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

Rok, strany: 2020, 641 - 656

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

$$ x_n+1=\dfrac{1+2y_n-k}{3+y_n-k},   y_n+1=\dfrac{1+2z_n-k}{3+z_n-k},   z_n+1=\dfrac{1+2x_n-k}{3+x_n-k}, $$

where $n, k\in \mathbb{N}₀$, the initial values $x_-k$, $x_-k+1,…, x₀$, $y_-k$, $y_-k+1,…, y₀$, $z_-k$, $z_-k+1,…, z₁$ and $z₀$ 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.

ISO 690:

Khelifa, A., Halim, Y., Bouchair, A., Berkal, M. 2020. On a system of three difference equations of higher order solved in terms of Lucas and Fibonacci numbers. In Mathematica Slovaca, vol. 70, no.3, pp. 641-656. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0378

APA:

Khelifa, A., Halim, Y., Bouchair, A., Berkal, M. (2020). On a system of three difference equations of higher order solved in terms of Lucas and Fibonacci numbers. Mathematica Slovaca, 70(3), 641-656. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0378

Vydavateľ: Mathematical Institute, Slovak Academy of Sciences, Bratislava