I've recently come across the following equality in a paper: suppose one defines an analytic function $L(n,x)$ which is equal to the $n$th Laguerre polynomial for $n\in\{0,1,\ldots\}$, and let* $L^{(1,0)}(n,x) = \frac{\partial}{\partial n}L(n,x)$. Then apparently
$L^{(1,0)}(-1,-x) = -[\gamma_E + \Gamma(0,-x) + \log(-x)]e^{-x}$
where $\gamma_E$ is the Euler-Mascheroni constant and $\Gamma$ is the incomplete gamma function.
Numerically, this checks out, but I would be interested in a symbolic proof that the equality holds. I've been playing around with the generating function for the Laguerre polynomials (and some other representations), but I haven't figured out a way to make the connection. Can anyone help me out?
*$L^{(1,0)}$ is called the "multivariate Laguerre polynomial" in the paper; I couldn't find a source to tell me whether this is a widely recognized function or not.