A new group in the Riordan family of matrix groups: the Sprugnoli group

TL;DR

This paper introduces a new group of lower-triangular matrices generalizing the Riordan group, characterized by three power series, with applications in combinatorics and a production matrix framework.

Contribution

It defines the Sprugnoli group, extending Riordan groups, and explores its structure, properties, and higher-order generalizations based on power series.

Findings

Defined the Sprugnoli group as a new matrix group

Provided a production matrix characterization of the group

Outlined higher order generalizations based on n-tuples of power series

Abstract

We define a group of lower-triangular matrices whose columns are defined by power series. This group can be seen as a generalization of the (ordinary) Riordan group and the double Riordan group. Elements of this group are defined by three power series. Sequence bisections and vertically stretched Riordan arrays play an important role in the formulation of this group. We give a production matrix characterization of this new group. We also indicate how higher order groups can be defined, based on $n$-tuples of power series. We have chosen to name this group in memory of Renzo Sprugnoli, who was a pioneer in the application of the Riordan group to combinatorial problems as well as contributing to an understanding of the rich structure of Riordan arrays.