Factorizations of some periodic linear recurrence systems

Abstract

Let $P$ and $Q$ be relatively prime integers. The Lucas sequences are defined by $U_0 = 0$, $U_1 = 1$, $V_0 = 2$, $V_1 = P$, and $$ U_n = P U_{n-1} - Q U_{n-2} \quad \text {and} \quad V_n = P V_{n-1} - Q V_{n-2} , $$ where $n \ge 2$. We show that $$\b