Generating Function for mth Terms of Sequence/Examples/m = 3, r = 1

Example of Generating Function for mth Terms of Sequence

Let $\map G z$ be the generating function for the sequence $\sequence {a_n}$.

Let $\omega = e^{2 i \pi / 3} = -\dfrac 1 2 + \dfrac {\sqrt 3} 2 i$.

Then:

$a_1 z + a_4 z^4 + a_7 z^7 + \cdots = \dfrac 1 3 \paren {\map G z + \omega^{-1} \map G {\omega z} + \omega^{-2} \map G {\omega^2 z} }$

From Generating Function for mth Terms of Sequence, for $r \in \Z$ such that $0 \le r < m$:

$\ds \sum_{n \bmod m \mathop = r} a_n z^n = \dfrac 1 m \sum_{0 \mathop \ge k \mathop < m} \omega^{-k r} \map G {\omega^k z}$

Setting $m = 3$ and $r - 1$:

$\ds \sum_{n \bmod 3 \mathop = 1} a_n z^n$

$=$

$\ds \dfrac 1 3 \sum_{0 \mathop \ge k \mathop < 3} \omega^{-k} \map G {\omega^k z}$

$\ds \leadsto \ \ $

$\ds a_1 z + a_4 z^4 + a_7 z^7 + \cdots$

$=$

$\ds \dfrac 1 3 \paren {\omega^{-0} \map G {\omega^0 z} + \omega^{-1} \map G {\omega^1 z} + \omega^{-2} \map G {\omega^2 z} }$

$\ds $

$=$

$\ds \dfrac 1 3 \paren {\map G z + \omega^{-1} \map G {\omega z} + \omega^{-2} \map G {\omega^2 z} }$

$\blacksquare$