On Simple Characterisations of Sheffer psi- polynomials and Related Propositions of the Calculus of Sequences

A “Calculus of Sequences” had started by the 1936 publication of Ward suggesting the possible range for extensions of operator calculus of Rota-Mullin, considered by several authors and after Ward. Because of convenience we shall call the Wards calculus of sequences in its afterwards elaborated form – a ψ-calculus. The notation used by Ward, Viskov, Markowsky and Roman is accommodated in conformity with Rota’s way of exposition. In this manner ψ-calculus becomes in parts almost automatic extension of finite operator calculus. The ψ-extension relies upon the notion of ∂ψ-shift invariance of operators. At the same time this calculus is an example of the algebraization of the analysis – here restricted to the algebra formal series. The efficiency of the notation used is further exemplified among others by easy proving of some Sheffer ψ-polynomials characterisation theorems as well as Spectral Theorem. ψ-calculus results may be extended to Markowsky “Q-umbral calculus”, where Q stands for a generalised difference operator (not necessarily ∂ψ-shift invariant) i.e. the one lowering the degree of any polynomial by one.