Closed Form Solution to finite sum I'm looking for a closed form solution to
$$
\sum_{n=0}^{m-1} n^p z^n
$$
, where p is an integer and z is complex.
Many thanks!

Hint:
Let $D$ be the operator 
$$D:=z\frac d{dz}.$$
We have
$$D\sum_{n=0}^{m-1}z^n=z\sum_{n=0}^{m-1}nz^{n-1}=\sum_{n=0}^{m-1}nz^n$$ and by recurrence
$$D^p\sum_{n=0}^{m-1}z^n=\sum_{n=0}^{m-1}n^pz^n.$$
Then
$$\sum_{n=0}^{m-1}n^pz^n=D^p\left(\frac{z^m-1}{z-1}\right).$$

The general solution will be of the form $$\frac{P(z)}{(z-1)^{p+1}}$$ where $P$ is a polynomial of degree $m+p$.
