# Orthogonality Of Legendre Polynomials

**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;