On a two-dimensional solvable system of difference equations

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.


Introduction
Let N, Z, R, C be the sets of natural, integer, real and complex numbers, respectively, and N l = {n ∈ Z : n ≥ l}, where l ∈ Z.Let k, l ∈ Z, k ≤ l, then instead of writing k ≤ j ≤ l, we will use the notation j = k, l.
Our first explanation of such a problem appeared in 2004, when we solved the following equation by a constructive method, explaining a closed-form formula for the case α = β = 1 previously presented in the literature.In [33,36,37] some extensions of the equation have been investigated later.The main point is that the previous equation is easily transformed to a solvable difference equation.After that we employed and developed successfully the method, e.g., in [6,38,39,[47][48][49].For some combinations of the method with other ones see, e.g., the following representative papers: [41,42,45,46,[50][51][52].
There has been also some recent interest in representation of solutions to difference equations and systems in terms of specially chosen sequences, for example, in terms of Fibonacci sequences (for some basics on the sequence see, e.g., [3,14,54]).Many papers present such results, but in the majority cases the results are essentially known.For some representative papers in the area see [40] and [53], where you can find some citations which have such results.
The following four systems of difference equations have been studied in recent paper [4], where some closed-form formulas for their solutions are given in terms of the initial values x −j , y −j , j = 0, 2, and some subsequences of the Fibonacci sequence.The closed-form formulas are only given and proved by induction.There are no theoretical explanations for the formulas.
A natural problem is to explain what is behind all the formulas given in [4].Since it is expected that the solvability is the main cause for this, we can try to use some of the ideas from our previous investigations, especially on rational difference equations and systems (e.g., the ones in [6,[36][37][38][39][47][48][49]).
Here we consider the following extension of the systems in (1.2) where parameters a, b, c, d and initial values x −j , y −j , j = 0, 2, are complex numbers.Our aim is to show that system (1.3) is solvable by getting its closed-form formulas in an elegant constructive way, and to show that all the closed-form formulas obtained in [4] easily follow from the ones in our present paper.

Main results
Assume that x n 0 = 0 for some n 0 ≥ −2.Then from the second equation in (1.3) it follows that y n 0 +1 = 0, and consequently dy n 0 +1 + cx n 0 = 0, from which it follows that y n 0 +3 is not defined.Now, assume that y n 1 = 0 for some n 1 ≥ −2.Then from the first equation in (1.3) it follows that x n 1 +1 = 0, and consequently bx n 1 +1 + ay n 1 = 0, from which it follows that x n 1 +3 is not defined.This means that the set 2 j=0 (x −j , y −j ) ∈ C 2 : x −j = 0 or y −j = 0 , is a subset of the domain of undefinable solutions to system (1.3).
Then system (2.2) can be written as S. Stević Then, from (2.5) we see that (u (j) m ) m≥−1 , j = 1, 2, are two solutions to the following difference equation m ) m≥−1 , j = 1, 2, are two solutions to the following difference equation Equations (2.7) and (2.8) are bilinear, so, solvable ones.Let where Then equation (2.7) becomes Let (s m ) m≥−1 be the solution to equation (2.10) such that Let λ 1 and λ 2 be the zeros of the characteristic polynomial Then general solution to equation (2.10) can be written in the following form [40] (here for m = −1 is involved the term s −2 , which is calculated by using the following relation From (2.9) and (2.12) it follows that 3), we obtain and for m ∈ N 0 .Let where Then equation (2.8) becomes Let ( s m ) m≥−1 be the solution to equation (2.18) such that Let λ 1 and λ 2 be the zeros of the characteristic polynomial Then general solution to equation (2.18) can be written in the following form (2.20) From (2.17) and (2.20) it follows that From (2.6) and (2.21) it follows that Using (2.22) in (2.4), we obtain and for m ∈ N 0 .From (2.15), (2.16), (2.23) and (2.24), we have ) ) for m ∈ N 0 .
From (2.31), since , and after some calculations we have From (2.32), (2.34) and after some calculations we have From the above consideration we see that the following result holds.
Theorem 2.1.Consider system (1.3).Let s n be the solution to equation (2.10) satisfying initial conditions (2.11), and s n be the solution to equation (2.18) satisfying initial conditions (2.19).Then, for every well-defined solution (x n , y n ) n≥−2 to the system the following representation formulas hold ) , (2.37) for n ∈ N 0 .

Some applications
As some applications we show how are obtained closed-form formulas for solutions to the systems in (1.2), which were presented in [4].
First result proved in [4] is the following.
Corollary 3.1.Let (x n , y n ) n≥−2 be a well-defined solution to the following system ) for n ∈ N 0 , where ( f n ) n≥−1 is the solution to the following difference equation satisfying the initial conditions f −1 = 0 and f 0 = 1.The following corollary is Theorem 3 in [4].
Proof.System (3.9) is obtained from system (1.From this and since s n is a solution to equation (3.17) we have from which along with (3.16) it follows that The following corollary is Theorem 4 in [4].
Proof.System (3.21) is obtained from system (1. where from which by some calculation it follows that Formula (3.35) shows that the sequence s n is six periodic.Namely, we have for m ≥ −1 (in fact, (3.36)-(3.38)hold for every m ∈ Z).Equalities (3.36)-(3.38)can be written as follows for m ≥ −1.
The following corollary is Theorem 5 in [4].
Corollary 3.4.Let (x n , y n ) n≥−2 be a well-defined solution to the following system ) ) ) for n ∈ N 0 .