A formal calculus on the Riordan near algebra

TL;DR

This paper introduces a formal calculus on the Riordan near algebra, extending the algebraic structure of the Riordan group and enabling the definition of generalized powers without convergence constraints.

Contribution

It develops a formal calculus on the Riordan near algebra, allowing algebraic manipulations similar to holomorphic calculus without radius of convergence limitations.

Findings

Established a formal calculus on the Riordan near algebra

Embedded the Riordan group as units within this algebra

Defined generalized powers in the Riordan group using the calculus

Abstract

The Riordan group is the semi-direct product of a multiplicative group of invertible series and a group, under substitution, of non units. The Riordan near algebra, as introduced in this paper, is the Cartesian product of the algebra of formal power series and its principal ideal of non units, equipped with a product that extends the multiplication of the Riordan group. The later is naturally embedded as a subgroup of units into the former. In this paper, we prove the existence of a formal calculus on the Riordan algebra. This formal calculus plays a role similar to those of holomorphic calculi in the Banach or Fr\'echet algebras setting, but without the constraint of a radius of convergence. Using this calculus, we define \emph{en passant} a notion of generalized powers in the Riordan group.