Selected facts on products of two involutions in the Riordan group

TL;DR

This paper investigates the properties of strongly reversible elements in the Riordan group, showing that not all reversible elements are strongly reversible and analyzing products of reversible elements.

Contribution

It provides new insights into the structure of the Riordan group by characterizing strongly reversible elements and exploring their differences from reversible elements.

Findings

Not all reversible elements are strongly reversible in the Riordan group.

Characterization of strongly reversible elements in the Riordan group.

Analysis of products of reversible elements in the Riordan group.

Abstract

An element of a group is called \emph{reversible} if it is conjugate to its inverse, and \emph{strongly reversible} if it can be expressed as a product of two involutions. We study strongly reversible elements in the Riordan group and in several of its important subgroups. We show that not every reversible element in the Riordan group is strongly reversible, and we investigate products of reversible elements in the Riordan group.