As mentioned in the introduction, the history of the umbral calculus goes back to the 17th century. The rise of the umbral calculus, however, takes place in the second half of the 19th century with the work of such mathematicians as Sylvester (who invented the name), Cayley and Blissard (see e.g., [6]). Although widely used, the umbral calculus was nothing more than a set of ''magic'' rules of lowering and raising indices (see e.g., [35]). These rules worked well in practice, but lacked a proper foundation. Attempts to put the umbral calculus on an axiomatic basis (see [7]) were unsuccesful. Although the mathematical world remained sceptical of the umbral calculus, it was used extensively (e.g., in Riordan's highly respected book on combinatorics).

A second line in the history of the umbral calculus in the form that we know today, is the theory of Sheffer polynomials. The history of Sheffer polynomials goes back to 1880 when Appell studied sequences of polynomials satisfying (see [2]). These sequences are nowadays called Appell polynomials. Although this class was widely studied (see the bibliography in [20] which is included in the bibliography of this survey), it was not until 1939 that Sheffer noticed the similarities with which the introduction of this survey starts. These similarities led him to extend the class of Appell polynomials which he called polynomials of type zero (see [109]), but which nowadays are called Sheffer polynomials. This class already appeared in [63]. Although Sheffer uses operators to study his polynomials, his theory is mainly based on formal power series. In 1941 the Danish actuary Steffensen also published a theory of Sheffer polynomials based on formal power series [117]. Steffensen uses the name poweroids for Sheffer polynomials (see also [111,110,117,118,119,116]). However, these theories were not adequate as they do not provide sufficient computational tools (expansion formulas etc.).

A third line in the history of the umbral calculus is the theory of abstract linear operators. This line goes back to the work of Pincherle starting in the 1890's (i.e., in the beginnings of functional analysis). His early work is laid down in the monumental monograph [81]. Pincherle went surprisingly far considering the state of functional analysis in those days, but his work lacked explicit examples. The same applies to papers by others in this field (see e.g., [19,20,132]).

A prelude to the merging of these three lines can be seen in [106], in which operators methods are used to free umbral calculus from its mystery. In [67] the ideas from [106] are extended to give a beautiful theory combining enriched functions, umbral methods and operator methods. However, only the subclass of polynomials of binomial type are treated in [67]. The extension to Sheffer polynomials is accomplished in [107]. The latter paper is much more geared towards special functions, while the former paper is a combinatorial paper.

The papers [67] and [107] were soon followed by papers that reacted directly on the new umbral calculus. E.g., Fillmore and Williamson showed that with equal ease the Rota umbral calculus could be situated in abstract vector spaces instead of the vector space of polynomials [25], Zeilberger noticed connections with Fourier analysis [138] and Garsia translated the operator methods of Rota back into formal power series [27].

We conclude this section with mentioning the remarkable paper
[108], in
which Rota and Taylor with an algebraic * tour de force * manage to make
sense of the
19th century umbral calculus (thereby fulfulling Bell's dream [6]; cf.
[84]).

sandro@win.tue.nl / loeb@labri.u-bordeaux.fr