Operational Umbral Calculus

Abstract

In this paper, we investigate the power of nearly purely operational techniques in the study of umbral calculus. We present a concise reconstruction of the theory based on a systematic use of linear operators, with particular attention to umbral operators. We also give an in-depth study of the generating functions associated to umbral calculus, and show how these lead to short proofs of several advanced results, including the Lagrange-B\"urmann inversion theorem. Finally, we discuss pseudoinverses for delta operators and illustrate our methods with a variety of examples.