The generalized Laguerre polynomials: Are there any expressions valid for any case?

Question

There are general expressions of the generalized Laguerre polynomials.

For example: $$ L_n^{(\alpha)}(x) = {}_{n+\alpha}\mathrm{C}_n\ {}_1F_1(-n, \alpha+1, x), \hspace{20pt}(1) $$ $$ L_n^{(\alpha)}(x) = \sum_{l=0}^n(-1)^l\ {}_{n+\alpha}\mathrm{C}_{n-l} \frac{x^l}{l!}. \hspace{35pt}(2) $$

On one hand, Eq(1) is invalid if $\alpha + 1$ is a negative integer due to a property of ${}_1F_1$, and Eq(2) is also invalid if $n + \alpha$ is an negative integer.

On the other hand, the expressions of $L_n^{(\alpha)}$ for each $n$ is, for example, $$ L_0^{(\alpha)}(x) = 1,\\ L_1^{(\alpha)}(x) = -x + \alpha + 1,\\ L_2^{(\alpha)}(x) = \frac{x^2}{2} - (\alpha + 2)x + \frac{(\alpha+1)(\alpha+2)}{2},\\ L_3^{(\alpha)}(x) = -\frac{x^3}{6} + \frac{(\alpha+3)x^2}{2} - \frac{(\alpha+3)(\alpha+2)x}{2} + \frac{(\alpha+3)(\alpha+2)(\alpha+1)}{6} $$ (from wikipedia: Laguerre polynomials). But in these expressions, $\alpha$ can be any real values, which include the invalid cases for Eq(1) and (2).

So I have the question described in the title.

Answer 1

Well, using the falling factorial $$ (z)_n=\prod_{k=0}^{n-1} (z-k) $$ you can write $$ L_n^{(\alpha)}(x)=\frac1{n!}\sum_{k=0}^n\frac{(-n)_k (k+\alpha+1)_{n-k}}{k!}x^k $$ which is valid for all $\alpha\in\mathbb{R}$ and all nonnegative integer $n$.