Legendre polynomial

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

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

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 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 orthogonalization  of the monomials $\{x^{i}\}$. The normalization used is $\langle P_{n}\|P_{n}\rangle=2/(2n+1)$, 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.

Title Legendre polynomial LegendrePolynomial 2013-03-22 15:12:10 2013-03-22 15:12:10 mathcam (2727) mathcam (2727) 14 mathcam (2727) Definition msc 33C45 OrthogonalPolynomials Rodrigues’s Formula Legendre’s Differential Equation  