Poweroids revisited - an old symbolic approach

TL;DR

This paper revisits and extends Jeffery's 1861 finite difference calculus approach to poweroids, providing a neat proof framework for binomial type polynomials and their properties, offering an alternative to traditional umbral methods.

Contribution

It extends Jeffery's finite difference calculus to general delta operators, offering a new proof approach for binomial type polynomials and their properties, and demonstrating its effectiveness as an alternative to umbral calculus.

Findings

Extended finite difference calculus to general delta operators.

Provided compact proofs for binomial property and connection constants.

Showed the approach as a legitimate alternative to umbral calculus.

Abstract

Jeffery's 1861 computations using finite difference calculus are resurrected and extended from forward differences to general delta operators and used to neatly prove theorems in the Rota--Mullins theory of polynomials of binomial type (Steffensen's poweroids) allowing, for example, compact treatments of umbral composition, the binomial property and the connection constants. It is shown that it forms a legitimate alternative to the usual umbral device and also anticipates a number of results obtained more recently.