Extended finite operator calculus - an example of algebraization of analysis
by
Ewa Borak
Institute of Computer Science; Białystok, Poland
Coauthors: A.K. Kwaśniewski

"A Calculus of Sequences" started in 1936 by Ward constitutes the general scheme for extensions of classical operator calculus of Rota-Mullin considered by many afterwards and after Ward. Because of the notation we shall call the Ward`s calculus of sequences in its afterwards elaborated form - a y-calculus.

The y-calculus in parts appears to be almost automatic, natural extension of classical operator calculus of Rota-Mullin or equivalently - of umbral calculus of Roman and Rota.

At the same time this calculus is an example of the algebraization of the analysis - here restricted to the algebra of polynomials. Many of the results of y-calculus may be extended to Markowsky Q-umbral calculus where Q stands for a generalized difference operator, i.e. the one lowering the degree of any polynomial by one.

This is a review article based on the recent author's relevant contributions. The article is supplemented by the short indicatory glossaries of terms and notation used by Ward, Viskov, Markowsky, Roman on one side and the Rota-oriented notation on the other side.

Date received: April 26, 2003