Umbral interpolation

An Algebraic Exposition of Umbral Calculus with Application to General Linear Interpolation Problem – a Survey

A systematic exposition of Sheffer polynomial sequences via determinantal form is given. A general linear interpolation problem related to Sheffer sequences is considered. It generalizes many known cases of linear interpolation. Numerical examples and conclusions close the paper. 1. The modern umbral calculus In the 1970s Rota and his collaborators [17,19,20] began to construct a completely rig...

The Umbral Refinement Equation

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...

Umbral Vade Mecum

*Correspondence: Cosmas K. Zachos, High Energy Physics Division 362, Argonne National Laboratory, Argonne, IL 60439-4815, USA e-mail: [email protected] In recent years the umbral calculus has emerged from the shadows to provide an elegant correspondence framework that automatically gives systematic solutions of ubiquitous difference equations—discretized versions of the differential cornerstones a...

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...