I'm working on an instructional video, and while I don't mind tossing out some hand-waving, and even some minor inaccuracies, I don't want to get things ENTIRELY wrong.

First question:

Quite a while ago, quite by accident, I discovered that one could solve linear shift-invariant difference equations via "the hard way" in much the same way that one solved linear differential equations.

In other words, given

y_k + a_1 *** y_{k-1} + a_0 *** y_{k-2} = f(k)

then one could solve the thing by first finding the homogeneous solution (arrived at by setting f(k) = 0), and then finding the homogeneous solution (assuming that f(k) was, by chance or design, equal to something "easy", i.e. some sum of A * k^n * d^m).

All of this was done by direct analogy to how it's done with differential equations.

Does anyone have any references to this? It can't be entirely my own invention. Even a name that coughs up results in a Google search would help.

Second question:

When you come up with the solution to a non-homogeneous solution (f(k) !=

0, above), the the range of functions that f(k) can be to keep the problem "easy" is constrained to functions that have "easy" z transforms (i.e., some limited sum of A * k^n * d^m). Is there a theorem associated with that? Again, references or a name I can Google (really, a name that I can set my viewers to Googling) would be cool.If I don't get any answers in a day or a few I'm just going to hand-wave, and leave people puzzled. But I'd much rather be able to point them in a fruitful direction.