Where does the following formula come from? Where does the following formula come from?
For a Laurent polynomial $f(z)=\sum a_j z^j$ and a positive integer $n$ we have $$\sum_{k\equiv \alpha\pmod n} a_k=\frac1n\sum_{\omega:\omega^n=1} \omega^{-\alpha}f(\omega).$$
I hope someone can answer this question or give some refferences about it ! Thanks a lot!
 A: first we shall see that:
$\frac1n\sum_{\omega:\omega^n=1} \omega^{-\alpha}f(\omega) = \frac1n\sum_{\omega:\omega^n=1}\omega^{-\alpha }\sum_{j}a_j\omega^j = \frac1n\sum_{\omega:\omega^n=1}\sum_{j}\omega^{j-\alpha }a_j =$
$ = \frac1n\sum_{j}\sum_{\omega:\omega^n=1}\omega^{j-\alpha }a_j$ 
I'm leaving for you to think why we can preform the last move (change the order).
also put notice that $\omega^{j-\alpha} = 1 \iff j \equiv \alpha \ (mod \ n)$ and also because there are n roots of unity in $\mathbb{C}$ we get:
$ \frac1n\sum_{j}\sum_{\omega:\omega^n=1}\omega^{j-\alpha }a_j  = \frac1n\sum_{j \equiv \alpha (mod \ n)}\sum_{\omega:\omega^n=1}a_j + \frac1n\sum_{j \neq \alpha (mod \ n)}\sum_{\omega:\omega^n=1}\omega^{j-\alpha }a_j = \sum_{j \equiv \alpha (mod \ n)} \frac{n}{n} a_j + \frac1n\sum_{j \neq \alpha (mod \ n)}a_j\sum_{\omega:\omega^n=1}\omega^{j-\alpha }$
but we know that $\forall \ 0<m< \ n: \sum_{\omega:\omega^n=1}\omega^{m } = 0 $ and therefor we get: $\frac1n\sum_{j \neq \alpha (mod \ n)}a_j\sum_{\omega:\omega^n=1}\omega^{j-\alpha } = 0 $ (as $0<j-\alpha < n$) and were done (-:
A: This formula comes from Thomas Simpson's Series Multisection Theory. Speak in the concrete，
a multisection of the series of an  analytic function
$$f(z) = \sum_{n=0}^\infty a_n\cdot z^n.$$
has a  closed-form expression  in terms of the function   $f(x)$:
$$\sum_{m=0}^\infty a_{qm+p}\cdot z^{qm+p} = \frac{1}{q}\cdot \sum_{k=0}^{q-1} \omega^{-kp}\cdot f(\omega^k\cdot z),$$
where $\omega = e^{\frac{2\pi i}{q}}$ is a  Primitive nth root of unity primitive ''q''-th root of unity .  This solution was first discovered by  Thomas Simpson.
