I don't know if this reference is of sufficient generality:
Finite Calculus: A Tutorial for Solving Nasty Sums
by David Gleich
https://www.cs.purdue.edu/homes/dgleich/publications/Gleich%202005%20-%20finite%20calculus.pdf
(The original link at http://www.stanford.edu/ no longer exists.)
It is only a paper, but it describes the methods of the so-called "umbral calculus": a really useful technique to know.