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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:

- P
_{0}(x) = 1 - P
_{1}(x) = x - P
_{n+1}(x) = ( (2n+1)xP_{n}(x)-nP_{n-1}(x) ) / (n+1)

where P_{n}(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:

- Fields that adhere to
**Laplace's equation**: ∇²*f*= 0, where*f*is a scalar field. - Phenomena subject to an inverse square law.
- Multipole expansions of gravitational potential for non-spherical masses (e.g., with non-zero J
_{2}). - General relativity.

(Items in this list of uses are not necessarily distinct from each other.)

https://en.wikipedia.org/wiki/Legendre_polynomials

https://en.wikipedia.org/wiki/Associated_Legendre_polynomials

http://hyperphysics.phy-astr.gsu.edu/hbase/Math/legend.html