Poweroids revisited - an old symbolic approach

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.