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November 5th, 2017, 11:23 PM  #1 
Senior Member Joined: Sep 2012 Posts: 201 Thanks: 1  Legendre reccurence relation
I am having a slight issue with generating function of Legendre polynomials and shifting the sum of the generating function. So here is an example: I need to derive the recurrence relation $lP_l(x)=(2l1)xP_{l1}(x)(l1)P_{l2}$ so I start with the following equation: $$(12xh+h^2)\frac{\partial\phi}{\partial h}=(xh)\phi $$ now taking the differential of the series of the Legendre generating function $$\frac{\partial}{\partial h}(\sum_{l=0}^{\infty} h^l P_l(x))$$ I make it equal to $$\sum_{l=1}^{\infty} lh^{l1} P_l(x))$$ Some books I have read ignore the shift and keep the sum $l=0$ which I can see because even at the $l=0$ the sums are the same. But when I expand the following is when I get in a bit of a mess. So expanding both sides I get the following: $$\sum_{l=1}^{\infty} lh^{l1} P_l(x))2x\sum_{l=1}^{\infty} lh^{l} P_l(x))+\sum_{l=1}^{\infty} lh^{l+1} P_l(x)=x\sum_{l=0}^{\infty} h^l P_l(x)\sum_{l=0}^{\infty} h^{l+1} P_l(x)[1]$$ So now if I make eq [1]=0 like so: $$\sum_{l=1}^{\infty} lh^{l1} P_l(x))2x\sum_{l=1}^{\infty} lh^{l} P_l(x))+\sum_{l=1}^{\infty} lh^{l+1} P_l(x)x\sum_{l=0}^{\infty} h^l P_l(x)+\sum_{l=0}^{\infty} h^{l+1} P_l(x)=0$$ So from here I know that I need to get all the powers of h to $h^{l1}$ So here what I do, which I believe is incorrect I jus't cant see why or to be honest understand why it wrong. My belief it has some thing to do with how the generating function works, but every book I have read and youtube video I have watched completely ignores the step that I get confused with, and without any explanation just gives the recurrence relation. Anyway, here's what I do. Looking at the series, I make all the powers of h equal to $l1$ $$\sum_{l=1}^{\infty} lh^{l1} P_l(x))2x\sum_{l=2}^{\infty} (l1)h^{l1} P_{l1}(x))+\sum_{l=3}^{\infty} (l2)h^{l1} P_{l2}(x)x\sum_{l=1}^{\infty} h^{l1} P_{l1}(x)+\sum_{l=2}^{\infty} h^{l1} P_{l2}(x)=0$$ Now youtube videos I have watched and books I have read just change the powers of h without make any shift to the series like soo. $$\sum_{l=0}^{\infty} lh^{l1} P_l(x))2x\sum_{l=0}^{\infty} (l1)h^{l1} P_{l1}(x))+\sum_{l=0}^{\infty} (l2)h^{l1} P_{l2}(x)x\sum_{l=0}^{\infty} h^{l1} P_{l1}(x)+\sum_{l=0}^{\infty} h^{l1} P_{l2}(x)=0$$ Then just factor out the sum and the h and you're left with the desired recurrence relation, albeit with a little simplifying. Whereas when I do it my way, I expand all the series until the are l=3 and then do the same thing. As I said, I believe this to be wrong, but I just can't see how that in the book I have read and video I have watched how they can just change the powers of the h without affecting the series itself. I would much appreciate it if someone could help me on understanding this. Last edited by skipjack; November 6th, 2017 at 06:31 AM. 

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legendre, realtion, reccurence, relation, series or generating 
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