Solutions Formulas for Three-dimensional Difference Equations System with Constant Coefficients

Abstract

In this paper, we study the following three-dimensional system of difference equations \begin{equation*} x_{n}=\frac{ax_{n-3}z_{n-2}+b}{cy_{n-1}z_{n-2}x_{n-3}}, \ y_{n}=\frac{ay_{n-3}x_{n-2}+b}{cz_{n-1}x_{n-2}y_{n-3}}, \ z_{n}=\frac{az_{n-3}y_{n-2}+b}{cx_{n-1}y_{n-2}z_{n-3}}, \ n\in \mathbb{N}_{0}, \end{equation*} where the parameters $a, b, c$ and the initial values $x_{-j},y_{-j},z_{-j}$, $j \in \{1,2,3\}$, are real numbers. We solve aforementioned system in explicit form. Then, we investigate the solutions in 3 different cases depending on whether the parameters are zero or non-zero. In addition, numerical examples are given to demonstrate the theoretical results. Finally, an application is given for solutions are related to Fibonacci numbers when $a=b=c=1$.

Keywords

• Explicit form of solution
• system of difference equation
• Fibonacci number
• periodicity

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