Finding an Isomorphism between the Riordan Group and a Subgroup of the Double Riordan Group

TL;DR

This paper establishes an isomorphism between the Riordan group and a specific subgroup of the Double Riordan group, addressing an open question about their algebraic relationship.

Contribution

The paper proves the existence of an isomorphism between the Riordan group and a subgroup of the Double Riordan group, clarifying their algebraic connection.

Findings

Established an explicit isomorphism between the groups.

Extended understanding of the algebraic structure of Riordan-related groups.

Addressed an open question in the literature.

Abstract

The Riordan group is a set of infinite lower-triangular matrices defined by two generating functions, $g$ and $f$. The elements of the group are called Riordan arrays, denoted by $(g,f)$, and the $k$th column of a Riordan array is given by the function $gf^k$. The Double Riordan group is defined similarly using three generating functions $g$, $f_1$, and $f_2$, where $g$ is an even function and $f_1$ and $f_2$ are odd functions. This group generalizes the Checkerboard subgroup of the Riordan group, where $g$ is even and $f$ is odd. An open question posed by Davenport, Shapiro, and Woodson was if there exists an isomorphism between the Riordan group and a subgroup of the Double Riordan group. This question is answered in this article.