Explicit Formulas and Periodic Behavior of Solutions for a Class of Rational Recursive Sequences
Abdualrazaq Sanbo *
General Studies Department, Jeddah College of Telecom and Electronics, TVTC, B.P. 2816, Jeddah 21461, Saudi Arabia.
*Author to whom correspondence should be addressed.
Abstract
This paper investigates the qualitative behavior and explicit solutions of a system of rational difference equations of the form
with arbitrary initial conditions u−2, u−1, u0, v−2, v−1, v0 ∈ \(\mathbb{R}\). The constants ai ∈ {−1, 1} for i = 1, 2, 3, 4. Four special cases of the system are examined based on the sign combinations in the denominators. For each case, we prove that all solutions are periodic with period twelve and derive explicit closed-form formulas for the solutions in terms of the initial values. The proofs are established using mathematical induction. Numerical examples are provided to verify and illustrate the theoretical results, demonstrating the periodic nature of the solutions. This work contributes to the understanding of higher-order rational difference equations and their periodic behavior.
Keywords: Difference equations, recursive sequences, periodicity, rational difference equations, system of difference equations