Rodrigues’ formula for the Laguerre polynomials

The Rodrigues’ formula for the Laguerre polynomials is a mathematical expression that provides a method to calculate any Laguerre polynomial using differentiation.

It is given by

$L_n(x)=\frac{e^x}{n!}\frac{d^n}{dx^n}(x^ne^{-x})\;\;\;\;\;\;\;\;428$

To prove eq428, we apply Leibniz’ theorem as follows:

$L_n(x)=\frac{e^x}{n!}\sum_{k=0}^n\begin{pmatrix}n\\k\end{pmatrix}\frac{d^kx^n}{dx^k}\frac{d^{n-k}e^{-x}}{dx^{n-k}}$

Substituting $\frac{d^kx^n}{dx^k}=\frac{n!}{(n-k)!}x^{n-k}$ and $\frac{d^{n-k}e^{-x}}{dx^{n-k}}=(-1)^{n-k}e^{-x}$ in the above equation and rearranging yields