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Legendre polynomials

(useful sequence of polynomials)

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.)

Further reading: