Ramus's Identity

Theorem
Let $k, m, n \in \Z_{\ge 0}$ be positive integers such that $0 \le k < m$.

Then:

Proof
Let $\omega := e^{2 \pi i / m}$.

Then by the Binomial Theorem:
 * $\displaystyle \sum_{0 \mathop \le j \mathop < m} \left({1 + \omega^j}\right)^n \omega^{-j k} = \sum_t \sum_{0 \mathop \le j \mathop < m} \binom n t \omega^{j \left({t - k}\right)}$

We have:

Thus the summation on the is:


 * $m \displaystyle \sum_{t \bmod m \mathop = k} \binom n t$

The summation on the is:

Because the is wholly real, so must the  be.

So, taking the real parts of the and equating it to the  reveals the result.