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
Umbra1 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 g