# Hockey-Stick Identity With Modified Top Term

As part of solving a problem I came across a summation which I'm having some difficulty simplifying. I'm not sure if it would even simplify into something nice.

$$\sum\limits_{i=0}^{p-1} {ip^e \choose k}$$

where $$p$$ is prime, $$0\leq e\in\mathbb{Z}$$, and $$k\in\mathbb{Z}$$, $$0\leq k\leq p^e(p-1)$$.

I'm aware that when $$e=0$$ this is just the hockey-stick identity, but working with the general case has been unfruitful.