Recurrence relation for associated Laguerre Polynomials: same index and argument

Question

The Laguerre polynomials $L_n(x)$ have the recurrence relation $$ L_{n+1}(x) = \frac{(2n+1-x)L_{n}(x) - nL_{n-1}(x)}{n+1}. $$

I'm looking for a similar relation between the associated Laguerre polynomials $L_n^k(x)$ when

1. $n$ and $x$ are constant,
2. $k$ and $x$ are constant.

Question: Are there any such relations?

Edit: I just stumbled upon the following formula for case 2 $$ L_{n+1}^k(x) = \frac{k + 2n + 1 - x}{n + 1}L_n^k(x) - \frac{k+n}{n+1}L_{n-1}^k(x), \quad (n>0)$$ with $L_0^k(x) = 1$ and $L_1^k(x) = 1 + k -x$. Now I would like to know if case 1 is possible?