SKEDSOFT

Orthogonality of Legendre Polynomials:

The differential equation and boundary condition satiesfies by the Legendre Polynomials forms a generalized system where the boundary condition amount to insisting on regularity of the solutions at the boundaries). They should therefore satiesfies the Orthogonality Relations.

If P_n and P_m are solutions of Legendre's equations then;

Integrating the combination P_m (C.5) - P_n (C.6) gives

as P_m,n and their derivatives are finite at x = ± 1. Hence, if n ≠ m.

Generating Functions for Legendery Polynomials:

We consider afunction of two variables G( x , t ) such that,

So that legendery polynomials are the coefficients in the Taylor Series of G( x , t ) about t = 0. Our first task is to identify what the function G( x , t )actually is.

What is  

We can evaluate this integral using Rodrigue's formula. We have

Integraing by parts give;

The complition of this arguments is left as an exercise. One way to proceed is to use the transformation s = (x 1)/2 to transform the integral and then use a reduction formula to show that;