I am studying the differential Hermite equation for two different books. And I noticed that they have a little difference in the expression for Hermite's EDO. I'll show you:
The first author I am using is a Brazilian author, Kleber Daum Machado. The name of the book is "Applied partial differential equations". Using the generalized Laguerre differential equation he makes the following development to arrive at the Hermite equation:
$$xy''+(\lambda+1-x)y'+my = 0$$
In the above expression it uses the following variable change $x = z^2$,and then through the chain rule relationships for the derivatives, it gets: $$\frac{dy}{dz} = \frac{1}{2z}\frac{dy}{dx}$$
and
$$\frac{d^2y}{dx^2}=\frac{1}{4z^2}\frac{d^2y}{dz^2}-\frac{1}{4z^3}\frac{dy}{dz}$$
Replacing both of the above expressions in the generalized Laguerre equation, the expression becomes:
$$\frac{1}{4}\frac{d^2y}{dz^2}-\frac{1}{4z}\frac{dy}{dz}+(\frac{\lambda + 1 - z^2}{2z})\frac{dy}{dz} + my = 0 $$
multiplying both sides by $4z$ and working the expression a little bit, it is possible to obtain:
$$z\frac{d^2y}{dx^2}-2z^2\frac{dy}{dx} + 4zmy = 0$$
my doubt starts here. After he gets this expression he makes the following change $4m =\mu$. And finally he gets:
$$\frac{d^2y}{dz^2}-2z\frac{dy}{dz}+\mu y=0$$ This last expression, finally, is the differential equation of Hermite
Another author i'm using is George Arfken (seventh edition). In chapter 18, he introduces the Hermite differential equation as follows:
$$H''_n(x) -2xH'_n(x)+2nH_n(x) = 0 $$
My question lies here. Notice that the difference in the expression that the other author reaches is in the last term, where we have a $4m= \mu$, on the other hand in the Hermite equation in the Aarfken book we have in the last term $2n$.
I was confused about this small difference. Maybe it's a dumb question, but I was curious about these differences from the last term. Does anyone know why Arfken wrote the equation that way?