Gessel, Ira M. Applications of the classical umbral calculus. (English) Zbl 1092.05005 Algebra Univers. 49, No. 4, 397-434 (2003). 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. Reviewer: Ivan Chajda (Olomouc) Cited in 1 ReviewCited in 48 Documents MSC: 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 Keywords:exponential generating function; bilinear generating function; Hermite polynomial; Charlier polynomial; Bernoulli number; Bell number; Kummer congruence PDF BibTeX XML Cite \textit{I. M. Gessel}, Algebra Univers. 49, No. 4, 397--434 (2003; Zbl 1092.05005) Full Text: DOI arXiv Digital Library of Mathematical Functions: §24.4(viii) Symbolic Operations ‣ §24.4 Basic Properties ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials Online Encyclopedia of Integer Sequences: Genocchi numbers of second kind (A005439) divided by 2^(n-1). Genocchi numbers (of first kind); unsigned coefficients give expansion of x*tan(x/2). Shifts one place left under 2nd-order binomial transform. Genocchi medians (or Genocchi numbers of second kind). E.g.f. exp( sinh(x) / exp(x) ) = exp( (1-exp(-2*x))/2 ). Triangle of coefficients of Laguerre polynomials n!*L_n(x) (rising powers of x).