Editing Associated Legendre polynomials (section)

==Generalizations==
The Legendre polynomials are closely related to [[hypergeometric series]]. In the form of spherical harmonics, they express the symmetry of the [[Riemann sphere|two-sphere]] under the action of the [[Lie group]] SO(3). There are many other Lie groups besides SO(3), and analogous generalizations of the  Legendre polynomials exist to express the symmetries of semi-simple Lie groups and [[Riemannian symmetric space]]s. 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.

By solving the Laplace equation in higher dimensions (with a potential that does not fall of <math>\sim 1/r</math>) Legendre Polynonials in higher than 3D can be defined.<ref>{{cite journal|first1=L. M. B. C.|last1=Campos|first2=F. S. R. P.|last2=Cunha|title=On hyperspherical Legendre polynomials and higher dimensional multipole expansions|journal=J. Inequal. Spec. Func.|year=2012|volume=3|number=3|url=http://ilirias.com/jiasf/repository/docs/JIASF3-3-1.pdf}}</ref>