Periodicities in the Riordan arrays of polynomials over finite fields

We study periodicity properties of the 2-D $(p_1(t)\mathrm{/}{p}_2(t),t{p}_3(t))$ and 3-D $(p_1(t)\mathrm{/}{p}_2(t),t{p}_3(t),p_4(t))$ Riordan arrays over a finite field ${\mathbb{F}}_q$, where each $p_i(t)$ is a polynomial with $p_i(0)\mathrm{\ne }0$. 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 $p_4(t)$.