2013/04/30 by Thomas L Curtright, Cosmas K Zachos
Physics and Astronomy · Mathematics · #math-ph #hep-th #math.MP
paper · pdf · doi:10.3389/fphy.2013.00015
published as Front. Physics 1 (2013) 15 · arXiv admin note: text overlap with arXiv:0710.2306
arxiv created 2013/06/27 · arxiv updated 2013/09/12
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.