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Zbl 1092.05005
Gessel, Ira M.
Applications of the classical umbral calculus. (English)
[J] Algebra Univers. 49, No. 4, 397-434 (2003). ISSN 0002-5240; ISSN 1420-8911/e

In the nineteenth century, Blissard developed a notation for manipulating sums involving binomial coefficients by expanding polynomials and then replacing exponents with subscripts. Blissard's notation has been known as the umbral calculus. The goal of the paper is to show, by numerous examples, how the umbral calculus can be used to prove interesting formulas not as easily proved by other methods. The applications are in three general areas: bilinear generating functions, identities for Bernoulli numbers and congruences for sequences such as Euler and Bell numbers.
[Ivan Chajda (Olomouc)]

MSC 2000:: *05A40 Umbral calculus
05A19 Combinatorial identities
05A10 Combinatorial functions
11B65 Binomial coefficients, etc.
11B68 Bernoulli numbers, etc.
11B73 Bell and Stirling numbers

Keywords: exponential generating function; bilinear generating function; Hermite polynomial; Charlier polynomial; Bernoulli number; Bell number; Kummer congruence

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