Kalnins, Ernie G.Miller, W., Jr.Tratnik, M.V.2018-11-081991-01-012018-11-081991Kalnins, E. G., Miller, W., Jr., & Tratnik, M. V. (1991). Families of Orthogonal and Biorthogonal Polynomials on the N-Sphere. SIAM Journal on Mathematical Analysis (SIMA), 22(1), 272–294. https://doi.org/10.1137/05220170036-1410https://hdl.handle.net/10289/12150The Laplace–Beltrami eigenvalue equation $H\Phi = \lambda \Phi $ on the n-sphere is studied, 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.application/pdfenThis is an author’s accepted version of an article published in the journal: SIAM Journal on Mathematical Analysis (SIMA). © 1991 Society for Industrial and Applied MathematicsScience & TechnologyPhysical SciencesMathematics, AppliedMathematicsMULTIVARIABLE ORTHOGONAL AND BIORTHOGONAL POLYNOMIALSTHE N-SPHEREJournal Article10.1137/0522017