A formal calculus on the Riordan near algebra

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.