Clever-Name
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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:
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} }
Where m is really |m|
I need to prove the following:
a)\ (2l+1)\omega P_{l}^{m}(\omega) = (l-m+1)P_{l+1}^{m}(\omega) + (l+m)P_{l-1}^{m}(\omega)
b)\ (2l+1)(1-\omega^{2})^{1/2}P_{l}^{m-1} = P_{l+1}^{m}(\omega) - P_{l-1}^{m}(\omega)
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)
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)
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 (1-2\omega s + s^{2})^{m+1/2} and define c_{m} = \frac{(2m)!s^{l}}{2^{m}(m!)}
Ignoring the summation sign, and the functional dependence of P on omega, for readability we have:
<br /> (1-2\omega s + s^{2})^{m+1/2}P_{l}^{m}s^{l} = c_{m}(1-\omega^{2})^{m/2}
Take derivative wrt omega:
<br /> (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}
Where P' denotes derivative wrt omega.
Dividing through by (1-2\omega s + s^{2})^{m+1/2} and multiplying through by (1-\omega^{2}) we get:
<br /> \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}}
But that last term is just m\omega P_{l}^{m}s^{l}, so we arrive at:
<br /> \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}
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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