Formulas of the solutions of a non-autonomous difference equation and two systems of difference equations

Abstract

In this work, we explicitly solve the following:

*   A higher-order non-autonomous difference equation:
\begin{equation*}
x_{n+1} = \alpha_{n} x_{n-k} + \frac{\beta_{n}}{x_{n} x_{n-1} \cdots x_{n-k+1}},
\end{equation*}
where $n \in \mathbb{N}_{0}$, $k \in \mathbb{N}$, the sequences $\left(\alpha_{n}\right)_{n \in \mathbb{N}_{0}}$ and $\left(\beta_{n}\right)_{n \in \mathbb{N}_{0}}$ are real, and the initial values $x_{-k}, x_{-k+1}, \ldots, x_{0}$ are nonzero real numbers.

*  A three-dimensional system of second-order difference equations:
\begin{equation*}
x_{n+1} = \frac{a_{1} y_{n-1} z_{n-1}}{a x_{n-1} + b y_{n-1} + c z_{n-1}}, \quad
y_{n+1} = \frac{a_{2} x_{n-1} z_{n-1}}{a x_{n-1} + b y_{n-1} + c z_{n-1}},
\end{equation*}
\begin{equation*}
z_{n+1} = \frac{a_{3} x_{n-1} y_{n-1}}{a x_{n-1} + b y_{n-1} + c z_{n-1}},
\end{equation*}
where $n \in \mathbb{N}_{0}$, the parameters $a, b, c, a_{1}, a_{2}, a_{3}$ are real numbers, and the initial values $x_{-1}, x_0, y_{-1}, y_0, z_{-1}, z_0$ are nonzero real numbers.

*  A three-dimensional system of first-order difference equations:
\begin{equation*}
x_{n+1} = \frac{a_{1} y_{n} z_{n}}{a x_{n} + b y_{n} + c z_{n}}, \quad
y_{n+1} = \frac{a_{2} x_{n} z_{n}}{a x_{n} + b y_{n} + c z_{n}}, \quad
z_{n+1} = \frac{a_{3} x_{n} y_{n}}{a x_{n} + b y_{n} + c z_{n}},
\end{equation*}
where $n \in \mathbb{N}_{0}$, the parameters $a, b, c, a_{1}, a_{2}, a_{3}$ are real numbers, and the initial values $x_0, y_0, z_0$ are nonzero real numbers.}