Umbral Vade Mecum

TL;DR

The paper discusses the umbral calculus as an elegant framework for systematically solving difference equations related to physics and engineering, by mapping them to well-known continuous functions.

Contribution

It introduces and illustrates the umbral calculus framework, highlighting its application to key equations and solitons in physics and engineering.

Findings

Provides systematic solutions to difference equations using umbral calculus

Demonstrates umbral mappings of special functions like Airy, Kummer, Whittaker

Applies umbral methods to solitons in integrable systems

Abstract

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 appearing in most areas of physics and engineering --- as maps of well-known continuous functions. This correspondence deftly sidesteps the use of more traditional methods to solve these difference equations. The umbral framework is discussed and illustrated here, with special attention given to umbral counterparts of the Airy, Kummer, and Whittaker equations, and to umbral maps of solitons for the Sine-Gordon, Korteweg--de Vries, and Toda systems.