The Vertical Recursive Relation of Riordan Arrays and Their Matrix Representation

TL;DR

This paper introduces a vertical recursive relation for Riordan arrays, providing new matrix representations, group structures, and extensions that facilitate the study of nonlinear recursive relations and identities in triangular matrices.

Contribution

It develops a vertical recursive relation approach for Riordan arrays, introduces the quasi-Riordan group, and extends recursive relations to study nonlinear recursions and identities.

Findings

Defined the vertical recursive relation for Riordan arrays.

Established the quasi-Riordan group of matrices.

Demonstrated extensions to nonlinear recursive relations.

Abstract

A vertical recursive relation approach to Riordan arrays is induced, while the horizontal recursive relation is represented by $A$- and $Z$-sequences. This vertical recursive approach gives a way to represent the entries of a Riordan array $(g,f)$ in terms of a recursive linear combinations of the coefficients of $g$. A matrix representation of the vertical recursive relation is also given. The set of all those matrices forms a group, called the quasi-Riordan group. The extensions of the horizontal recursive relation and the vertical recursive relation in terms of $c$- and $C$- Riordan arrays are defined with illustrations by using the rook triangle and the Laguerre triangle. Those extensions represent a way to study nonlinear recursive relations of the entries of some triangular matrices from linear recursive relations of the entries of Riordan arrays. In addition, the matrix representation of the vertical recursive relation of Riordan arrays provides transforms between lower order and high order finite Riordan arrays, where the $m$th order Riordan array is defined by $(g,f)_m=(d_{n,k})_{m\geq n,k\geq 0}$. Furthermore, the vertical relation approach to Riordan arrays provides a unified approach to construct identities.