harmonic analysis – laguerre polynomials variations

The generalized Laguerre polynomial is written by the following form:
$$L_{n}^{(alpha)}(x)=sum_{k=0}^{n} begin{pmatrix} n+ alpha \ n-k end{pmatrix} (-1)^{k} frac{x^{k}}{k!}$$
I have found the value of this type for $x=0$ ($L_{n}^{(alpha)}(0)$) on the paper “Some identities for the generalized Laguerre polynomial 2016, W.Shao, Y.He, J. Pan”. But i didn’t know where can i deduce this generalized Laguerre polynomial for other values $x=1,2$. My suggestion is to compute $L_{n}^{(alpha)}(x)$ for $x=1,2$ without depending on special functions or hypergeometric functions. Please i need a clear response.
Thanks and best regards.