Could someone please explain why the following is true?
Suppose we are computing the Taylor coefficient of $z^{tA}$ in $$\frac{1}{1-z^{A_1}} \frac{1}{1-z^{A_2}} \cdots \frac{1}{1-z^{A_n}} \frac{1}{1-z} $$ where all $A_k, A \in \mathbb{Z}$. Why is this equivalent to finding $$\text{Res} \left( \frac{z^{-tA-1}}{(1-z^{A_1})(1-z^{A_2}) \cdots (1 - z^{A_n})(1-z)}, z=0 \right)? $$
And a further question, why is that equal to $$\text{Res} \left( \frac{ z^{-tA} - 1}{ (1-z^{A_1})(1-z^{A_2}) \cdots (1 - z^{A_n})(1-z)z}, z=0 \right) +1? $$
The author of https://arxiv.org/pdf/math/0306035.pdf entirely glosses over this step, saying it's "clearly" true...