Spherical Harmonics: Evaluating 2lth Derivative

[eep]

Hi,

I'm trying to get the $Y_l^l$ spherical harmonic and I'm running into problems evaluating the following expression:

$$\frac{d^{2l}(\cos^2(\theta) - 1)^l}{d\cos(\theta)^{2l}}$$

The 2lth derivative with respect to cos theta of cos squared theta - 1 to the lth power
it just seems like I'm going to end up with more and more terms as I keep differentiating..

 

[quantumdude]

First of all, let's not make this more complicated then it has to be. Do a change of variables: $x=\cos(\theta)$.

Now try to work it out for the first few values of $l$. You should notice a simple pattern emerging. (Hint: I would expand the expression $(x^2-1)^l$ before differentiating.