Formal theory of (some) generating functions in $t$ and $t^{-1}$? I am interested in using series of the form $\sum_{n=-\infty}^{\infty} a_nt^n$ (where $a_n\in\mathbb C$) as generating functions. In general, multiplication of such series goes against the "formal power series" philosophy since the coefficients of the product are themselves infinite sums. However, the results are sometimes intelligible. Here is an example:
Let $G(t) = \frac 1{2-t} = \frac 12 + \frac 14t + \frac 18t^2 + \cdots$ be the usual probability-generating function for a random variable with Geometric($\frac 12$) distribution. If we have two independent Geometric($\frac 12$) random variables, $X$ and $Y$, then we can correctly read the distribution of $X-Y$ from the expansion of $G(t)\cdot G(t^{-1})$:
$$G(t)\cdot G(t^{-1}) = \cdots + \frac 1{12}t^{-2} + \frac 16t^{-1} + \frac 13 + \frac 16t + \frac 1{12}t^2 + \cdots$$
Although each coefficient is an infinite sum, those sums are absolutely convergent and so there is no doubt about how to evaluate them.
We can go further with this example. Suppose we write
\begin{align*}
G(t)\cdot G(t^{-1}) &= \frac 1{2-t}\cdot\frac 1{2-\frac 1t} \\
&= \frac 1{5-2t-\frac 2t} \\
&= \frac 15\cdot\frac 1{1-\frac 25(t+\frac 1t)} \\
&= \frac 15\sum_{n=0}^\infty \left[\frac 25\left(t+\frac 1t\right)\right]^n
\end{align*}
and attempt to extract the coefficients. Then the results appear to be accurate; for example, the constant term is $\frac 15\sum_{n=0}^\infty \binom{2n}n\left(\frac 25\right)^{2n}$, which does equal $\frac 13$. But this is a delicate game: if we multiply by $\frac tt$ at some point in the computation above, and try to expand $\frac t{5t-2t^2-2}$ as an ordinary power series, the results are (obviously) different.
Here's my question: Is there a theory that formalizes useful computations with series in $\mathbb C[[t,t^{-1}]]$ while excluding contradictory computations (much like the usual theory of formal power series does for computations in $\mathbb C[[t]]$)? References gratefully accepted.
 A: There is a natural space in which such operations can be carried out; however, necessarily, its definition is functional-analytic rather than algebraic. 
Let $H=L^2(S^1)$. Any $f\in H$ has a unique Fourier expansions of the form
$$
f(\theta)=\sum_{n\in\Bbb{Z}}a_n t^n, \quad \theta\in\Bbb{R}/2\pi\Bbb{Z},
\quad t=e^{i\theta}\tag1
$$
whose coefficients satisfy
$$
\sum_{n\in\Bbb{Z}}|a_n|^2<\infty.\tag2
$$
Conversely, every formal expansion of the form (1) whose coefficients satisfy (2) defines $f\in H$. In fact, $H$ is a commutative Banach algebra and it is a standard fact that addition and multiplication can be carried out at the level of formal expansions (1). Thus for
$$
g(\theta)=\sum_{n\in\Bbb{Z}}b_n t^n,\tag3
$$
the expansion of the product $fg$ is given by
$$
f(\theta)g(\theta)=\sum_{n\in\Bbb{Z}}\left(\sum_{k\in\Bbb{Z}}a_k b_{n-k}\right)t^n,\tag4
$$
where each coefficient in the expansion has a well-defined complex value due to the absolute convergence of the corresponding series (the latter property follows from the Cauchy-Schwartz inequality). Clearly, the multiplication is associative, due to the corresponding property in $H=L^2(S^1)$ and uniqueness of the Fourier expansion.
On the other hand, let us start with a general expansion of the form $(1)$ and impose a fairly mild requirement that the constant term of the product 
$$
\left(\sum_{n\in\Bbb{Z}}a_n t^n\right)\left(\sum_{n\in\Bbb{Z}}a_n t^{-n}\right),\tag5
$$
where the second factor is obtained by substituting $t^{-1}$ for $t$ in $(1)$,
be given by an absolutely convergent series. Using $(4)$ with $b_n=a_{-n}$, one easily sees that this requirement translates precisely to condition $(2)$ on the coefficients. So if we are interested in the products of type $(5)$, as in the example in the original posting, we might as well restrict attention to the elements of $H$ (i.e. impose condition $(2)$ on the coefficients) from the get-go. This indicates that $H$ is indeed a natural realm for usual aritmetic operations with such doubly-infinite expansions.
To summarize, we found a natural space $H$ where addition and multiplication of formal expansions of type $(1)$ and $(3)$ are well-defined. The definition of $H$ involves analytic condition $(2)$. The resulting object is a commutative Banach algebra. 
