# Legendre's equation

## Homework Statement

Obtain the recurrene relation between the coefficient ar in the series solution

y= (between r=0 and $$\infty$$) $$\Sigma$$ arxr

to (1-x2)y''-2xy'+k(k+1)y=0
Deduce that if k is a positive integer, then ak+2=0, so that the equation possesses a solution which is a polynomial of degree k.

## Homework Equations

(there is more to this question....but i think ill try getting through this bit first!)

## The Attempt at a Solution

I have managed to get the correct recurrance relation

(n+2)(n+1)an+2-an(n(n+1)-k(k+1))=0

But i have no idea what to do now (the show if k is a positive integer) etc :-(

HallsofIvy
$$a_{n+2}= a_n\frac{n(n+1)- k(k+1)}{(n+1)(n+2)}$$