Operational Umbral Calculus
Combinatorics
2025-12-05 v4 Number Theory
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.
Cite
@article{arxiv.2407.16348,
title = {Operational Umbral Calculus},
author = {Kei Beauduin},
journal= {arXiv preprint arXiv:2407.16348},
year = {2025}
}
Comments
27 pages, to be published