Laguerre polynomials

Remarks

For $\alpha = 0$ the coefficient is a binomial coefficient: $L_n(x) = \sum_{k=0}^{n} \binom{n}{k} \frac{(-x)^k}{k!}$. Example: $L_1^{(\alpha)}(x) = 1 + \alpha - x$ and $L_2(x) = 1 - 2x + \frac{x^2}{2}$. Rodrigues' formula: $L_n^{(\alpha)}(x) = \frac{x^{-\alpha} e^x}{n!} \frac{d^n}{dx^n} \left( e^{-x} x^{n + \alpha} \right)$. Orthogonality: $\int_0^\infty L_m^{(\alpha)}(x) L_n^{(\alpha)}(x) \, x^\alpha e^{-x} \, dx = \frac{\Gamma(n + \alpha + 1)}{n!} \delta_{mn}$. The polynomial $L_n^{(\alpha)}$ solves the Laguerre equation $x y'' + (\alpha + 1 - x) y' + n y = 0$. Check for $n = 1$: $y = 1 + \alpha - x$ gives $0 - (\alpha + 1 - x) + (1 + \alpha - x) = 0$. Non-example: the clause $\alpha > -1$ is needed for orthogonality, because for $\alpha \le -1$ the weight $x^\alpha e^{-x}$ is not integrable near $0$.