TL;DR
This paper explores operational methods in umbral calculus, emphasizing linear operators and generating functions to simplify proofs and extend the theory, including pseudoinverses and examples.
Contribution
It provides a systematic, operator-based reconstruction of umbral calculus, offering new proofs and insights into advanced results like the Lagrange-Bürmann inversion theorem.
Findings
Short proofs of advanced umbral results
Systematic use of linear operators and generating functions
Illustrative examples of pseudoinverses for delta operators
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.
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