Rodrigues’ Formula for Laguerre equation

Question

This is exercise 12.1.2 a from Arfken's Mathematical Methods for Physicists 7th edition :

Starting from the Laguerre ODE,

$xy''+(1-x)y'+\lambda y =0 $,

obtain the Rodrigues formula for its polynomial solutions $L_n (x)$

According to Arfken (equation 12.9 ,chapter 12) the Rodrigues formula is :

$ y_n(x) = \frac {1}{w(x)}(\frac{d}{dx})^n[w(x)p(x)^n]$

I found that $w(x) = e^{-x}$ and then :

$L_n(x) = e^x (\frac{d}{dx})^n[e^{-x}x^n]$

But the answer is ,according to Arfken and everywhere else I look,is :

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

I can't figure out exactly how $ \frac{1}{n!}$ appeared.
I think it might be related to the fact that $ L_n(x) =\sum_{k=0}^n \binom{n}{k} \frac{(-x)^k}{k!} \quad $

Any help will be appreciated , thank you

Answer 1

Dividing $L_n$ as you have found it, by $n!$ gives it a unit norm with respect to weight function $e^{-x}$ on $(0,\infty)$.

This is established in particular here.