Generalized Riordan Groups and Operators on Polynomials

TL;DR

This paper introduces a unified approach to generalized Riordan arrays using group operations on lower triangular matrices, revealing their isomorphisms and properties of weighted Sheffer sequences.

Contribution

It provides a new group-theoretic framework for generalized Riordan arrays, simplifying proofs and establishing isomorphisms among different weighted groups.

Findings

All groups from different weights are isomorphic via conjugation.

Direct proofs of properties of weighted Sheffer sequences are achieved.

A new result on the intersection of generalized Riordan groups with different weights.

Abstract

We present an approach to generalized Riordan arrays which is based on operations in one large group of lower triangular matrices. This allows for direct proofs of many properties of weighted Sheffer sequences, and shows that all the groups arising from different weights are isomorphic since they are conjugate. We also prove a result about the intersection of two generalized Riordan with different weights.