
Rassias, Th. M. (ed.) et al., Topics in polynomials of one and several variables and their applications. Volume dedicated to the memory of P. L. Chebyshev (1821-1894). Singapore: World Scientific. 299-322 (1993).
Summary: In this expository paper, we describe the families of orthogonal and biorthogonal polynomials associated with the Laplace-Beltrami eigenvalue equation $H{\Phi }=\lambda {\Phi }$ on the $n$-sphere, with an added vector potential term motivated by the differential equations for the polynomial Lauricella functions ${F}_{A}$. The operator $H$ is self-adjoint with respect to the natural inner product induced on the sphere and, in certain special coordinates, it admits a spectral decomposition with eigenspaces composed entirely of polynomials. The eigenvalues are degenerate but the degeneracy can be broken through use of the possible separable coordinate systems on the $n$-sphere. Then a basis for each eigenspace can be selected in terms of the simultaneous eigenfunctions of a family of commuting second order differential operators that also commute with $H$. The results provide a multiplicity of $n$-variable orthogonal and biorthogonal families of polynomials that generalize classical results for one and two variable families of Jacobi polynomials on intervals, disks and paraboloids. We look carefully at the problem of expanding the (product of) Heun polynomial basis for the 2-sphere, in terms of the (product of) Jacobi polynomials basis.
##### MSC:
 33C55 Spherical harmonics 42C05 General theory of orthogonal functions and polynomials 42-06 Proceedings of conferences (Fourier analysis)