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!

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.
