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.