For questions about (associated) Laguerre polynomials, which arise in quantum physics.
The Laguerre polynomials are solutions of Laguerre differential equation: $$xy'' + (1 - x)y' + ny = 0, \text{ where } n \in \mathbb{N} \cup \{0\}, \tag1 \label{eq1}$$ which is a second-order linear differential equation.
Equation \eqref{eq1} is a special case of a more general "associated Laguerre differential equation", defined by $$ xy''+(\alpha+1-x)y'+\lambda y=0, \tag2 \label{eq2},$$ where $\lambda$ and $\alpha$ are real numbers with $\alpha = 0$ and $\lambda = n$.
These polynomials, usually denoted $L_0$, $L_1$, $\dots$, are a polynomial sequence which may be defined by the Rodrigues formula,
$$ L_{n}(x)={\frac {e^{x}}{n!}}{\frac {d^{n}}{dx^{n}}}\left(e^{-x}x^{n}\right)={\frac {1}{n!}}\left({\frac {d}{dx}}-1\right)^{n}x^{n}. $$
The Laguerre polynomials are also used for Gaussian quadrature to numerically compute integrals of the form $$ \int _{0}^{\infty }f(x)e^{-x}\,dx.$$
