On a Solvable System of Difference Equations in Terms of Generalized Fibonacci Numbers

Abstract

ABSTRACTIn this paper, we represent that the following three-dimensional system of difference equationsxn+1=αyn+aynyn−βzn−1, yn+1=βzn+bznzn−γxn−1, zn+1=γxn+cxnxn−αyn−1,  n∈ℕ0,$$\matrix{{{x_{n + 1}} = \alpha {y_n} + {{a{y_n}} \over {{y_n} - \beta {z_{n - 1}}}},\quad {y_{n + 1}} = \beta {z_n} + {{b{z_n}} \over {{z_n} - \gamma {x_{n - 1}}}},\quad {z_{n + 1}} = \gamma {x_n} + {{c{x_n}} \over {{x_n} - \alpha {y_{n - 1}}}},\qquad n \in {{\mathbb N}_0},} \cr} $$where the parametersa, b, c, α, β, γand the initial valuesx−i,y−i,z−i,i∈ {0, 1}, are real numbers, can be solved in closed form by using transformation. We analyzed the solutions in 10 different cases depending on whether the parameters are zero or nonzero. It is noteworthy to depict that the solutions of some particular cases of this system are presented in terms of generalized Fibonacci numbers. Note that our results considerably extend and improve some recent results in the literature.