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:

<br /> \frac{d^{2l}(\cos^2(\theta) - 1)^l}{d\cos(\theta)^{2l}}<br />

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.