Periodic Column Partial Sums in the Riordan Array of a Polynomial

TL;DR

This paper studies the periodicity of column partial sums in Riordan arrays generated by polynomials, classifying cases where these sums exhibit eventual periodicity and exploring their structural properties.

Contribution

It provides a classification of polynomials for which the Riordan array's column partial sums are eventually periodic, extending understanding of their combinatorial structure.

Findings

Column partial sums are eventually periodic for certain polynomial-generated Riordan arrays.

Classification of linear and quadratic polynomials with periodic partial sums.

Identification of polynomial families of higher degrees with periodic partial sums.

Abstract

When $p(t)$ is a polynomial of degree $d$, $k$-th column of the Riordan array $\bigl(1/(1 - t^{d+1}), tp(t)\bigr)$ is an eventually periodic sequence with the repeating part beginning at the $1 + (k-1)(d+1)$-st term. The pre-periodic terms add up to the $(k-1)(d+1)$-st partial sum of the corresponding formal power series, and thus the Riordan array of $p(t)$ generates a sequence of column partial sums. We classify linear and quadratic polynomials, and present a particular family of polynomials of higher degrees, for which such sequences of column partial sums are eventually periodic.