This monograph is intended to be an elementary introduction to the modern umbral calculus. Since we have in mind the largest possible audience, the only prerequisite is an acquaintance with the basic notions of algebra, and perhaps a dose of applied mathematics (such as differential equations) to help put the theory in some mathematical perspective.
The title of this work really should have been The Modern Classical Umbral Calculus, Within the past few years many, indeed infinitely many, distinct umbral calculi have begun to be studied. Actually, the existence of distinct umbral calculi was recognized in a vague way as early as the 1930s but seems to have remained largely ignored until the past decade.
In any case, we shall occupy the vast majority of our time in studying one particular umbral calculus—the one that dates back to the 1850s and that has received the attention (both good and bad) of mathematicians up to the present time. For this, we use the term classical umbral calculus. Only in the last chapter do we glimpse the newer, much less well established, nonclassi-cal umbral calculi.
The classical umbral calculus, as it was from 1850 to about 1970, consisted primarily of a symbolic technique for the manipulation of sequences, whose mathematical rigor left much to be desired. To drive this point home one need only look at Eric Temple Bell's unsuccessful attempt (in 1940) to convince the mathematical community to accept the umbral calculus as a legitimate mathematical tool. (Even now some are still trying to achieve Eric Temple Bell's original goal.) This old-style umbral calculus was, however, useful in deriving certain mathematical results; but unfortunately these results had to be verified by a different, more rigorous method.
In the 1970s Gian-Carlo Rota, a mathematician with a superlative talent for handling just this sort of situation, began to construct a completely rigorous foundation for the theory—one that was based on the relatively
