Periodicities in the Riordan arrays of polynomials over finite fields

Abstract

We study periodicity properties of the 2-D $\bigl(p_1(t)/p_2(t),\, tp_3(t)\bigr)$ and 3-D $\bigl(p_1(t)/p_2(t),\, tp_3(t),\, p_4(t)\bigr)$ Riordan arrays over a finite field ${\mathbb F}_q$, where each $p_i(t)$ is a polynomial with $p_i(0)\neq 0$. We show that the columns of the 2-D Riordan array are eventually periodic sequences, where a circulant matrix generated by the coefficients of $p_3(t)$ determines the behavior of this periodicity as the column index grows indefinitely. Furthermore, we prove that the preperiodic column partial sums of the 2-D array are periodic, and present a family of the Riordan arrays for which such sequences of partial sums are identically zero. We also show that the layers of the 3-D Riordan array contain periodic orbits related to each other via powers of a circulant matrix generated by the coefficients of $p_4(t)$.

Disclosure

“sums is identically zero. Acknowledgement This research was funded by the Center for Undergraduate Research and Creative Activity (CURCA) at Siena University under the Summer Scholars Program. The authors also acknowledge the use of AI tools for scholarly literature exploration, proof verification, and manuscript preparation. 2 Preliminaries 2.1 Riordan Arrays For the detailed introduction to the subject, the reader is referred to the books by Barry [2] and Shapiro”

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2026-08-23T18:56:06.399654Z

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