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 $lP_l(x)=(2l-1)xP_{l-1}(x)-(l-1)P_{l-2}$
so I start with the following equation:
$$(1-2xh+h^2)\frac{\partial\phi}{\partial h}=(x-h)\phi $$
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 $$\sum_{l=1}^{\infty} lh^{l-1} P_l(x))$$
Some book 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 side I get the following:
$$\sum_{l=1}^{\infty} lh^{l-1} P_l(x))-2x\sum_{l=1}^{\infty} lh^{l} P_l(x))+\sum_{l=1}^{\infty} lh^{l+1} P_l(x)=x\sum_{l=0}^{\infty} h^l P_l(x)-\sum_{l=0}^{\infty} h^{l+1} P_l(x)[1]$$
So now if I make eq [1]=0 like so:
$$\sum_{l=1}^{\infty} lh^{l-1} P_l(x))-2x\sum_{l=1}^{\infty} lh^{l} P_l(x))+\sum_{l=1}^{\infty} lh^{l+1} P_l(x)-x\sum_{l=0}^{\infty} h^l P_l(x)+\sum_{l=0}^{\infty} h^{l+1} P_l(x)=0$$
So from here I know that I need to get all the powers of h to $h^{l-1}$
So here what I do, which I believe is incorrect I just cant see why or to be honest understand why it wrong. My belief it it has some thing to do with how the generating function works, but every book I have read and youtube video I have watch completely ignores the step that I get confused with, and without any explanation just give the recurrence relation. Anyway here what I do.
looking at the series I make all the powers of $h$ equal to $l-1$
$$\sum_{l=1}^{\infty} lh^{l-1} P_l(x))-2x\sum_{l=2}^{\infty} (l-1)h^{l-1} P_{l-1}(x))+\sum_{l=3}^{\infty} (l-2)h^{l-1} P_{l-2}(x)-x\sum_{l=1}^{\infty} h^{l-1} P_{l-1}(x)+\sum_{l=2}^{\infty} h^{l-1} P_{l-2}(x)=0$$
Now youtube videos I have watched and book I have read just change the powers of h without make any shitf to the series like soo.
$$\sum_{l=0}^{\infty} lh^{l-1} P_l(x))-2x\sum_{l=0}^{\infty} (l-1)h^{l-1} P_{l-1}(x))+\sum_{l=0}^{\infty} (l-2)h^{l-1} P_{l-2}(x)-x\sum_{l=0}^{\infty} h^{l-1} P_{l-1}(x)+\sum_{l=0}^{\infty} h^{l-1} P_{l-2}(x)=0$$
Then just factor out the sum and the h and your left with the desired recurrence relation albeit with a little simplifying.
Whereas when I do it my way I expand all the series until $l=3$ and then do the same thing. But as I said I believe this to be wrong, but I just cant see how that in the book I have read and video I have watched how they can just change the powers of the $h$ with out affecting the series itself.
I would much appreciate if someone could help me on understanding this.