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.

does dependon the circle (or annulus) on which the decomposition is made (once you pass through a pole, the corresponding part flips). So, you just want the expansion near the unit circle rather than near the origin or near infinity. I'm not sure if you had something more interesting in mind, so I'll stop here. $\endgroup$ – fedja Sep 17 '16 at 17:43