On a system of three difference equations of higher order solved in terms of Lucas and Fibonacci numbers

Abstract

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 equationsxn+1=1+2yn−k3+yn−k,yn+1=1+2zn−k3+zn−k,zn+1=1+2xn−k3+xn−k,$$\begin{array}{} \displaystyle x_{n+1} = \dfrac{1+2y_{n-k}}{3+y_{n-k}},\qquad y_{n+1} = \dfrac{1+2z_{n-k}}{3+z_{n-k}},\qquad z_{n+1} = \dfrac{1+2x_{n-k}}{3+x_{n-k}}, \end{array}$$wheren,k∈ ℕ0, the initial valuesx−k,x−k+1, …,x0,y−k,y−k+1, …,y0,z−k,z−k+1, …,z1andz0are 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.