The Triple Riordan Group

TL;DR

This paper introduces the triple Riordan group, a new algebraic structure extending the double Riordan group, with potential for further generalizations in combinatorial and algebraic contexts.

Contribution

It defines the triple Riordan group, generalizing the double Riordan group, and establishes a framework for future higher-order Riordan group structures.

Findings

Defines the triple Riordan group with specific power series conditions

Generalizes the construction of the double Riordan group

Provides a foundation for further extensions of Riordan groups

Abstract

We define the triple Riordan group, whose elements consist of $4$-tuples of power series $(g, f_1, f_2, f_3)$ with $g\in \mathbf{R}[[x^3]]$, and $f_1, f_2, f_3 \in x\mathbf{R}[[x^3]]$, for an appropriate ring $\mathbf{R}$. The construction of this group generalizes that of the double Riordan group, and lays the pattern for further generalizations.