On a system of difference equations of third order solved in closed form

2019/10/31 by Youssouf Akrour, Akrour, Youssouf, Nouressadat Touafek +3
Mathematics · Medicine · #Dynamical Systems (math.DS) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Mathematical and Theoretical Epidemiology and Ecology Models #Nonlinear Differential Equations Analysis #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.1910.14365

openalex publication_date 2019/10/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this note we show the the system of difference equations x_n+1=\dfracay_n-2x_n-1y_n+bx_n-1y_n-2+cy_n-2+dy_n-2x_n-1y_n, y_n+1=\dfracax_n-2y_n-1x_n+by_n-1x_n-2+cx_n-2+dx_n-2y_n-1x_n, where n∈ ℕ₀, the initial values x_-2, x_-1, x₀, y_-2, y_-1 and y₀ are arbitrary nonzero real numbers and the parameters a, b, c and d are arbitrary real numbers with d≠ 0, can be solved in a closed form. We will see that when a=b=c=d=1 the solutions are expressed using the famous Teteranacci numbers. In particular, the results obtained here extend those in our work \citearxiv.

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