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## Homework Statement

I'm working on problem 6.11 in Bransden and Joachain's QM. I have to prove 4 different recurrence relations for the associate legendre polynomials. I have managed to do the first two, but can't get anywhere for the last 2

## Homework Equations

Generating Function:

[tex] T(\omega, \phi) = \sum_{l} P_{l}^{m}(\omega)s^{l} = \frac{(2m)!(1-\omega^{2})^{m/2}s^{m}}{2^{m}(m!)(1-2\omega s + s^{2})^{m+1/2} }[/tex]

Where m is really [itex]|m|[/itex]

I need to prove the following:

[tex] a)\ (2l+1)\omega P_{l}^{m}(\omega) = (l-m+1)P_{l+1}^{m}(\omega) + (l+m)P_{l-1}^{m}(\omega)[/tex]

[tex] b)\ (2l+1)(1-\omega^{2})^{1/2}P_{l}^{m-1} = P_{l+1}^{m}(\omega) - P_{l-1}^{m}(\omega)[/tex]

[tex] c) (1-\omega^{2})\frac{dP_{l}^{m}(\omega)}{d\omega} = (l+1)\omega P_{l}^{m}(\omega) - (l+1-m)P_{l+1}^{m}(\omega)[/tex]

[tex] d) (1-\omega^{2})\frac{dP_{l}^{m}(\omega)}{d\omega} = -(1-\omega^{2})^{1/2}(l+m)(l-m+1)P_{l}^{m-1}(\omega) + m\omega P_{l}^{m}(\omega)[/tex]

## The Attempt at a Solution

I was able to prove A and B by differentiating the generating function with respect to s and doing some rearranging and such, but c and d just aren't working for me. I started by differentiating with respect to omega and here is what I have so far:

First multiply through by [itex](1-2\omega s + s^{2})^{m+1/2}[/itex] and define [itex] c_{m} = \frac{(2m)!s^{l}}{2^{m}(m!)}[/itex]

Ignoring the summation sign, and the functional dependence of P on omega, for readability we have:

[tex]

(1-2\omega s + s^{2})^{m+1/2}P_{l}^{m}s^{l} = c_{m}(1-\omega^{2})^{m/2}[/tex]

Take derivative wrt omega:

[tex]

(m+1/2)(-2s)(1-2\omega s + s^{2})^{m-1/2}P_{l}^{m}s^{l} + (1-2s\omega + s^{2})^{m+1/2}P'_{l}^{m}s^{l} = c_{m}(m/2)(-2\omega)(1-\omega^{2})^{m/2-1}[/tex]

Where P' denotes derivative wrt omega.

Dividing through by [itex](1-2\omega s + s^{2})^{m+1/2}[/itex] and multiplying through by [itex](1-\omega^{2})[/itex] we get:

[tex]

\frac{-(1-\omega^{2})(2m+1)sP_{l}^{m}s^{l}}{1-2\omega s +s^{2}} + (1-\omega^{2})P'_{l}^{m}s^{l} = \frac{-c_{m}m\omega(1-\omega^{2})^{m/2}}{(1-2\omega s + s^{2})^{m/2}}[/tex]

But that last term is just [itex]m\omega P_{l}^{m}s^{l}[/itex], so we arrive at:

[tex]

\frac{-(1-\omega^{2})(2m+1)sP_{l}^{m}s^{l}}{1-2\omega s +s^{2}} + (1-\omega^{2})P'_{l}^{m}s^{l} = -m\omega P_{l}^{m}s^{l}[/tex]

Which is fairly close to c. I've tried subbing in a or b here but it just gets messier and I can't make it simplify.

Any suggestions? Or does anyone know of a textbook or other resource that goes though these derivations in detail?

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