radial part is typically of the form and hence the solutions are spherical harmonics. Generalizations The Legendre polynomials are closely related to hypergeometric series. In the form of spherical harmonics, they express the symmetry of the two-sphere under the action of the Lie group SO(3). There are many other Lie groups besides SO(3), and an analogous generalization of the Legendre polynomials exist to express the symmetries of semi-simple Lie groups and Riemannian symmetric spaces. Crudely speaking, one may define a Laplacian on symmetric spaces; the eigenfunctions of the Laplacian can be thought of as generalizations of the spherical harmonics to other settings. See also Angular momentum Gaussian quadrature Legendre polynomials Spherical harmonics Whipple's transformation of Legendre functions Laguerre polynomials Hermite polynomials Notes and references ; Section 12.5. (Uses a different sign convention.) . ; Chapter 3. . ; Chapter 2. . Schach, S. R. (1973) New Identities for Legendre Associated Functions of Integral Order and Degree , Society for Industrial and Applied Mathematics Journal on Mathematical Analysis, 1976, Vol. 7, No. 1 : pp. 59–69 External links Associated Legendre polynomials in MathWorld Legendre polynomials in MathWorld Legendre and Related Functions in DLMF Atomic physics Orthogonal polynomials