Rodrigues’ formula for the associated Laguerre polynomials

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

It is given by

$L_n^m(x)=\frac{e^xx^{-m}}{n!}\frac{d^n}{dx^n}(x^{n+m}e^{-x})\;\;\;\;\;\;\;\;446$

To prove eq446, we apply Leibniz’ theorem as follows:

$L_n^m(x)=\frac{e^xx^{-m}}{n!}\sum_{k=0}^n\frac{n!}{k!(n-k)!}\frac{d^kx^{n+m}}{dx^k}\frac{d^{n-k}e^{-x}}{dx^{n-k}}\;\;\;\;\;\;\;\;446$

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

$L_n^m(x)=\sum_{k=0}^n\frac{(-1)^{n-k}}{k!(n-k)!}\frac{(n+m)!}{(n+m-k)!}x^{n-k}$

Letting $r=n-k$ , we have $L_n^m(x)=\sum_{r=0}^n\frac{(-1)^{r}}{r!(n-r)!}\frac{(n+m)!}{(r+m)!}x^{r}$ , which is the expression for the associated Laguerre polynomials.