Physicist here, so apologies in advance for any wishy-washy use of notation, etc.
A bit of context for my actual question - I'm currently working on a research project related to quantum-classical analogs, and we're taking a phase-space approach to the quantum harmonic oscillator using (among other things) a Wigner transform.
Given a wavefunction $\psi_n(x, t)$, we can compute the Wigner transform $W(x, p, t)$ - $$ W(x, p, t) = \frac{1}{2 \pi \hbar} \int \psi_n \left( x + \frac{y}{2} \right) \psi_n^* \left( x - \frac{y}{2} \right) exp \left[ -i \frac{py}{\hbar} \right] dy $$
Using a change of variables, we can write - $$ W(x, p, t) = W(x', p', t = 0)$$
Where $x' \neq x$, $p' \neq p$ are I guess what you might call "generalized positon/momenta," both of which are time-dependent. The above Wigner function for harmonic oscillator eigenstates has an exact solution that looks something like - $$W^{HO}(x, p, t) = f(x, p, t) \hspace{1mm} L_n \left[ g(x, p, t) \right] $$
Where $L_n(x)$ is a Laguerre polynomial. The fidelity $F(t)$ ("overlap," roughly) between the state at $t = 0$ and a later time $t > 0$ is then - $$ F(t) = \int \int W_0 W_t \hspace{1mm} dx \hspace{1mm} dp $$ $$ = \int \int W^{HO}(x, p, t = 0) W^{HO}(x', p', t = 0) \hspace{1mm} dx \hspace{1mm} dp$$ $$ = \int \int f(x, p, 0) f(x', p', 0) \hspace{1mm} L_n \left[ g(x, p, 0) \right] \hspace{1mm} L_n \left[ g(x', p', 0) \right] \hspace{1mm} dx \hspace{1mm} dp $$
Now for my actual question - I've been led to believe that the integral of the product of multiple Laguerre polynomials is only defined when their arguments are the same. I've been looking at the generating function - $$ h(u, x) = \sum_n u^n L_n(x) = \frac{1}{1-u} exp \left[ - \frac{ux}{1-u} \right] $$
Which is essentially a Taylor series where - $$ L_n(x) = \frac{1}{n!} \frac{\partial^n}{\partial u^n} h(u, x)\bigg|_{u = 0}$$
Which seems... sort of helpful, but doesn't leave me any closer to a reasonable expression for $F(t)$ that I'd be able to work with (I don't think).
I suppose my question is - why are generating functions useful? Finding $L_n(x)$ this way seems like it's just shifting the intractability of the problem from an integral that isn't defined to a nasty partial derivative. I understand their use in discrete math/combinatorics, but I'm struggling to see how it's helpful to me in this context. Surely I just don't know enough about them and/or there's something I'm missing. Thanks in advance.