Solving Legendre's Equation | Recurrence Relation

[KateyLou]

Homework Statement

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

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

to (1-x²)y''-2xy'+k(k+1)y=0
Deduce that if k is a positive integer, then a_k+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)a_n+2-a_n(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]

So
$$a_{n+2}= a_n\frac{n(n+1)- k(k+1)}{(n+1)(n+2)}$$
Assume that a₀ and a_{1[/sup] are given. What is a2, a3, a4 in terms of a0 and a1. Can you guess a form for the general an form? Can you then prove it by, say, induction?}

 

[KateyLou]

i am not too sure what induction is...(sorry!) guessing a form for a_n - do you is this not given by the recurrance relation?