Let $n \geq 1$ be an integer. For $A \subset \mathbb{R}^n$, let $E_A = \sum_{\gamma \in A \cap \mathbb{Z}^n} x^\gamma \in \mathbb{C}[[\mathbb{Z}^n]]$. Let $\gamma_1, \ldots, \gamma_k$ be a family of vectors in $\mathbb{Z}^n \subset \mathbb{R}^n$ and $$C(\gamma_1, \ldots, \gamma_k) = \left\{\sum_{i=1}^k t_i \gamma_i : (t_1, \ldots, t_k) \in [0, +\infty[^k\right\}.$$ Show that if $\gamma_1, \ldots, \gamma_k$ is a free family, $E_{v + C(\gamma_1, \ldots, \gamma_k)}$ is rational for all $v \in \mathbb{R}^n$.
Let $n \geq 1$ be an integer. For $A \subset \mathbb{R}^n$, let $E_A = \sum_{\gamma \in A \cap \mathbb{Z}^n} x^\gamma \in \mathbb{C}[[\mathbb{Z}^n]]$.
Let $\gamma_1, \ldots, \gamma_k$ be a family of vectors in $\mathbb{Z}^n \subset \mathbb{R}^n$ and
$$C(\gamma_1, \ldots, \gamma_k) = \left\{\sum_{i=1}^k t_i \gamma_i : (t_1, \ldots, t_k) \in [0, +\infty[^k\right\}.$$
Show that if $\gamma_1, \ldots, \gamma_k$ is a free family, $E_{v + C(\gamma_1, \ldots, \gamma_k)}$ is rational for all $v \in \mathbb{R}^n$.