From How to show orthogonality of associated Laguerre polynomials? or Finding a generating function for the Laguerre polynomials it appears that the expression, $L^q_p(x)$ is that of an associated Laguerre polynomial and defined as
$$ L^q_p(x)=\frac{x^{−q}\exp{(x)}}{p!}\frac{d^p}{dx^p}\left(x^{(p+q)}\exp{(−x)}\right). $$
However I am not quite sure what the $q$th order derivative means, for a non-integer $q$.
I am looking for the limiting behaviour of $L_q^p(x)$ as $x\rightarrow\pm\infty$, or as $x\rightarrow0$. I am most interested in the cases that $p$ is a half integer (that is, $p=\frac m2$ and $q=\pm\frac12$. I found some help (for my special cases) at Associated Laguerre polynomials of half-integer parameters but the answer says it is just an empirical formula with unestablished formal validity.
Any suggestions/help on how to proceed? Just a reference or two, or some suggested step would be extremely helpful, and much apprecited.