Sardonicus,
Here is something else which is pertinent to my question, the Blissard (Umbral) Calculus and Abrahamson's nonstandard analysis. Years ago I came across H. Dorrie's book, "100 Great Problems..." and attempted to solve all the questions. Bernoulli's Power Sum problem was one success and I expanded and generalized this problem with the methods I developed. I later found out that Blissard had developed what I came up with many years earlier. I was very impressed with what the librarian was able to dredge from the archives in university bowels regarding his published works. I prefer his insights over what is presently known as the Umbral Calculus. This methodology is a good tool when nothing else seems to work. The NSA (nonstandard analysis) perspective can be useful if you are trying to be clever rigorously.
