Show that a function $P : \mathbb{Z} \rightarrow \mathbb{C}$ is quasi-polynomial if and only if there exist an integer $m \in \mathbb{N}^*$ and $m$ polynomials $P_0, \ldots, P_{m-1}$ with complex coefficients such that for all $j \in \{0, \ldots, m-1\}$ and for all $n \in \mathbb{Z}$ congruent to $j$ modulo $m$, we have $P(n) = P_j(n)$.