# Properties Of Legendre Polynomials

**Properties of Legendre Polynomials:**

The Legendre Polynomials are the everywhere regular solutions of Legendre’s Equation,

(1 − x^{2})u′′ − 2xu′ mu = [ (1 − x^{2})u′ ] ′ mu = 0,

which are possible only if

m = n(n 1), n = 0, 1, 2,……

We write the solution for a particular value of n as P_{n}(x). It is a polynomial of degree n. If n is even/odd then the polynomial is even/odd. They are normalized such that P_{n}(1) = 1.

P_{0}(x) = 1,

P_{1}(x) = x,

P_{2}(x) = (3x^{2}− 1)/2,

P_{0}(x) = (5x^{3}− 3x)/2.

__Rodrigue’s Formula:__

They can also be represented using Rodrigue’s Formula:

This can be demonstrated through the following observations which are given as:

**1. It’s a Polynomial.**

**2. it take the value 1 at 1 if,
**

**3. It satiesfies the equation.
**

Finally;

Now differenciate n 1 times, using Lebnitz to get;

As the equations is linear and satisfies the equation also.