On the Central Description of the Group of Riordan Arrays

TL;DR

This paper introduces a new 'central' description of Riordan arrays using two power series, relating it to the traditional form and exploring its properties, including product rules and inverses, primarily for ordinary generating functions.

Contribution

It presents an alternative 'central' framework for Riordan arrays, connecting it to central coefficients and extending the description to exponential generating functions.

Findings

New 'central' description links to central coefficients.

Product and inverse rules are established in the new framework.

Extension to exponential generating functions is briefly discussed.

Abstract

We provide an alternative description of the group of Riordan arrays, by using two power series of the form $\sum_{n=0}^{\infty} g_n x^n$, where $g_0 \ne 0$ to build a typical element of the constructed group. We relate these elements to Riordan arrays in the usual description, showing that each newly constructed element is the vertical half of a "usual" element. The product rules and the construction of the inverse are given in this new description, which we call a "central" description, because of links to the central coefficients of Riordan arrays. This is done for the case of ordinary generating functions. Finally, we briefly look at the exponential case.