Legendre polynomials

A class of orthogonal polynomials that are solutions to the differential equation:

        2
    2  d y      dy
(1-x ) --- - 2x --  + l(l+1)y = 0
         2      dx
       dx

The Legendre polynomials are generated by the Rodrigues formula:

                  l
           1     d     2     l
P (x) = ------- ---- (x  + 1)
 l        l        l
         2  l!   dx

Which leads to the sum formula:

                       k
         1   ⌊l/2⌋ (-1) (2l-2k)!     l-2k
P (x) = --- ∑     ---------------- x
 l        l  k=0   k!(l-k)!(l-2k)!
         2

The Legendre polynomials satisfy the following three-term recurrence relation:

(l+1)P_l+1(x) = (2l + 1)xP_l - lP_l-1(x)

which can be used to generate the polynomials given that P₀ = 1, and P₁ = x.

The Legendre polynomials are orthogonal over the interval -1 < x < 1 with weighting function 1, satisfying:

 1                    2
∫  P (x) P (x) dx = ------ δ
-1  n     m         2n + 1  mn

where δ_mn is the Kronecker delta.

These polynomials can also be generated by applying the Gram-Schmidt theorem on the sequence of all powers of x, (a fundamental sequence of the separable space of all infinitely differentiable functions) using this inner product.

The Legendre polynomials are a special case of the Jacobi polynomials for which the parameters α and β are both zero. The Legendre polynomials are also expressible in terms of hypergeometric functions as:

                        1
P (x) =  F (-n, n+1; 1; - (1-x))
 n      2 1             2

where ₂F₁ is the Gaussian hypergeometric function. Note that this is only a polynomial if and only if n is an integer.

The first few Legendre polynomials are:

P (x) = 1
 0
P (x) = x
 1
        1    2
P (x) = - (3x  - 1)
 2      2
        1    3
P (x) = - (5x  - 3)
 3      2
        1     4      2
P (x) = - (35x  - 30x  + 3)
 4      8

The associated Legendre polynomials are solutions to the associated Legendre equation:

                              2
d (    2  dy)   (            m   )
--((1-x ) --) + (l(l+1) - -------)y = 0
dx(       dx)   (         1 - x^2)

Note that the regular Legendre DE is a special case of this; when m = 0. They can be given in terms of the regular Legendre polynomials by:

                         m
 m          m    2 m/2  d
P (x) = (-1) (1-x )    --- P (x)
 l                       m  l
                       dx

If m = 0, they reduce to the unassociated Legendre polynomials.

These polynomials are very important in applied mathematics and have many applications in mathematical physics, because many problems with spherical symmetry produce equations that may be transformed into the differential equations satisfied by the Legendre polynomials. They often arise when partial differential equations with spherically symmetric boundary conditions are solved by separation of variables.