Legendre polynomial

The Legendre polynomialsDlmfDlmfMathworldPlanetmath are a set of polynomials {Pn}n=0 each of order n that satisfy Legendre’s ODE:


Alternatively Pn is an eigenfunctionMathworldPlanetmath of the self-adjointMathworldPlanetmathPlanetmathPlanetmath differential operator ddx(1-x2)ddx with eigenvalueMathworldPlanetmathPlanetmathPlanetmathPlanetmath -n(n+1).

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

By Sturm-Liouville theory, this means they’re orthogonalMathworldPlanetmath over some interval with some 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 orthogonalizationPlanetmathPlanetmath of the monomials {xi}. The normalization used is PnPn=2/(2n+1), which makes Pn(±1)=(±1)n

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


The first few explicitly are:

P0(x) = 1
P1(x) = x
P2(x) = 12(3x2-1)
P3(x) = 12(5x3-3x)
P4(x) = 18(35x4-30x2+3)

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


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 harmonicsDlmfDlmfMathworld.

Title Legendre polynomial
Canonical name LegendrePolynomial
Date of creation 2013-03-22 15:12:10
Last modified on 2013-03-22 15:12:10
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 14
Author mathcam (2727)
Entry type Definition
Classification msc 33C45
Related topic OrthogonalPolynomials
Defines Rodrigues’s Formula
Defines Legendre’s Differential EquationMathworldPlanetmath