Periodicity and Circulant Matrices in the Riordan Array of a Polynomial

TL;DR

This paper explores Riordan arrays defined by specific power series and polynomials, revealing their periodic column structures and using circulant matrices to analyze their long-term behavior, with connections to combinatorial sequences.

Contribution

It introduces a class of Riordan arrays characterized by periodic columns and employs circulant matrices to study their asymptotic properties, linking to combinatorial sequences.

Findings

Columns are eventually periodic with period d+1

Circulant matrices describe long-term periodic behavior

Connections to interesting combinatorial sequences

Abstract

We consider Riordan arrays $\bigl(1/(1-t^{d+1}), ~ tp(t)\bigr)$. These are infinite lower triangular matrices determined by the formal power series $1/(1-t^{d+1})$ and a polynomial $p(t)$ of degree $d$. Columns of such matrix are eventually periodic sequences with a period of $d + 1$, and circulant matrices are used to describe the long term behavior of such periodicity when the column's index grows indefinitely. We also discuss some combinatorially interesting sequences that appear through the corresponding A - and Z - sequences of such Riordan arrays.