Understanding the Legendre Recurrence Relation for Generating Functions

[Taylor_1989]

Homework Statement

I am having a slight issue with generating function of legendre polynomials and shifting the sum of the genertaing function.

So here is an example:

I need to derive the recurence 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 ignor the shift and keep the sum $l=0$ which I can see beacuse 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 can't 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 equalt 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.

Where as when I do it my way I expand all the series unitl the are l=3 and then do the same thing. But 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 if someone could help me on understanding this.

 

[haruspex]

The recurrence relation should state a range of l to which it applies. That range should exclude the terms which are different between your sum equation and the "book" one.
The book one, as you quote it, is clearly wrong since it includes h^-1 and h^-2 terms.