On solution of a difference equation via generalized Fibonacci sequence

Articles in Press

Document Type : Research Paper

Authors
Ömer Aktaş 1
Merve Kara 2
1 Instutite of Science, Karamanoğlu Mehmetbey University, Turkey
2 Department of Mathematics, Kamil Ozdag Science Faculty, Turkey
10.22075/ijnaa.2025.39381.5561
Abstract
We consider the following the difference equation
\begin{equation*}
w_n=\frac{w^{r+1}_{n-2}w_{n-3}}{w_{n-1}\left(\gamma_nw_{n-4}^r+\delta_nw_{n-2}w_{n-3}\right)},\quad n\in\mathbb{N}_0,
\end{equation*}
where $r\in\mathbb{N},$ the initial conditions $w_{-j}$, $j=\overline{1,4}$ are non zero real numbers and $\left(\gamma_n\right)_{n\in\mathbb{N}_0}$, $\left(\delta_n\right)_{n\in\mathbb{N}_0}$ are non zero real number sequences. In addition, the
solution of a more general difference equation defined by one to one continuous function is obtained. The solution of the mentioned equation is gained via a generalized Fibonacci sequence. Finally, we obtain the solution of the above difference equation when the sequences $\left(\gamma_n\right)_{n\in\mathbb{N}_0}$, $\left(\delta_n\right)_{n\in\mathbb{N}_0}$ are constant.
Keywords
Fibonacci numbers
difference equation
solution
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International Journal of Nonlinear Analysis and Applications

Articles in Press, Corrected Proof
Available Online from 19 June 2026

  • Receive Date 14 October 2025
  • Revise Date 15 November 2025
  • Accept Date 16 November 2025