Let $k \in \mathbb{N}^*$ and $(a_1, \ldots, a_k) \in (\mathbb{N}^*)^k$ a $k$-tuple of strictly positive integers. When $k \geq 2$, we assume they are coprime as a set. We define a function $P : \mathbb{N} \rightarrow \mathbb{C}$ by setting for all $n \in \mathbb{N}$: $$P(n) = \operatorname{Card}\left\{(n_1, \ldots, n_k) \in \mathbb{N}^k : n_1 a_1 + \cdots + n_k a_k = n\right\}.$$ We assume $k = 2$. We set $a = a_1, b = a_2, \omega_a = \exp(2\mathrm{i}\pi/a), \omega_b = \exp(2\mathrm{i}\pi/b)$. From a partial fraction decomposition of the fraction $\frac{1}{(1 - x^a)(1 - x^b)}$, show the formula $$P(n) = \frac{1}{2a} + \frac{1}{2b} + \frac{n}{ab} + \frac{1}{a} \sum_{j=1}^{a-1} \frac{\omega_a^{-jn}}{1 - \omega_a^{jb}} + \frac{1}{b} \sum_{k=1}^{b-1} \frac{\omega_b^{-kn}}{1 - \omega_b^{ka}}$$ for all integer $n \geq 0$.
Let $k \in \mathbb{N}^*$ and $(a_1, \ldots, a_k) \in (\mathbb{N}^*)^k$ a $k$-tuple of strictly positive integers. When $k \geq 2$, we assume they are coprime as a set. We define a function $P : \mathbb{N} \rightarrow \mathbb{C}$ by setting for all $n \in \mathbb{N}$:
$$P(n) = \operatorname{Card}\left\{(n_1, \ldots, n_k) \in \mathbb{N}^k : n_1 a_1 + \cdots + n_k a_k = n\right\}.$$
We assume $k = 2$. We set $a = a_1, b = a_2, \omega_a = \exp(2\mathrm{i}\pi/a), \omega_b = \exp(2\mathrm{i}\pi/b)$. From a partial fraction decomposition of the fraction $\frac{1}{(1 - x^a)(1 - x^b)}$, show the formula
$$P(n) = \frac{1}{2a} + \frac{1}{2b} + \frac{n}{ab} + \frac{1}{a} \sum_{j=1}^{a-1} \frac{\omega_a^{-jn}}{1 - \omega_a^{jb}} + \frac{1}{b} \sum_{k=1}^{b-1} \frac{\omega_b^{-kn}}{1 - \omega_b^{ka}}$$
for all integer $n \geq 0$.