On a system of three difference equations of higher order solved in terms of Lucas and Fibonacci numbers

Abstract

<jats:title>Abstract</jats:title><jats:p>In this paper we give some theoretical explanations related to the representation for the general solution of the system of the higher-order rational difference equations</jats:p><jats:p><jats:disp-formula id="j_ms-2017-0378_eq_001_w2aab3b7e1644b1b6b1aab1c15b2aAa"><jats:alternatives><jats:tex-math>$$\begin{array}{} \displaystyle x_{n+1} = \dfrac{1+2y_{n-k}}{3+y_{n-k}},\qquad y_{n+1} = \dfrac{1+2z_{n-k}}{3+z_{n-k}},\qquad z_{n+1} = \dfrac{1+2x_{n-k}}{3+x_{n-k}}, \end{array}$$</jats:tex-math></jats:alternatives></jats:disp-formula></jats:p><jats:p>where <jats:italic>n</jats:italic>, <jats:italic>k</jats:italic>∈ ℕ<jats:sub>0</jats:sub>, the initial values <jats:italic>x</jats:italic><jats:sub>−<jats:italic>k</jats:italic></jats:sub>, <jats:italic>x</jats:italic><jats:sub>−<jats:italic>k</jats:italic>+1</jats:sub>, …, <jats:italic>x</jats:italic><jats:sub>0</jats:sub>, <jats:italic>y</jats:italic><jats:sub>−<jats:italic>k</jats:italic></jats:sub>, <jats:italic>y</jats:italic><jats:sub>−<jats:italic>k</jats:italic>+1</jats:sub>, …, <jats:italic>y</jats:italic><jats:sub>0</jats:sub>, <jats:italic>z</jats:italic><jats:sub>−<jats:italic>k</jats:italic></jats:sub>, <jats:italic>z</jats:italic><jats:sub>−<jats:italic>k</jats:italic>+1</jats:sub>, …, <jats:italic>z</jats:italic><jats:sub>1</jats:sub> and <jats:italic>z</jats:italic><jats:sub>0</jats:sub> are arbitrary real numbers do not equal −3. This system can be solved in a closed-form and we will see that the solutions are expressed using the famous Fibonacci and Lucas numbers.</jats:p>

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

“…Clearly if take p = 1 in the system (1.9) we get the system (1.8). So our results generalizes the results obtained in [19].…”

“…Also, we presented some results about the general behavior of solutions of system (1.9) and some numerical examples are carried out to support the analysis results. Our system generalized the systems studied in [18] and [19].…”

“…Khelifa et al in [19] gave some theoretical explanations related to the representation for the general solution of the system of three higher-order rational difference equations…”

“…Motivated by the paper [19], we represents the well-defined solutions of the system of (2p + 1) higher-order rational difference equations…”

Bazi Fark Denklemleri̇ni̇n Ayarlanmiş Jacobsthal-Padovan Sayilari İle İli̇şki̇li̇ Tam Çözümleri̇

“…Because, some of the solution forms of these equations are even expressible in terms of well-known integer sequences. As it can be seen from the references, there are many papers on such these studies from several authors, see [6,7,8,9,10,11,12,13,21].…”