What does the linear differential operator, $Q$, defined as
$Q = (1 + \frac{1}{\alpha}\frac{d}{dt})^2$
mean exactly? That is, in the context of ODEs.
Does it mean the following?
$Q = 1 + \frac{2}{\alpha}\frac{d}{dt} + \frac{1}{\alpha^2}\frac{d^2}{dt^2}$
or this?
$Q = 1 + \frac{2}{\alpha}\frac{d}{dt} + \frac{1}{\alpha^2}(\frac{d}{dt})^2$
It means applying the operator twice. Remember, $1$ and $\frac{1}{\alpha}$ are not simply numbers, they are operators, so you can't just multiply them that way (at least, usually you can't). Here it works out since all the operators commute.
To your question, that just looks like different ways of writing the same thing, since $\left(\frac{d}{dt}\right)^2$ is simply differentiating twice with respect to $t$, as is $\left(\frac{d^2}{dt^2}\right)$. Again, it's just notational conventions.