Bell matrices and Bell polynomials

[JmsNxn]

I need to understand umbral calculus better as I believe there is a neat representation of iterated functions there.

Skip umbral calculus and go straight to Ramanujan. I've studied both, and umbral calculus is just a naive, early, little rigor, version of Ramanujan's work. Ramanujan, singlehandedly, showed how right umbral calculus was, but additionally did it with new tools. Upon which, more rigorous mathematicians soldified all of Ramanujan's observations. A lot of umbral calculus is "coincidences we can't explain"--Ramanujan gave the explanation.