Legendre 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
• P_n+1(x) = ( (2n+1)xP_n(x)-nP_n-1(x) ) / (n+1)

­where P_n(x) is the Legendre polynomial with degree 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 by 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: ∇²f = 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 J₂).
• General relativity.

(These cited uses are not necessarily distinct from each other.)