On a two-dimensional solvable system of difference equations

摘要

Here we solve the following system of difference equations xn+1 = ynyn?2 bxn?1 + ayn?2 , yn+1 = xnxn?2 dyn?1 + cxn?2 , n ∈ N0, where parameters a, b, c, d and initial values x?j , y?j , j = 0, 2, are complex numbers, and give a representation of its general solution in terms of two specially chosen solutions to two homogeneous linear difference equations with constant coefficients associated to the system. As some applications of the representation formula for the general solution we obtain solutions to four very special cases of the system recently presented in the literature and proved by induction, without any theoretical explanation how they can be obtained in a constructive way. Our procedure presented here gives some theoretical explanations not only how the general solutions to the special cases are obtained, but how is obtained general solution to the general system.