I would like to simplify (if possible)
$$ \sum_{k=0}^\infty(-\alpha)^k\frac{(2k)!\:L(2k,-\beta)}{k!} $$
where $L(n,x)$ is the $n$-th Laguerre polynomial evaluated at $x$. In this case, I know that $0<\alpha<1$ and $\beta>0$, but this sum only seems to converge numerically for very small $\alpha$ (which is fine, I'm assuming there are restrictions on these constants).
EDIT: This appears to not even converge.