Laguerre diferential equation for the radial part of the Hydrogen atom

I am working on understanding the solution to the Schrödinger equation for the Hydrogen atom and I am now stuck on something concerning the Radial part of the equation.

After the change of variables $rho = kr$ with $k^2 = – frac{2 mu E}{hbar^2}$ and some work on the equations I got

$$dfrac{d^2 u(rho)}{drho^2} rho + left[ (2l+1) +1 – 2rho right] frac{du(rho)}{drho} + 2(n-l-1) u(rho) = 0$$

I know that the solution to this equation is the Laguerre polynomial $u(rho) = L_{n-l-1}^{2l+1}(2rho)$. Aaccording to the equation on wikipedia the Laguerre polynomial is the solution to the following differential equation

$$ frac{d^2 L_m^alpha(x)}{dx^2} x + (alpha+1-x)frac{d L_m^alpha(x)}{dx} + m L_m^alpha(x) = 0 $$

I can identify that $m = n-l-1 $ and $alpha = 2l+1$ but there is a slight difference in my equation (the factor $2$ in front of $n-l-1$ and $rho$ that should be $2rho$ in the first term of the equation). I am however confident that my equation is correct as it is the same as the one I found in some textbooks, and I would very much appreciate some explanation on my issue.