MMN-3385

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

**A. Khelifa**, University of Mohamed Seddik Ben Yahia, LMAM Laboratory and Department of Mathematics, Jijel, Algeria,`amkhelifa@yahoo.com`

**Y. Halim**, Abdelhafid Boussouf University of Mila, Department of Mathematics and Computer Science, and LMAM Laboratory, University of Mohamed Seddik Ben Yahia, Jijel, Algeria,`halyacine@yahoo.fr`

**M. Berkal**, University of Alicante, Department of Applied Mathematics, Alicante, Spain,`mb299@gcluad.ua.es`

## Abstract

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.

Vol. 22 (2021), No. 1, pp. 331-350

DOI: 10.18514/MMN.2021.3385