The umbral calculus. Even after Gian-Carlo Rota revived it, its significance was misunderstood by many. No less a mathematician than Ira Gessel admitted this publicly, in his paper, Applications of the classical umbral calculus (arXiv version).

When I first encountered umbral notation it seemed to me that this was all there was to it; it was simply a notation for dealing with exponential generating functions, or to put it bluntly, it was a method for avoiding the use of exponential generating functions when they really ought to be used. The point of this paper is that my first impression was wrong: none of the results proved here (with the exception of Theorem 7.1, and perhaps a few other results in section 7) can be easily proved by straightforward manipulation of exponential generating functions. The sequences that we consider here are defined by exponential generating functions, and their most fundamental properties can be proved in a straightforward way using these exponential generating functions. What is surprising is that these sequences satisfy additional relations whose proofs require other methods. The classical umbral calculus is a powerful but specialized tool that can be used to prove these more esoteric formulas.