Definition:Legendre Polynomial

Definition
The Legendre polynomials are the solutions to Legendre's differential equation.

These solutions form a polynomial sequence of orthogonal polynomials on the interval $\left[{-1 \,.\,.\, 1}\right]$.

Bonnet's Recursion Formula
Legendre polynomials can be found using Bonnet's Recursion Formula.


 * $\left({n + 1}\right) P_{n + 1} \left({x}\right) = \left({2 n + 1}\right) x P_n \left({x}\right) - n P_{n - 1} \left({x}\right)$

Length of Legendre Polynomial
$\displaystyle \left\Vert{P_n \left({x}\right)}\right\Vert = \sqrt {\int_{-1}^1 \left(P_n \left({x}\right)\right)^2 \, \mathrm d x} = \sqrt{\frac 2 {2 n + 1}}$

Proof of Length
Applying Bonnet's Recursion Formula for $n - 1$:


 * $n P_n \left({x}\right) = \left({2 n - 1}\right) x P_{n - 1} \left({x}\right) - \left({n - 1}\right) P_{n - 2} \left({x}\right)$

so:
 * $P_n \left({x}\right) = \dfrac {2 n - 1} n x P_{n - 1} \left({x}\right) - \dfrac {n - 1} n P_{n - 2} \left({x}\right)$

Substituting for $P_n \left({x}\right)$:

From Orthogonality of Legendre Polynomials:
 * $\displaystyle \int_{-1}^1 P_n \left({x}\right) P_m \left({x}\right) \, \mathrm d x = 0 \iff n \ne m$

so:
 * $\displaystyle (1): \quad \left\Vert{P_n \left({x}\right)}\right\Vert^2 = \frac {2 n - 1} n \int_{-1}^1 x P_n \left({x}\right) P_{n-1} \left({x}\right) \, \mathrm d x$

From Bonnet's Recursion Formula:


 * $\displaystyle x P_n \left({x}\right) = \frac {n + 1} {2 n + 1} P_{n + 1} \left({x}\right) + \frac n {2 n + 1} P_{n - 1} \left({x}\right)$

Substituting for $x P_n \left({x}\right)$ in $(1)$:

Thus:

Most of this cancels out, leaving:


 * $\left\Vert{P_n \left({x}\right)}\right\Vert^2 = \dfrac {\left\Vert{P_0 \left({x}\right)}\right\Vert^2} {2 n + 1}$

It remains to compute the length of the first Legendre polynomial:

Thus:
 * $\left\Vert{P_n \left({x}\right)}\right\Vert^2 = \dfrac 2 {2n + 1}$

and so taking the square root:
 * $\left\Vert{P_n \left({x}\right)}\right\Vert = \sqrt{\dfrac 2 {2 n + 1} }$