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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.

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