Families of Orthogonal and Biorthogonal Polynomials on the N-Sphere
Kalnins, 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/10.1137/0522017
Permanent Research Commons link: https://hdl.handle.net/10289/12150
The 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.
This 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 Mathematics