On the solutions of a system of (2p+1) difference equations of higher order

[摘要] In this paper we represents the well-defined solutions of the system of the higher-order rational difference equations \begin{equation*} x^{(j)}_{n+1}=\dfrac{1+2x^{(j+1)mod(2p+1)}_{n-k}}{3+x^{(j+1)mod(2p+1)}_{n-k}},\quad n, k, p \in \mathbb{N}_{0} \end{equation*} in terms of Fibonacci and Lucase sequences, where the initial values $x^{(j)}_{-k}, x^{(j)}_{-k+1}$,\\$\ldots, x^{(-1)}_0$ and $x^{(i)}_0$, $j=1,2,\ldots,2p+1$, do not equal -3. Some theoretical explanations related to the representation for the general solution are also given.

[学科分类] 数学（综合）

[关键词] Fibonacci sequence;Lucas sequence;system of difference equations;representation of solutions