Umbral Vade Mecum
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 wellknown 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 SineGordon, Kortewegde Vries, and Toda systems.
 Publication:

Frontiers in Physics
 Pub Date:
 October 2013
 DOI:
 10.3389/fphy.2013.00015
 arXiv:
 arXiv:1304.0429
 Bibcode:
 2013FrP.....1...15C
 Keywords:

 umbral correspondence;
 discretization;
 difference equations;
 umbral transform;
 hypergeometric functions;
 Mathematical Physics;
 High Energy Physics  Theory
 EPrint:
 arXiv admin note: text overlap with arXiv:0710.2306