In this paper we present the well-defined solution of the following system of higher-order rational difference equations:

$$\begin{aligned} x^{(j)}_{n+1}=\frac{x^{(j+1)\pmod {p}}_{n-k}}{a+bx^{(j+1)\pmod {p}}_{n-k}},\quad n, p, k\in \mathbb {N}_0 , j=\overline{1,p}, \end{aligned}$$

where the parameters a, b are nonzero real numbers and the initial values \(x^{(j)}_{-k}\), \(x^{(j)}_{-k+1}\),\(\ldots \), \(x^{(j)}_{-1}\) and \(x^{(j)}_0,\) \(j=\overline{1,p}\), do not equal \(-\frac{a}{b}\). Some theoretical explanations related to the representation for the general solution are also given.

中文翻译：

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关于高阶 p 个差分方程的可解系统

在本文中，我们提出了以下高阶有理差分方程组的明确解：

$$\begin{aligned} x^{(j)}_{n+1}=\frac{x^{(j+1)\pmod {p}}_{nk}}{a+bx^{( j+1)\pmod {p}}_{nk}},\quad n, p, k\in \mathbb {N}_0 , j=\overline{1,p}, \end{aligned}$$

其中参数a ,  b是非零实数和初始值\(x^{(j)}_{-k}\) , \(x^{(j)}_{-k+1}\) , \(\ldots \) , \(x^{(j)}_{-1}\)和\(x^{(j)}_0,\) \(j=\overline{1,p}\) , 不等于\(-\frac{a}{b}\)。还给出了与一般解的表示相关的一些理论解释。