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.
A proof of the curious binomial coefficient identity which is connected with the fibonacci numbers
