Representation of solutions of bilinear difference equations in terms of generalized Fibonacci sequences

Well-defined solutions of the bilinear difference equation are represented in terms of generalized Fibonacci sequences and the initial value. Our results extend and give natural explanations of some recent results in the literature. Some applications concerning a two-dimensional system of bilinear difference equations are also given.


Introduction
Studying difference equations and systems which are not closely related to differential ones is a topic of recent interest (see, ).Solvable difference equations attract attention of mathematicians for a long time.Some classical classes of solvable difference equations and methods for solving them can be found, for example, in [14].Recently, there has been an increasing interest in the topic (see, for example, [1-4, 6-8, 17, 20, 21, 23-41] and the related references therein).Some of the recent papers give formulas for solutions to some very special difference equations or systems of difference equations and prove them by using only the method of induction (quite frequently the proofs of some statements are even omitted or incomplete).However, the formulas are not justified by some theoretical explanations.
In paper [20] we gave a theoretical explanation for the formula of solutions of the following difference equation given in [7] (in fact, a generalization of equation (1.1) was treated in [20]).Paper [20] attracted some attention among the experts in difference equations and trigged off a new interest in the area.For some results regarding solutions of various types of extensions of equation (1.1), Email: sstevic@ptt.rs 2 S. Stević see, for example, [1,17,24,25,27,37].Papers [3] and [4] consider also an extension of equation (1.1), but do not use formulas for their solutions.Some other explanations for the formulas of some special difference equations or systems of difference equations appearing in recent literature, can be found, for example, in papers [23], [28], [36] and [38].
Recent paper [40] is also one of those which give some formulas and prove them by the induction, but does not use any other mathematical technique in explaining the formulas.
Namely, the authors of [40] represented the general solution of the following difference equation in terms of the initial value x 0 and the Fibonacci sequence, that is, the sequence defined as follows More precisely, it was proved by induction that every well-defined solution of equation (1.2) can be written in the following form However, the authors of [40] did not explain how they come up with the formula and did not support it by any mathematical theory.They also proved that every well-defined solution of the equation can be written in the following form where the terms of the Fibonacci sequence with negative indices are calculated by the formula and where, of course, is assumed that f 0 = 0 and f 1 = 1 (recurrence relation (1.7) is obtained from (1.3) when we replace n by −n + 1).As in the case of equation (1.2), they also did not explain how they come up with formula (1.6) nor gave any theoretical explanation for it.
The other results in [40] are folklore, that is, follow easily from well-known ones.Formulas (1.4) and (1.6) could be also known, but we are not able to find some specific references for them at the moment.Nevertheless, in our opinion, these two formulas are interesting and motivated us to explain them theoretically.Actually, our aim is to obtain, in a natural way, similar representation for a more general difference equation which includes into itself equations (1.2) and (1.5).As some applications of our main results we give explanations of some results in [8], and also obtain related results for a two-dimensional system of bilinear difference equations.

Preliminaries and some basic solvable difference equations
In this section we present some known difference equations and results related to them, and also introduce some notions which will be used in the proofs of our main results.

Linear first-order difference equation
Probably, the most known difference equation which can be solved is the linear first-order difference equation, i.e.
x n+1 = p n x n + q n , n ∈ N 0 , ( where (p n ) n∈N 0 and (q n ) n∈N 0 are arbitrary real (or complex) sequences and x 0 ∈ R (or x 0 ∈ C).Equation (2.1) can be solved in closed form in many ways, and its general solution is For example, if p n = 0, n ∈ N 0 , by dividing both sides of (2.1) by ∏ n j=0 p j , we obtain Summing equalities in (2.3) from 0 to n − 1, we get , from which formula (2.2) easily follows.

Generalized Fibonacci sequence
Here we define an extension of the Fibonacci sequence in the following way ) and we will call it the generalized Fibonacci sequence (note that for a = b = 1 is obtained the Fibonacci sequence).We assume that b = 0, otherwise, equation (2.5) becomes a special case of the linear first-order difference equation (2.1).Note that the characteristic polynomial associated to equation (2.4) is so that the characteristic roots are and if a 2 + 4b = 0, then the solution of equation (2.4) satisfying conditions (2.5) is The main motivation for introducing the generalized Fibonacci sequence are representations (1.4) and (1.6) of solutions of equations (1.2) and (1.5).

Linear second order difference equation with constant coefficients.
As is well-known, the equation (the homogeneous linear second order difference equation with constant coefficients), where a, x 0 , x 1 ∈ R, and b ∈ R \ {0}, is usually solved by using the characteristic roots λ 1 and λ 2 of the characteristic polynomial λ 2 − aλ − b = 0.This standard method along with some calculations easily gives formulas for its general solution (see formulas (2.11) and (2.12)).To demonstrate the importance of equation (2.1), for the completeness and the benefit of the reader, recall, that the formulas can be also obtained by using formula (2.2).Namely, since a = λ 1 + λ 2 and b = −λ 1 λ 2 , we have that Using the change of variables which is equation (2.1) with p n = λ 2 and q n = 0, n ∈ N, so its solution is Equation (2.10) is also equation (2.1), but with p n = λ 1 and 2) is obtained that the general solution of equation (2.8) is from which for the case λ 1 = λ 2 is easily obtained (2.11) (2.12) Formulas (2.11) and (2.12) are well-known, but what is interesting to note is the fact that solution (2.11) can be written in the following form and that the same formula also holds for the case λ 1 = λ 2 , with Remark 2.1.Note that representation (2.13) holds also for n = 0, if we assume that Now, we have all the ingredients for formulating and proving the main results in this paper.

Extensions of formulas (1.4) and (1.6) and their consequences
A natural extension of equations (1.2) and (1.5) is the bilinear difference equation where parameters α, β, γ, δ and initial value z 0 are real numbers.We will assume that γ = 0, since for γ = 0 equation (3.1) is reduced to a special case of equation (2.1).Beside this, we will also assume that αδ = βγ, since otherwise is obtained the trivial equation (case γ = δ = 0 is excluded by the first assumption).For some recent applications of equation (3.1), see, for example, [6] and [34].
Our aim is to obtain an extension of formula (1.4), for the solutions of difference equation (3.1), in terms of the initial value and a sequence of type in (2.4) satisfying the conditions in (2.5).We also want to obtain an extension of formula (1.6) for the solutions of equation (3.1).
Note that equation (3.1) can be written in the form from which it follows that Since we are interested in well-defined solutions of equation (3.1) we may assume that Hence we can use the change of variables

S. Stević
By using (3.7) in (3.3) we get Hence, by using (3.6) it follows that From all above mentioned we see that the following theorem holds.
Theorem 3.1.Consider equation (3.1), with γ = 0 and αδ = βγ.Then every well-defined solution of the equation can be written in the following form where (s n ) n∈N 0 is the sequence satisfying difference equation (3.6) with the initial conditions s 0 = 0 and s 1 = 1.
If α = 0, then from (3.8) we get Hence, for β = γ = δ = 1 we have that s n = f n , n ∈ N 0 , and consequently we get formula (1.4), giving a natural explanation for it.where (s n ) n∈N 0 is the sequence satisfying difference equation (3.6) with the initial conditions s 0 = 0 and s 1 = 1.
Proof.We have from which (3.10) follows.
Bilinear difference equations 7 Difference equation (2.4) can be naturally extended for negative indices by using the following recurrence relation where s 0 = 0 and s 1 = 1.
It is known that its solution is from which it follows that Hence, for the case of difference equation (3.6), we have that Using (3.12) into (3.8)we get which is a representation of well-defined solutions of equation (3.1) in terms of the generalized Fibonacci sequence with negative indices.Hence we have that the following theorem holds., n ∈ N 0 , (3.14) where (s −n ) n≥−1 is the sequence satisfying recurrent relation (3.11) with the initial conditions s 0 = 0 and s 1 = 1.

Some applications
As some applications of our main results, in this section we give theoretical explanations for the formulas presented in Theorems 4-6 in [8], and obtain some related results for a twodimensional system of bilinear difference equations.The author of [8] formulated, among others, the following three results and proved them by induction.However, none theoretical explanations are given therein and it was also not explained how the formulas for solutions of the difference equations therein are obtained, especially since the forms of the solutions do not look simple.
Theorem 4.1.Let (x n ) n≥−1 be a solution of the following difference equation Then Theorem 4.2.Let (x n ) n≥−1 be a solution of the following difference equation Theorem 4.3.Let (x n ) n≥−1 be a solution of the following difference equation Now we give theoretical explanations for the formulas presented in Theorems 4.1-4.3,based on our main results.Before this, note that the author of [8] under solutions seems tacitly understands well-defined solutions.Hence, we will assume that the solutions we deal with are of this type.For some results in the area, see, e.g.[29].

Case of equation (4.1)
First note that we may assume that x n = 0 for every n ∈ N. Otherwise, if there is an n 0 ∈ N such that x n 0 = 0, then if x n 0 +1 is defined, from (4.1) we would have x n 0 +1 = 0, which would imply that x n 0 +2 is not defined (since in this case x n 0 + x n 0 +1 = 0).We may also assume that x −1 = 0, for if x −1 = 0 and x 0 = 0, and solution (x n ) n≥−1 is well-defined, then we can consider equation (4.1) for n ∈ N, that is, to reduce the case to the previous one by scaling indices backward for one.
Hence, we can use the change of variables and transform equation (4.1) into the following one which is a special case of equation (3.1), with α = β = γ = 1 and δ = 2. Clearly, from (4.7) we have that By using Theorem 3.1 we have that every well-defined solution of equation (4.8) can be written in the form where (s n ) n∈N 0 is the sequence satisfying the difference equation with the initial conditions s 0 = 0 and s 1 = 1.Employing formula (2.6) or (2.7) we have , n ∈ N 0 . (4.12) Now note that Using this in (4.12) we obtain

Case of equation (4.3)
Note that we may also assume that x n = 0 for every n ∈ N. Otherwise, if there is an n 1 ∈ N such that x n 1 = 0, then if x n 1 +1 is defined, from (4.3) we would have x n 1 +1 = 0, which would imply that x n 1 +2 is not defined (since in this case x n 1 +1 − x n 1 = 0).We may also assume that x −1 = 0, for if x −1 = 0 and x 0 = 0, and solution (x n ) n≥−1 is well-defined, then we can consider equation (4.3) for n ∈ N, that is, to reduce the case to the previous one by scaling indices backward for one.Hence, we can use the change of variables and transform equation (4.3) into the following one which is a special case of equation (3.1), with β = δ = −1, γ = 1 and α = 2. Clearly, from (4.16) we have that By using Theorem 3.1 we have that every well-defined solution of equation (4.17) can be written in the form where (s n ) n∈N 0 is the sequence satisfying the difference equation with the initial conditions s 0 = 0 and s 1 = 1.This means that (s n ) n∈N 0 is the Fibonacci sequence.

Case of equation (4.5)
As in the case of equation (4.1) it is shown that in this case we may also assume that x n = 0 for every n ≥ −1.Hence, we can use the change of variables in (4.16), so that equation (4.5) is transformed into the following equation and we have that relation (4.18) holds.By using Theorem 3.1 (or formula (1.4)) we have that

On a bilinear system of difference equations
A natural system of difference equations related to equation (3.1) is the following where s 0 = 0 and s 1 = 1.Applying Theorem 3.1 for the case of equations (4.23) and (4.24), and using the relations which are obtained from the equations in (4.22) with n = 0, after some calculation we obtain the following result.The following system is a special case of system (4.22) and is a natural generalization of equation (1.2).Corollary 4.5.Consider the system of difference equations where z 0 and w 0 are real numbers.Then for every well-defined solution of the system the following relations hold ) w 2n = w 0 f 2n−1 + f 2n w 0 f 2n + f 2n+1 , n ∈ N 0 , (4.28) , n ∈ N 0 .(4.29) ) in (3.2) and obtain b n+1 = 1 α + δ + (βγ − αδ)b n , n ∈ N 0 .(3.4)If we use the following change of variables get c n+1 − (α + δ)c n + (αδ − βγ)c n−1 = 0, n ∈ N. (3.6)Now note that equation (3.6) is nothing but equation (2.4) with a = α + δ and b = βγ − αδ.Hence, by using representation (2.13) we see that the general solution of equation (3.6) in terms of the sequence s n := s n (α + δ, βγ − αδ), and initial values c 0 and c 1 is

Corollary 3 . 2 .
Consider equation (3.1), with βγ = 0 and α = 0. Then for every well-defined solution of the equation the following formula holds n