Here is the generating function I'm studying.

$f=\prod^N_{j=1}\left(1+e^{i\cdot j\varphi}z\right)$.

$\varphi$ is a phase related to a quantum optics problem.

And I want to know the analytical result of:

$\lim_{z\to 0}\frac{1}{X!}\frac{d^Xf}{dz^X}$

with $1\le X\le N$.