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 incorporating an additional parameter, commonly indicated by P with both a subscript and a superscript to indicate the additional parameter.
Uses of Legendre polynomials and associated Legendre polynomials include:
(Items in this list of applications are not necessarily distinct from each other.)