Overview
The Legendre Polynomials are defined such that {% P_n(x) %} is a polynomial of degree {% n %}.
{% \displaystyle \int_{-1}^1 P_m(x) P_n(x) dx = 0 %}
for {% m \neq n %}
{% \displaystyle \int_{-1}^1 P_n(x) P_n(x) dx = 2/(2n + 1) %}
Legendre Polynomials Library
Rodrigues Formula
The Rodrigues formula gives an explicit way to calculate the Legendre polynomial of any order.
{% P_n(x) = \frac{1}{2^2 n!} \frac{d^n}{dx^n} (x^2 -1) %}