Applications of the Classical Umbral Calculus

ai. Blissard’s notation has been known variously as Lucas’s method, the symbolic method (or symbolic notation), and the umbral calculus. We shall use Rota and Taylor’s term “classical umbral calculus” [36] to distinguish it from the more elaborate mathematical edifice that the term “umbral calculus” has come to encompass [31, 32, 34]. The goal of this article is to show, by numerous examples, how the classical umbral calculus can be used to prove interesting formulas not as easily proved by other methods. Our applications are in three general areas: bilinear generating functions, identities for Bernoulli numbers and their relatives, and congruences for sequences such as Euler and Bell numbers. The classical umbral calculus is intimately connected with exponential generating functions; thus an = an is equivalent to e = ∞