On Some Solvable Difference Equations and Systems of Difference Equations

Abstract

Here, we give explicit formulae for solutions of some systems of difference equations, which extend some very particular recent results in the literature and give natural explanations for them, which were omitted in the previous literature.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Stevo Stevic 1 and Josef Diblík 2, 3 and Bratislav Iricanin 4 and Zdenek Smarda 3

Recommended by Svatoslav Stanek

1, Mathematical Institute of the Serbian Academy of Sciences and Arts, Knez Mihailova 36/III, 11000 Beograd, Serbia
2, Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Brno University of Technology, 60200 Brno, Czech Republic
3, Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, 61600 Brno, Czech Republic
4, Faculty of Electrical Engineering, University of Belgrade, Bulevar Kralja Aleksandra 73, 11000 Beograd, Serbia

Received 8 August 2012; Accepted 27 September 2012

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Recently, there has been a great interest in difference equations and systems (see, e.g., [ 1- 25]), and among them in those ones which can be solved explicitly (see, e.g., [ 1- 5, 9- 11, 15, 16, 18- 24] and the related references therein). For some classical results in the topic see, for example, [ 7].

Beside the above-mentioned papers, there are some papers which give formulae of some very particular equations and systems which are proved by induction, but without any explanation how these formulae are obtained and how these authors came across the equations and systems. Our explanation of such a formula that we gave in [ 10] has re-attracted attention to solvable difference equations.

Our aim here is to give theoretical explanations for some of the formulae recently appearing in the literature, as well as to give some extensions of their equations.

Before we formulate our results, we would like to say that the system of difference equations [figure omitted; refer to PDF] where a ,b ,c , α , β , and γ are real numbers, was completely solved in [ 15], that is, we found formulae for all well-defined solutions of system ( 1.1).

2. Scaling Indices

In the recent paper [ 25] were given some formulae for the solutions of the following systems of difference equations: [figure omitted; refer to PDF]

Now we show that the results regarding system ( 2.1) easily follow from known ones. Indeed, if we use the change of variables [figure omitted; refer to PDF] then the systems in ( 2.1) are reduced to the next systems [figure omitted; refer to PDF] This means that ( ^{u n (i )} , ^{v n (i )} _{) n ...5; -2} , i =1,2 , are two (independent) solutions of the systems of difference equations [figure omitted; refer to PDF] However, all the systems of difference equations in ( 2.4) are particular cases of system ( 1.1). Hence, formulae for the solutions of systems ( 2.1) given in [ 25] follow directly from those in [ 15].

2.1. An Extension of Systems ( 2.1)

Systems ( 2.1) can be extended as follows: [figure omitted; refer to PDF] where k is a fixed natural number.

If we use the change of variables [figure omitted; refer to PDF] then system ( 2.5) is reduced to the following k systems of difference equations: [figure omitted; refer to PDF] i =1 , ... ,k . This means that _{( ^{u n (i )} , ^{v n (i )} ) n ...5; -2} , i =1 , ... ,k , are k (independent) solutions of system ( 1.1), and solutions of system ( 2.5) are obtained by interlacing solutions of systems ( 2.7), i =1 , ... ,k .

For example, a natural extension of the systems in ( 2.1) is obtained for taking k =3 , b /a = ±1 , c /a = ±1 , β / α = ±1 , and γ / α = ±1 in ( 2.5), that is, the system becomes [figure omitted; refer to PDF]

In this way it can be obtained countable many, at first sight different, systems of difference equations. Systems of difference equations in ( 2.1) are artificially obtained in this way. This method can be applied to any equation or system of difference equations, and one can get papers with putative "new" results.

3. Some Third-Order Systems of Difference Equations Related to ( 1.1)

The following third-order systems of difference equations [figure omitted; refer to PDF] n ∈ _{... 0} , have been studied recently (see [ 6] and the references therein).

As is directly seen, the first two equations in systems ( 3.1)-( 3.3) are the same, and they form a particular case of system ( 1.1) which is solved in [ 15].

Since we know solutions for _{x n} and _{y n} , it is only needed to find explicit solutions for _{z n} in the third equations in systems ( 3.1)-( 3.3), that is, in all three equations, the only unknown sequence is _{z n} . The joint feature for all three cases is that _{z n} can be solved in closed form.

Now we discuss systems of difference equations given in ( 3.1)-( 3.3).

Case of System ( 3.1)

From the third equation in ( 3.1), we get [figure omitted; refer to PDF] from which it follows that [figure omitted; refer to PDF]

Case of System ( 3.3)

From the third equation in ( 3.3), we get [figure omitted; refer to PDF] from which it follows that [figure omitted; refer to PDF]

Remark 3.1.

The third equations in systems ( 3.1) and ( 3.3) are particular cases of the following difference equation (up to the shifting indices): [figure omitted; refer to PDF] where k ∈ ... . From ( 3.8), it follows that [figure omitted; refer to PDF] and consequently, [figure omitted; refer to PDF]

Case of System ( 3.2)

If we use the change of variables _{v n} =1 / _{z n} , the third equation in ( 3.2) becomes [figure omitted; refer to PDF] Hence, [figure omitted; refer to PDF] from which it follows that [figure omitted; refer to PDF] so that [figure omitted; refer to PDF]

Remark 3.2.

The third equation in system ( 3.2) is a particular case of the following difference equation: [figure omitted; refer to PDF] which, by the change of variables _{z n} =1 / _{v n} , is transformed into [figure omitted; refer to PDF] from which it follows that [figure omitted; refer to PDF] so by a well-known formula, we have that [figure omitted; refer to PDF] for m ∈ _{... 0} and i =0,1 , ... ,k -1 , and consequently, [figure omitted; refer to PDF] for m ∈ _{... 0} and i =0,1 , ... ,k -1 .

Remark 3.3.

Note that ( 3.16) suggests that the third equation in ( 3.2) can be also of the form [figure omitted; refer to PDF] where k ∈ ... is fixed, that is, to be an equation which consists of k (independent) linear first-order difference equation, which is solvable. In fact, the third equation in systems ( 3.1)-( 3.3) can be any difference equation which can be solved in _{z n} , and in this way we can obtain numerous putative "new" results.

4. A Generalization of System ( 1.1)

Consider the following system of difference equations: [figure omitted; refer to PDF] where g ,h : ... [arrow right] ... are increasing functions such that [figure omitted; refer to PDF]

Now we will find formulae for all well-defined solutions of system ( 4.1), that is, for the solutions ( _{x n} , _{y n} ) , n ...5; -1 , such that [figure omitted; refer to PDF] for every n ∈ _{... 0} .

If _{x -1} =0 , then from ( 4.1), ( 4.2), and ( 4.3), and by the method of induction, we get _{x 2n +1} =0 , n ∈ _{... 0} . Also, if _{x 0} =0 , then from ( 4.1), ( 4.2), and ( 4.3), and by the method of induction, we get _{x 2n} =0 , n ∈ _{... 0} . Similarly, if _{y -1} =0 , then we get _{y 2n +1} =0 , n ∈ _{... 0} , while if _{y 0} =0 , then we get _{y 2n} =0 , n ∈ _{... 0} .

If _{x n 0} =0 for some _{n 0} ∈ ... , then from ( 4.1)-( 4.3) it follows that _{x n 0 -2k} =0 , for each k ∈ _{... 0} such that _{n 0} -2k ...5; -1 . Hence, in this case we have that _{x -1} =0 or _{x 0} =0 . Similarly, if _{y n 1} =0 for some _{n 1} ∈ ... , then from ( 4.1)-( 4.3) it follows that _{y n 1 -2k} =0 , for each k ∈ _{... 0} such that _{n 1} -2k ...5; -1 . Hence, in this case we have that _{y -1} =0 or _{y 0} =0 . Thus, in both cases we arrive at a situation explained in the previous paragraph.

Hence, from now on, we assume that none of the initial values _{x -1} , _{x 0} , _{y -1} , and _{y 0} is equal to zero. Then, for every well-defined solution of system ( 4.1), we have that _{x n} ...0;0 and _{y n} ...0;0 , for every n ...5; -1 , and consequently g ( _{x n} ) ...0;0 and h ( _{y n} ) ...0;0 , for every n ...5; -1 .

Let [figure omitted; refer to PDF] then by taking function g to the first equation in system ( 4.1) and function h to the second one, then multiplying the first equation in such obtained system by h ( _{y n} ) and the second by g ( _{x n} ) , system ( 4.1) is transformed into: [figure omitted; refer to PDF] from which it follows that [figure omitted; refer to PDF]

Hence, if bd ...0;1 , we have that [figure omitted; refer to PDF] while if bd =1 , we have that [figure omitted; refer to PDF]

We have also that [figure omitted; refer to PDF] while if bd =1 , we have that [figure omitted; refer to PDF]

From ( 4.5), we have that [figure omitted; refer to PDF]

Using the relations ( 4.12), we get [figure omitted; refer to PDF]

Example 4.1.

If we choose g (t ) = ^{t 2k +1} and h (t ) = ^{t 2l +1} for some k ,l ∈ _{... 0} , then conditions ( 4.2) and ( 4.3) are obviously satisfied, and system ( 4.1) can be written in the form [figure omitted; refer to PDF] and from ( 4.13), we have that its solutions are given by [figure omitted; refer to PDF]

Remark 4.2.

System ( 4.1) can be generalized by using the method of scaling indices from Section 2.1, that is, the following system is also solvable: [figure omitted; refer to PDF] n ∈ _{... 0} , where k ∈ ... and g ,h : ... [arrow right] ... are increasing functions satisfying conditions ( 4.2) and ( 4.3).

It is easy to see that the change of variables in ( 2.6) leads to the following k systems of difference equations: [figure omitted; refer to PDF] for i =1 , ... ,k .

Remark 4.3.

Well-defined solutions of the following system of difference equations: [figure omitted; refer to PDF] where g ,h : ... [arrow right] ... are increasing functions satisfying conditions ( 4.2) and ( 4.3) and _{a n} , _{b n} , _{c n} , and _{d n} , n ∈ _{... 0} , are real sequences, can be found similarly. We omit the details.

5. Solutions of a Generalization of a Recent Equation

Explaining some recent formulae appearing in the literature, in our recent paper [ 24], we have found formulae for well-defined solutions of the following difference equation: [figure omitted; refer to PDF] where k ∈ ... and the parameters a , b as well as initial values _{x -i} , i = 0 ,k ¯ are real numbers.

Equation ( 5.1) can be extended naturally in the following way: [figure omitted; refer to PDF] where k ∈ ... and the sequences ( _{a n ) n ∈ ... 0} , ( _{b n ) n ∈ ... 0} , as well as initial values _{x -i} , i = 0 ,k ¯ , are real numbers.

Employing the change of variables [figure omitted; refer to PDF] Equation ( 5.2) is transformed into the linear first-order difference equation [figure omitted; refer to PDF] whose general solution is [figure omitted; refer to PDF]

From ( 5.3), we have that [figure omitted; refer to PDF] for n ...5;k , which yields [figure omitted; refer to PDF] for every m ∈ _{... 0} and i ∈ {k ,k +1 , ... ,3k -1 } .

Using ( 5.5) in ( 5.7), we get [figure omitted; refer to PDF] for every m ∈ _{... 0} and i ∈ {k ,k +1 , ... ,3k -1 } .

Formula ( 5.8) generalizes the main formulae obtained in our paper [ 24].

Acknowledgments

The second author is supported by the Grant P201/10/1032 of the Czech Grant Agency (Prague) and the "Operational Programme Research and Development for Innovations," no. CZ.1.05/2.1.00/03.0097, as an activity of the regional Centre AdMaS. The fourth author is supported by the Grant FEKT-S-11-2-921 of Faculty of Electrical Engineering and Communication, Brno University of Technology and by the Grant P201/11/0768 of the Czech Grant Agency (Prague). This paper is also supported by the Serbian Ministry of Science projects III 41025, III 44006, and OI 171007.

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