I want to solve this equation $${\frac{1}{(\sqrt{2m})^n}\Bigr(\frac{\hbar}{i}\frac{d}{dx} - im\omega x\Bigr) }^n\psi(x) =0$$
See more about Hermite function recursion relation $(x-\partial) (e^{x^2/2}\partial^n e^{-x^2})=- (e^{x^2/2}\partial^{n+1} e^{-x^2})$.
Here's $\psi$ for some value of n
For $n=0$ , $\psi(x) = 0$
$n=1, \psi(x) = C e^{-\frac{m\omega }{2\hbar}x^2}$
$n=2$
$f(x) = \psi, a = \hbar/i, b = m \omega x$
Is there a more general solution for all values of n?