Families of Orthogonal and Biorthogonal Polynomials on theN-Sphere

Abstract

We study the Laplace-Beltrami eigenvalue equation H \Phi = \Phi on then-sphere, with an added vector potential term motivated by the differential equationsfor the polynomial Lauricella functions FA . The operator H is self-adjoint withrespect to the natural inner product induced on the sphere and, in certain specialcoordinates, it admits a spectral decomposition with eigenspaces composed entirelyof polynomials. The eigenvalues are degenerate but the degeneracy can be brokenthrough use of ...