Abstract 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 xn+1=1+2yn−k3+yn−k,yn+1=1+2zn−k3+zn−k,zn+1=1+2xn−k3+xn−k, $$\begin{array}{} \displaystyle x_{n+1} = \dfrac{1+2y_{n-k}}{3+y_{n-k}},\qquad y_{n+1} = \dfrac{1+2z_{n-k}}{3+z_{n-k}},\qquad z_{n+1} = \dfrac{1+2x_{n-k}}{3+x_{n-k}}, \end{array}$$ 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.

中文翻译：

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关于用卢卡斯数和斐波那契数解的三个高阶差分方程组

摘要 本文给出了高阶有理差分方程组xn+1=1+2yn−k3+yn−k,yn+1=1+2zn−的系统通解表示的一些理论解释。 k3+zn−k,zn+1=1+2xn−k3+xn−k, $$\begin{array}{} \displaystyle x_{n+1} = \dfrac{1+2y_{nk}}{3 +y_{nk}},\qquad y_{n+1} = \dfrac{1+2z_{nk}}{3+z_{nk}},\qquad z_{n+1} = \dfrac{1+2x_ {nk}}{3+x_{nk}}, \end{array}$$ 其中 n, k∈ ℕ0, 初始值 x−k, x−k+1, …, x0, y−k, y− k+1, ..., y0, z−k, z−k+1, ..., z1 和 z0 是不等于 -3 的任意实数。该系统可以以封闭形式求解，我们将看到这些解是使用著名的斐波那契数和卢卡斯数表示的。