I'm trying to follow a document describing a 'Perfectly Matched Layer' (an FDTD boundary condition), however it uses partial derivative notation in a context I'm not familiar with:
$$ \psi^n_i = a_i\psi^{n-1}_i + b_i\bigg(\frac{\partial}{\partial i}\bigg)^n $$
I don't know what the numerator-less partial derivative in the parentheses is, it appears to be partial derivative of nothing in respect to $i$. The authors use partial-derivative-with-no-numerator notation in a context I'm more familiar with later on in the document:
$$ \frac{\partial^2 P}{\partial i^2} = \frac{\partial}{\partial i}\bigg(\frac{\partial P}{\partial i}\bigg)$$
I should point out that $i$ is a basis vector in 3D Cartesian space i.e. $i \epsilon \{x, y, z\}$.
As you may have realised, mathematics is not a strong subject of mine, but I'll try to answer any queries you have.
That is an equality of operators. You have to apply it to some function $f$, like in
$$\psi^n_i f = a_i\psi^{n-1}_i f + b_i\bigg(\frac{\partial}{\partial i}\bigg)^n f = a_i\psi^{n-1}_i f + b_i \frac{\partial^n f}{\partial i^n}.$$