The classical umbral calculus: Sheffer sequences

TL;DR

This paper introduces a new umbral calculus framework for Sheffer sequences, simplifying computations and clarifying concepts, with applications to connection constants, Lagrange inversion, and recurrence relations.

Contribution

It develops an innovative umbral approach to Sheffer sequences, enhancing computational efficiency and conceptual understanding over previous methods.

Findings

Simplified computation of Sheffer sequences.

New insights into connection constants and Lagrange inversion.

Effective solutions to recurrence relations.

Abstract

Following the approach of Rota and Taylor \cite{SIAM}, we present an innovative theory of Sheffer sequences in which the main properties are encoded by using umbrae. This syntax allows us noteworthy computational simplifications and conceptual clarifications in many results involving Sheffer sequences. To give an indication of the effectiveness of the theory, we describe applications to the well-known connection constants problem, to Lagrange inversion formula and to solving some recurrence relations.