Sum over k of r-kt choose k by z^k/Proof 1

Proof
From Sum over $k$ of $\dbinom r k$ by $\dbinom {s - k t} r$ by $\left({-1}\right)^k$ and renaming variables:


 * $\displaystyle \sum_j \left({-1}\right)^j \binom k j \binom {r - j t} k = t^k$

Thus: