I was wandering if it does make any sense study "difference operator" and "sum operator" using the framework of functional analysis.
As example (i just made it up)
$$\left( a_k\Delta^2 + b_k \Delta + c_k \right)u_k = \nu_k \;\; \forall k \geq 0$$ where $a_k, b_k, c_k$ and $\nu_k$ are fixed sequences.
Likewise an equation like
$$\sum_{n=0}^{+\infty}k_{m,n}u_n = x_m$$
and so on...
I have a lot of material the speaks about differential operator but much stuff about difference equations, except some book about "difference calculus" but it doesn't sounds like what i mean in such question...