I am currently studying Legendre polynomials and I am trying to prove its generating function specifically using the recurrence relation
$$ (n+1)L_{n+1}(x) = x(2n+1)L_n(x) - nL_{n-1}(x), \quad \forall n\geqslant 0.$$ where $L_n(x)$ represents the $n-$order Legendre polynomial. I have searched quite some places to find a proof like this but wasn't able to find something concrete on it. Most of the proofs I saw of the generating function make use of the binomial theorem.
With this in mind, I believe I am doing a good work so far. With some easy calculation, one can easily check that the recurrence relation presented above yields
$$ xL_n(x) - L_{n-1}(x) = (n+1)L_{n+1}(x) - 2xnL_n(x) + (n-1)L_{n-1}(x), \quad \forall n \geqslant 0.$$
Now, I was wondering: It would be really cool if we could just guarantee something like this (for some $w?$):
$$ \sum_{n=0}^{\infty} w^n (xL_n(x) - L_{n-1}(x)) = \sum_{n=0}^{\infty}w^n\Big[(n+1)L_{n+1}(x) -2xnL_n(x) + (n-1)L_{n-1}(x)\Big].$$ As a matter fact, assuming convergence of both series, I was able to prove the generating function based off this result. Thus, what is left to do now is to show that we can actually assume this convergence.
First thing first. It suffices to show that one of the series converge, since by the second recurrence relation above these two series are exactly the same thing.
Second things second. Let us now try to show convergence of one of the series: for example, let us grab the L.H.S. First, we study the series
$$ \sum_{n=0}^{\infty} x L_n(x)w^n \quad \text{ and } \quad \sum_{n=0}^\infty L_{n-1}(x)w^n.$$
Note that
$$ \sum_{n=0}^\infty x L_n(x)w^n = x \sum_{n=0}^\infty L_n(x)w^n = x(1+xw+\frac{1}{2}(3x^2-1)w^2+\dots), $$ which clearly represents a geometric series with a non-constant coefficient. How can one show that such series converges?
I would be really gratefull for hints.
Hint: Regard $x$ as a parameter. You can quickly determine the radius $R$ of convergence of the series $\sum_k L_k w^k$ by using the ratio test, which involves estimating the limiting behavior of $|L_n/L_{n+1}|\to R$. From the recurrence equation, after dividing everything by the LHS and taking limits you get a quadratic equation that $R$ must satisfy.