On the Group of Almost-Riordan Arrays

TL;DR

This paper explores an extended group of Riordan arrays called almost-Riordan arrays, identifying subgroup structures and demonstrating their applications in Hankel transform properties.

Contribution

It introduces a super group of Riordan arrays, characterizes subgroups and normal subgroups, and connects almost-Riordan arrays to Hankel matrix transformations.

Findings

Certain subsets form subgroups

A normal subgroup is identified with cosets as Riordan arrays

Examples show applications to Hankel and Toeplitz matrices

Abstract

We study a super group of the group of Riordan arrays, where the elements of the group are given by a triple of power series. We show that certain subsets are subgroups, and we identify a normal subgroup whose cosets correspond to Riordan arrays. We give an example of an almost-Riordan array that has been studied in the context of Hankel and Hankel plus Toepliz matrices, and we show that suitably chosen almost-Riordan arrays can lead to transformations that have interesting Hankel transform properties.