Closed Form Solution to finite sum

Question

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!

Answer 1

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

$$\sum_{n=0}^{m-1}n^pz^n=D^p\left(\frac{z^m-1}{z-1}\right).$$

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