Let's say we have a sequence $a\_n$ that is defined for all $n\in\mathbb{Z}$
and i want to work with its GF $$A(z)=\sum\_{n\in\mathbb{Z}}a\_nz^n$$
But there are some problems with convergence. For example for $a\_n=1$:


$$A(z)=\cdots\frac{1}{z^3}+\frac{1}{z^2}+\frac{1}{z}+1+z+z^2+\cdots=\frac{1}{1-\frac{1}{z}}+\frac{z}{1-z}=\frac{z}{z-1}+\frac{z}{1-z}=0$$


We get an error.((


Is there any way to work with such generating functions?



How to work with this power series?
Yes, you may consider them as a module over the ring of polynomials. In other words, you may sum up such series and multiply them by polynomials. Also, you may differentiate such series, and usual rules work. For example, your series $\sum\_{n\in \mathbb{Z}} z^n=A(z)$ satisfies $(1-z)A(z)=0$. An application of this module which shows how to work with it may be found, for example, [here](https://mathoverflow.net/questions/182705/how-to-prove-an-elementary-functional-equation-for-polylogarithms/182735#182735).


