### 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:

• P0(x) = 1
• P1(x) = x
• Pn+1(x) = ( (2n+1)xPn(x)-nPn-1(x) ) / (n+1)

­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.)

(mathematics,gravity)