Roman describes umbral calculus as the study of the class of Sheffer sequences. Umbral calculus provides a formalism for the systematic derivation and classification of almost all classical combinatorial identities for polynomial sequences, along with associated generating functions, expansions, duplication formulas, recurrence relations, inversions, Rodrigues representation, etc., (e.g., the Euler-Maclaurin integration formulas, Boole's summation formula, the Chu-Vandermonde identity, Newton's divided difference interpolation formula, Gregory's formula, Lagrange inversion).

Appell sequence | binomial theorem | Chu-Vandermonde identity | combinatorics | Faà di Bruno's formula | finite difference | Sheffer sequence