Generalized Laguerre function $ L^{\alpha}_n(x)$ fo $\alpha$ not necessarily a positive integer

Question

Is what we can define the generalized Laguerre function $ L^{\alpha}_n(x)$ for $\alpha=-1$ for all $n\geq 0$ and $x>0$?

Answer 1

The corresponding ODE for $L^{-1}_n(x)$ is $xy''-xy'+ny=0$ .

According to http://www.wolframalpha.com/input/?i=xy%22-xy'%2Bny%3D0 , it seem that $L^{-1}_n(x)\propto xM(1-n,2,x)$ .