Legendre polynomial

The Legendre polynomials are a set of polynomials $\\{P_{n}\\}_{n=0}^{\\infty}$ each of order $n$ that satisfy Legendre’s ODE:

\\frac{d}{dx}[(1-x^{2})P_{n}^{\\prime}(x)]+n(n+1)P_{n}(x)=0.

Alternatively $P_{n}$ is an eigenfunction of the self-adjoint differential operator $\\frac{d}{dx}(1-x^{2})\\frac{d}{dx}$ with eigenvalue $-n(n+1)$ .

The Legendre polynomials are also known as Legendre functions of the first kind.

By Sturm-Liouville theory, this means they’re orthogonal over some interval withsome weight function. In fact it can be shown that they’re orthogonal on $[-1,1]$ with weight function $W(x)=1$ . As with any set of orthogonal polynomials, this can be used to generate them (up to normalization) by Gram-Schmidt orthogonalization of the monomials $\\{x^{i}\\}$ . The normalization usedis $\\langle P_{n}\\|P_{n}\\rangle=2/(2n+1)$ , which makes $P_{n}(\\pm 1)=(\\pm 1)^{n}$

Rodrigues’s Formula (which can be generalized to some other polynomial sets) is a sometimes convenient form of $P_{n}$ in terms of derivatives:

P_{n}(x)=\\frac{1}{2^{n}n!}\\left(\\frac{d}{dx}\\right)^{n}(x^{2}-1)^{n}

The first few explicitly are:

$\\displaystyle P_{0}(x)$	$\\displaystyle=$	$\\displaystyle 1$
$\\displaystyle P_{1}(x)$	$\\displaystyle=$	$\\displaystyle x$
$\\displaystyle P_{2}(x)$	$\\displaystyle=$	$\\displaystyle\\frac{1}{2}(3x^{2}-1)$
$\\displaystyle P_{3}(x)$	$\\displaystyle=$	$\\displaystyle\\frac{1}{2}(5x^{3}-3x)$
$\\displaystyle P_{4}(x)$	$\\displaystyle=$	$\\displaystyle\\frac{1}{8}(35x^{4}-30x^{2}+3)$
$\\displaystyle...$

As all orthogonal polynomials do, these satisfy a three-term recurrence relation:

(n+1)P_{n+1}(x)=(2n+1)xP_{n}(x)-(n)P_{n-1}(x)

The Legendre functions of the second kind also satisfy the Legendre ODE but are not regular at the origin.

Related are the associated Legendre functions, and spherical harmonics.