The Legendre polynomials are a sequence of ever-longer polynomials, one of each non-negative-integer polynomial degree, in a pattern analogous to the pattern of binomial coefficients, but more complicated. One representation:
where Pn(x) is the Legendre polynomial to the degree of n. They occur in the series solutions (in the manner of a Taylor series) to some useful differential equations.
Associated Legendre polynomials (aka associated Legendre functions) are a generalization of Legendre polynomials that incorporate an additional parameter. Each associated Legendre polynomial is symbolized with a P with both a subscript and a superscript, the superscript indicating the additional parameter:
m P l
Uses of Legendre polynomials and associated Legendre polynomials include spherical harmonics and their uses:
(Items in this list of uses are not necessarily distinct from each other.)