AbstractIn this paper we give some theoretical explanations related to the representation for the general solution of the system of the higher-order rational difference equations$$\begin{array}{} \displaystyle x_{n+1} = \dfrac{1+2y_{n-k}}{3+y_{n-k}},\qquad y_{n+1} = \dfrac{1+2z_{n-k}}{3+z_{n-k}},\qquad z_{n+1} = \dfrac{1+2x_{n-k}}{3+x_{n-k}}, \end{array}$$where n, k∈ ℕ0, the initial values x−k, x−k+1, …, x0, y−k, y−k+1, …, y0, z−k, z−k+1, …, z1 and z0 are arbitrary real numbers do not equal −3. This system can be solved in a closed-form and we will see that the solutions are expressed using the famous Fibonacci and Lucas numbers.

In this paper we give some theoretical explanations related to the representation for the general solution of the system of the higher-order rational difference equations x n+1 = 5y n−k − 5 y n−k , y n+1 = 5x n−k − 5 x n−k , n, k ∈ N 0 , where N 0 = N ∪ {0}, and the initial conditions x −k , x −k+1 ,. . ., x 0 , y −k , y −k+1 ,. . ., y 0 are non zero real numbers such that their solutions are associated to Lucas numbers. We also study the stability character and asymptotic behavior of this system.