Generating Function for mth Terms of Sequence

Theorem
Let $G \left({z}\right)$ be the generating function for the sequence $\left\langle{a_n}\right\rangle$.

Let $m \in \Z_{>0}$ be a (strictly) positive integer.

Let $\omega = e^{2 i \pi / m} = \cos \dfrac {2 \pi} m + i \sin \dfrac {2 \pi} m$.

Then for $r \in \Z$ such that $0 \le r < m$:
 * $\displaystyle \sum_{n \bmod m \mathop = r} a_n z^n = \dfrac 1 m \sum_{0 \mathop \ge k \mathop < m} \omega^{-k r} G \left({\omega^k z}\right)$