Resource on Infinite Systems of Difference Equations I have asked this question previously at Math.stackexchange, but it seems to receive little attention there. 
In my efforts (somewhere on the boundary of discrete mathematics and theoretical computer science), I have come upon the necessity of solving (or at least finding out some of the solution's properties) infinite systems of difference (NOT differential) equations (linear, autonomous, first-order). Solving countable systems is sufficient for my purpose.
However, I have almost no idea about how to solve such systems (and what are the solvability conditions). I have a feeling that a possible approach is to seek generalizations of the finite-dimensional method to infinite-dimensional spaces via the spectral theory of linear operators, but I have no idea about the details.
So my question is whether there are any resources that deal systematically with infinite systems of difference equations. I would be grateful either for a resource that can be read also without a knowledge of advanced mathematics (that is something readable for non-mathematicians) or, alternatively, for a more advanced resource with a recommendation where to learn the prerequisities. My mathematical background is unfortunately quite basic (I am a computer science major): possibly relevant fields I have a background in are calculus, linear algebra, some fundamentals of modern analysis, difference/differential equations, and basics of functional analysis (however I have very poor knowledge of spectral theory).
I would appreciate any resources dealing with this issue, as well as any recommendations about which parts of mathematics I am supposed to learn (preferably with some recommended resources). I am willing to invest a considerable amount of time into the study, but I would like to follow some recommended path that leads to the goal. In the case that the theory of infinite systems of differential equations is similar, I would appreciate also resources dealing with them.
 A: If your equations look like $x_{i+1} = Ax_i$, where each $x_j$ is an element of a (possibly infinite dimensional) vector space and $A$ is a linear operator thereon, then the general solution is $x_j = A^j x_0 = \exp(j\log(A))x_0$, with $x_0$ specifying the initial conditions. The properties of $A^j$, or equivalently $\exp(j\log(A))$, can be deduced from a knowledge of the spectrum and invariant subspaces of $A$. This is precisely spectral theory. So I guess you just have to bite the bullet and start reading up on spectral theory.
The easiest situation is when the $x_j$ are vectors in a Hilbert space and $A$ is a bounded self-adjoint operator. This case is covered in essentially every introductory book on functional analysis. There is a long list of references of introductory and higher level reference books on Wikipedia.
If your $A$ is not of the above form, then you need to be more specific about the structure of your difference equation to get a more relevant suggestion.
A: There is a branch of mathematics called difference algebra. It deals systematically with quite general (possibly infinite) systems of algebraic difference equations. The Wikipedia entry will give you some idea.
A recently appeared monograph is

A. Levin, Difference algebra, Springer, 2008.

In difference algebra, there is an analog of Hilbert's basis theorem, which, roughly speaking, states that for every infinite system of difference equations, there exists a finite subsystem having precisely the same set of solutions.
However, different people have a different idea of what actually is a difference equation and it is quite crucial to specify where one is looking for the solutions. I am afraid the above reference is actually way too general to be really useful for you and your problem, but it definitely is a resource that deals systematically with infinite systems of difference equations. 
