Let $L_n^{(\alpha)}(x)$ denote a generalized Laguerre polynomial. I would like to have a closed form for the expression $$\frac{L_n^{(\alpha)}(1)}{L_{n-1}^{(\alpha)}(1)};$$ where we set $x=1$ and both $n$ and $\alpha$ are positive integers.
I'm aware, as explained in this answer on Math.SE, that the quotient can be expressed as a continued fraction. But this holds for any $x$, and I was hoping to get a closed form in my case, given its specificity. Mathematica doesn't seem to be of any help.
I would be grateful for any comment or idea.