Applications of the classical umbral calculus. (English) Zbl 1092.05005

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.


05A40 Umbral calculus
05A19 Combinatorial identities, bijective combinatorics
05A10 Factorials, binomial coefficients, combinatorial functions
11B65 Binomial coefficients; factorials; \(q\)-identities
11B68 Bernoulli and Euler numbers and polynomials
11B73 Bell and Stirling numbers
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