Umbral calculus via integral transforms

Diierence Equations via the Classical Umbral Calculus

Formal Calculus and Umbral Calculus

We use the viewpoint of the formal calculus underlying vertex operator algebra theory to study certain aspects of the classical umbral calculus. We begin by calculating the exponential generating function of the higher derivatives of a composite function, following a very short proof which naturally arose as a motivating computation related to a certain crucial “associativity” property of an im...

Applied Umbral Calculus

Common ground to the three concepts are special polynomial sequences, called She er sequences. A polynomial sequence (sm (x))m2N0 is a sequence of polynomials sm (x) 2 K [x] such that deg sm = m, s0 6= 0. It is convenient to de ne sm = 0 for negative m. The coe cient ring K is assumed to be an integral domain. For this introduction to Finite Operator Calculus it su ces to choose K as R [!], the...

Universal Constructions in Umbral Calculus

Modern umbral calculus is steadily approaching maturity, as applications develop in several areas of mathematics. To maximize this utility it is important to work in the most general (as opposed to the most abstract) setting. The origins of the 19th century theory lie in analysis. In a beautiful recent article [13] Rota and Taylor have returned to these roots, and their bibliography details man...

Algebras and the Umbral Calculus ∗

We apply Baxter algebras to the study of the umbral calculus. We give a characterization of the umbral calculus in terms of Baxter algebra. This characterization leads to a natural generalization of the umbral calculus that include the classical umbral calculus in a family of λ-umbral calculi parameterized by λ in the base ring.