Periodicities in the Riordan arrays of polynomials over finite fields

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 $p_4(t)$.