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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?

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    $\begingroup$ The radius of convergence is not the same for the two series. For the series $1/(1-z)$ the radius of convergence is $|z|<1$ and for $1/(1-1/z))$ is $|z|>1$. $\endgroup$
    – Alan
    Commented Oct 12, 2016 at 16:12
  • $\begingroup$ thank you, but i know it, my question is:Is there ANY WAY to work with such generating functions? $\endgroup$
    – Radmir Sultamuratov
    Commented Oct 12, 2016 at 16:13
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    $\begingroup$ This is known as a "Laurent series". The region of the complex plane where it converges is, in general, an annulus centered at $0$. Of course the annulus may be (as in your example) empty. $\endgroup$
    – Gerald Edgar
    Commented Oct 12, 2016 at 17:04
  • $\begingroup$ This Laurent series DOES converge---on the unit circle (the limit is the Dirac delta function, with singularity at the point "1"). Convergence is, of course, not in the classical but in the distributional sense. The details of this were worked out by Gottfried Köthe in 1952---for a reference, see my answer below. $\endgroup$
    – vltava
    Commented Oct 12, 2016 at 19:47
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    $\begingroup$ See web.mit.edu/~darij/www/algebra/va3.pdf (Sections 1-3) for some fundamentals of these "doubly-infinite" formal series. (This is all folklore material, but few people have bothered writing it down. Note that my writeup is complicated by the extra lengths I went to make things work natively in positive characteristic.) $\endgroup$
    – darij grinberg
    Commented Oct 12, 2016 at 21:10

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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.

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On the other hand, there is no non-trivial way to define multiplication for these "doubly infinite" power series. At least not in a way that would endow them with a ring structure for which the addition operation is the usual one.

Indeed, the ring in question would then be $\mathbb{Z}[[X,Y]]/(XY-1)$, but this is the zero ring, since $f=1-XY \in \mathbb{Z}[[X,Y]]$ has inverse $1+XY+(XY)^2+\ldots$.

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There is a rigorous way to deal with such series analytically, within the framework of distribution theory, more precisely of their representation as boundary values of anslytic functions. The theory was initiated by Gottfried Köthe in a paper entitled "Die Randverteiling analytischer Funktionen" which can easily be found online (Math. Zeitschr. 1952. (Translation of the title: The boundary values of analytic functions). The author shows how distributions on the unit disc---essentially, periodic ones---can be regarded as (generalised) boundary values of functions which are analytic on its complement in the complex plane. In fact, the example in the question is just such a representation for the Dirac delta function. I would guess that this was already known to und used by mathematical physicists (on a non-rigorous basis). If I recall correctly, this material is expounded in his classical monograph and so can be found in an english version.

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