I would like to know if there's an exact expression for this integral in terms of known elementary or special functions:
$$\int_0^\infty \exp \left(-\frac{a+b+c}{2}x \right) L_j (a x) L_k (b x) L_l (c x)~ x ~dx$$
Where $L_j$ are Laguerre polynomials.
$$L_j (v)=\sum_{p=0}^j \binom{j}{p} \frac{(-1)^p v^p}{p!}$$
Technically, we could use the triple sum and the explicit expressions for the polynomials, but that's a little computationally expensive. I would prefer some kind of closed form, or at least a single finite sum. I will already be summing the result over one of the indices.
I have searched through the known literature on definite integrals and on Laguerre polynomials, and haven't been able to find an exact formula.
$a,b,c$ and $j,k,l$ are different in the general case. Just to be clear, $a,b,c$ are real and positive, while $j,k,l$ are nonnegative integers.
Here's a relevant paper: https://www.jstor.org/stable/2003738, but it just offers a multiple sum solution.
We can of course, 'get rid' of one parameter by changing the variable, for example: $$x=\frac{t}{a} \\ b'=\frac{b}{a}\\ c'=\frac{c}{a}$$