On a three-dimensional solvable system of difference equations

Abstract

In this paper we solve the following system of difference equations xn+1 = zn−1 a + bynzn−1 , yn+1 = xn−1 a + bznxn−1 , zn+1 = yn−1 a + bxnyn−1 , n ∈ N0 where parameters a, b and initial values x−1, x0, y−1, y0, z−1, z0 are nonzero real numbers, and give a representation of its general solution in terms of a specially chosen solutions to homogeneous linear difference equation with constant coefficients associated to the system.

On The Solutions of Three-Dimensional Difference Equation Systems Via Pell Numbers

“…For example; Sahinkaya et al [4], in their study, starting conditions are real numbers, they defined the difference equation system 𝑥 𝑛+1 = 𝑥 𝑛 𝑦 𝑛 + 𝑎 𝑥 𝑛 + 𝑦 𝑛 , 𝑦 𝑛+1 = 𝑦 𝑛 𝑧 𝑛 + 𝑎 𝑦 𝑛 + 𝑧 𝑛 , 𝑧 𝑛+1 = 𝑧 𝑛 𝑥 𝑛 + 𝑎 𝑧 𝑛 + 𝑥 𝑛 (1.4) They also obtained the general solutions in closed form. Halim et al [5], in their study; investigated the solutions of on a three-dimensional system of difference equations 𝑥 𝑛+1 = 𝑧 𝑛−1 𝑎 + 𝑏𝑦 𝑛 𝑧 𝑛−1 , 𝑦 𝑛+1 = 𝑥 𝑛−1 𝑎 + 𝑏𝑧 𝑛 𝑥 𝑛−1 , 𝑧 𝑛+1 = 𝑦 𝑛−1 𝑎 + 𝑏𝑥 𝑛 𝑦 𝑛−1 (1.5) In addition, in the studies [13]- [20], there have been approaches from different perspectives on the solutions of difference equation systems. There are also studies that related the solutions of difference equations or systems of difference equations with some integer sequences (see, for example, [20]- [26] and the related references therein).…”

On The Solutions of Three-Dimensional Difference Equation Systems Via Pell Numbers

“…For example; Sahinkaya et al [4], in their study, starting conditions are real numbers, they defined the difference equation system 𝑥 𝑛+1 = 𝑥 𝑛 𝑦 𝑛 + 𝑎 𝑥 𝑛 + 𝑦 𝑛 , 𝑦 𝑛+1 = 𝑦 𝑛 𝑧 𝑛 + 𝑎 𝑦 𝑛 + 𝑧 𝑛 , 𝑧 𝑛+1 = 𝑧 𝑛 𝑥 𝑛 + 𝑎 𝑧 𝑛 + 𝑥 𝑛 (1.4) They also obtained the general solutions in closed form. Halim et al [5], in their study; investigated the solutions of on a three-dimensional system of difference equations 𝑥 𝑛+1 = 𝑧 𝑛−1 𝑎 + 𝑏𝑦 𝑛 𝑧 𝑛−1 , 𝑦 𝑛+1 = 𝑥 𝑛−1 𝑎 + 𝑏𝑧 𝑛 𝑥 𝑛−1 , 𝑧 𝑛+1 = 𝑦 𝑛−1 𝑎 + 𝑏𝑥 𝑛 𝑦 𝑛−1 (1.5) In addition, in the studies [13]- [20], there have been approaches from different perspectives on the solutions of difference equation systems. There are also studies that related the solutions of difference equations or systems of difference equations with some integer sequences (see, for example, [20]- [26] and the related references therein).…”