AI 中文总结
研究有限域上二维和三维里奥丹阵列的周期性，证明二维阵列列及前周期列部分和的周期性，给出部分和为零的阵列类，还表明三维阵列层含相关周期轨道，核心方法是利用多项式系数生成循环矩阵研究。
AI 中文摘要
我们研究了有限域${\mathbb F}_q$上二维$\bigl(p_1(t)/p_2(t),\, tp_3(t)\bigr)$和三维$\bigl(p_1(t)/p_2(t),\, tp_3(t),\, p_4(t)\bigr)$里奥丹阵列的周期性性质，其中每个$p_i(t)$是满足$p_i(0)\neq 0$的多项式。我们表明二维里奥丹阵列的列最终是周期序列，由$p_3(t)$系数生成的循环矩阵决定了随着列索引无限增长时这种周期性的行为。此外，我们证明了二维阵列的前周期列部分和是周期的，并给出了一类里奥丹阵列，其部分和序列恒为零。我们还表明三维里奥丹阵列的层包含通过由$p_4(t)$系数生成的循环矩阵的幂相互关联的周期轨道。
英文摘要
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 $p_4(t)$.
Comments4 figures
