Extended finite operator calculus as an example of algebraization of analysis

TL;DR

This paper reviews the development of extended finite operator calculus, illustrating how algebraic methods can generalize classical analysis within the framework of formal series, highlighting recent contributions and notation conventions.

Contribution

It provides a comprehensive review of the algebraization of analysis through extended finite operator calculus, emphasizing recent advances and clarifying notation used by key researchers.

Findings

Demonstrates the algebraic structure underlying the calculus of sequences

Highlights the role of formal series in extending classical analysis

Summarizes recent contributions and notation in the domain

Abstract

A wardian calculus of sequences started almost seventy years ago constitutes the general scheme for extensions of the classical umbral operator calculus considered by many afterwards . At the same time this calculus is an example of the algebraization of the analysis here restricted to the algebra of formal series. This is a review article based on the recent first author contributions. As the survey article it is supplemented by the short indicatory glossaries of notation and terms used by prominent contributors to the domain.