In an engineering setting, I reduced my problem to calculating the following sum:

$$\sum_{n=0}^\infty \frac{n!}{(k+n)!}\left[\int_0^a \left(\frac{x}{u}\right)^kL_n^{(k)}\left(\frac{x^2}{u^2}\right)\exp\left[-\frac{x^2}{2u^2}\right]J_n(vx)\mathrm{d}x\right]^2,$$

where $L_n^{(k)}(\cdot)$ is the generalized Laguerre polynomial, $J_n(\cdot)$ is $n$-th order Bessel function of the first kind, and $a>0$,$u>0$, $v>0$, and $k$ is a positive integer. I am not sure what to do with this beast. I will be happy with a tight upper bound, or a fast method to calculate this numerically (using numerical integration for each element in the sum is extremely slow, as I have to evaluate this sum for many values of $u$, $v$, and $k$).

Does anyone have any ideas?