# Representations of solutions to linear and bilinear difference equations and systems of bilinear difference equations

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## Abstract

We represent general solution to a homogeneous linear difference equation of second order in terms of a specially chosen solution to the equation and apply it to get a representation of general solution to the bilinear difference equation in terms of a solution to an associate difference equation of second order, considerably generalizing some recent results in an elegant way. We also present the corresponding representations for some systems of bilinear difference equations. Many historical notes not so known to wide audience are also presented, and we offer an answer to an open question regarding the attribution of the bilinear difference equation.

## Keywords

Linear difference equation Bilinear difference equation General solution Representation of solutions System of bilinear difference equations History of difference equations## MSC

39A05 39A06## 1 Introduction

It has now been 300 years since the area of difference equations/recurrent relations started to attract a serious interest of scientist working in various branches of science. During these three centuries scientists have obtained many nice results on the equations, and the area has been growing constantly. However, it has been also noticed that more and more researchers interested in the area work on some difference equations and systems without knowing some basic facts on them, so it frequently happens that some basic results are rediscovered and published. Here, among other things, we will demonstrate it on a concrete example of a nonlinear difference equation. One of the aims of the paper, among other ones, is also to give many useful mathematical and historical information to the experts, as well as to everyone who is interested in the area. The reader can find here many old sources, some of which are most probably original ones, which were forgotten during centuries of investigations of recurrent relations, and can see that some known results and formulas are wrongly attributed, a thing which frequently happens.

### 1.1 Some history

Since the time of de Moivre, recurrent relations have been attracting interest of mathematicians. The notion recurrent series/sequence was most probably coined in [1], although such sequences had already appeared in [2]. From [1] and [2] we see that de Moivre had all necessary ingredients for solving homogeneous linear difference equations with constant coefficients in closed form, although some explicit formulas for the solutions to the equations of small orders appeared later in [3] (see also [4]). The results by de Moivre can be regarded as a starting point for a serious investigation of recurrence relations.

There had been few results on recurrent relations before de Moivre, such as Cassini’s formula for Fibonacci numbers from 1680, but de Moivre seems to be the first who presented some general methods for dealing with the relations. Of course, it is well known that recurrent relations in some descriptive ways had been already known to Fibonacci [5], and certainly to many other ancient researchers, but important analytic results were not known for a long time.

That recurrent relations can be written in the form of difference equations and vice-versa was realized during the eighteenth century by other researchers. Laplace could be the first who noted the connection (see [6]). This is why many researchers have been using both notions interchangeably since that time. This will be also the case in our paper.

*m*and

*p*are the zeros of the polynomial

A more systematic presentation of the de Moivre methods and ideas can be found in Euler’s known book [7], where solutions to some other linear recurrent relations can also be found.

Hence, the representation of solutions to equation (1) given in (5) is a very simple consequence of some results by de Moivre. The simple consequence we could not find in de Moivre’s papers, but our observation shows that the representation is/must be a matter of folklore. We have used recently this representation in [8]. The usefulness of solutions (4) has been demonstrated in several recent papers of us on product-type difference equations and systems (see, e.g., [9, 10] and the references therein).

*bilinear difference equation*or

*linear fractional difference equation*(here we use the first name).

The change of variables can already be found in papers by Laplace, see, e.g., [6]. Hence, the use of the change of variables is a common method which has been applied since (see, e.g., [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27]). For a different approach in solving equation (8), which uses a linear system of difference equations, see [15, 28]. A related method has been recently used in [29] for solving another system of difference equations. See also [30]. Methods for solving some linear systems of difference equations appeared also in [6]. We have to mention that both methods transform equation (8) to an equation of the form (1). For some applications of equation (8), see also [31, 32, 33].

*nonhomogeneous linear first-order difference equation with variable coefficients*

There are three standard methods for solving equation (18) which correspond to the three methods for solving the linear differential equation of first order. A presentation of the methods can be found in [26]. Many books on difference equations which deal with equation (18) solve it by using one of the methods [15, 20, 22, 23, 24, 35]. Usefulness of the equation in solving many classes of nonlinear difference equations and systems has been recently demonstrated in many papers [32, 36, 37, 38, 39, 40, 41, 42] (see also the references therein; see also [43, 44] and compare the methods therein with the solvability ones). For some applications and qualitative analysis of the solutions to equation (18), see [23, 45, 46, 47].

For some other solvable difference equations and systems or the ones which have invariant integrals and their applications, see, e.g., [48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58] and the references therein.

### 1.2 On a representation of an equation of form (8)

*generalized Fibonacci sequences*as those solutions to equation (1) satisfying the initial conditions

### 1.3 On a recent extension of representation (21)

In [8] we generalized representation formula (21) for the case of solutions to general equation (8) by proving the following result.

### Theorem 1

*Consider equation*(8),

*with*\(\gamma \ne0\)

*and*

*Let*\((s_{n})_{n\in\mathbb{N} _{0}}\)

*be the solution to equation*

*satisfying the initial conditions*

*Then every well*-

*defined solution to equation*(8)

*has the following representation*:

*for*\(n\in\mathbb{N} _{0}\).

Using formulas (16) and (17) and some calculations it is easily proved that Theorem 1 also holds in the case when \(\gamma =0\) [62]. Bearing in mind this fact we see that the following somewhat more general result holds.

### 1.4 Problems treated in the paper

*v⃗*, \(x_{0}\), and \(x_{1}\).

Another aim of ours here is to give a positive answer to the question and to apply the obtained result in getting representations of solutions to equation (8) as well as to the corresponding representations for some bilinear systems of difference equations. Besides, we offer here an answer to an open problem circulating among some experts regarding the attribution of equation (8).

## 2 General representation of solutions to equation (8)

In this section we prove our main results. The first one is a generalization of Theorem 2.

### Theorem 3

*Let*\(\vec{v}=(v_{0}, v_{1})\in {\mathbb {R}}^{2}\) (

*or*\(\in {\mathbb {C}}^{2}\))

*and*\((s_{n}(\vec{v}))_{n\in\mathbb{N} _{0}}\)

*be the solution to equation*(1)

*such that*

*Then all the solutions to equation*(1)

*have representation*(29)

*if and only if*

*Further*,

*if*(31)

*holds*,

*then for every solution*\((x_{n})_{n \in\mathbb{N} _{0}}\)

*to equation*(1)

*the following representation holds*:

*for*\(n\in\mathbb{N} _{0}\).

### Proof

### Remark 1

*S̃*consisting of all \(\vec{v}\in {\mathbb {C}}^{2}\) such that (31) does not hold consists of the zero vector \(\vec{0}\in {\mathbb {C}}^{2}\), and of those

*v⃗*such that

By representation (32) a generalization of the representation of solutions to difference equation (8) given in (28) can be obtained. Namely, the following theorem holds.

### Theorem 4

*Consider equation*(8)

*with*\(\alpha \delta \ne \beta \gamma \).

*Assume that*\(\vec{v}=(v_{0}, v_{1})\in {\mathbb {R}}^{2}\) (

*or*\(\in {\mathbb {C}}^{2}\))

*satisfies the following condition*:

*Let*\((s_{n})_{n\in\mathbb{N} _{0}}\)

*be the solution to equation*(26)

*such that*\(s_{0}=v_{0}\)

*and*\(s_{1}=v_{1}\).

*Then every well*-

*defined solution*\((z_{n})_{n\in\mathbb{N} _{0}}\)

*to equation*(8)

*has the following representation*:

*for*\(n\in\mathbb{N} _{0}\).

### Proof

*δ*cannot be equal to zero so that equation (41) is defined).

*α*and

*δ*are the zeros of the characteristic polynomial

From (43) and (53) we see that (50) holds also in this case, finishing the proof of the theorem. □

### 2.1 On a bilinear system of difference equations

It is a natural problem to see if there is a representation similar to the one in Theorem 4 for the case of bilinear systems of difference equations.

### Theorem 5

*Consider system*(54)

*with*\((\alpha \delta -\beta \gamma )(ad-bc)\ne0\).

*Assume that*\(\vec{v}=(v_{0}, v_{1}) \in {\mathbb {R}}^{2}\) (

*or*\(\in {\mathbb {C}}^{2}\))

*satisfies the condition*

*Let*\((s_{n})_{n\in\mathbb{N} _{0}}\)

*be the solution to equation*(59)

*such that*\(s_{0}=v_{0}\)

*and*\(s_{1}=v_{1}\).

*Then every well*-

*defined solution*\((z_{n}, w_{n})_{n\in\mathbb{N} _{0}}\)

*to system*(54)

*has the following representation*:

*where*

*α̃*,

*β̃*,

*γ̃*,

*δ̃*,

*α̂*,

*β̂*,

*γ̂*,

*δ̂*

*are defined in*(56)

*and*(58).

### 2.2 On a three-dimensional generalization of equation (8)

Here we show that it is also possible to generalize Theorem 4 for the case of a three-dimensional close-to-cyclic system of bilinear difference equations which extends difference equation (8).

### Theorem 6

*Consider system*(65).

*Assume that*

*or*\(\in {\mathbb {C}}^{2}\))

*satisfies the condition*

*Let*\((s_{n})_{n\in\mathbb{N} _{0}}\)

*be the solution to equation*(81)

*such that*\(s_{0}=v_{0}\)

*and*\(s_{1}=v_{1}\).

*Then every well*-

*defined solution*\((z_{n}, w_{n}, u_{n})_{n\in\mathbb{N} _{0}}\)

*to system*(65)

*has the following representation*:

*for every*\(n\in\mathbb{N} _{0}\)

*and each*\(j=0,1,2\),

*where*

*ã*,

*b̃*,

*c̃*,

*d̃*,

*ẽ*,

*f̃*,

*g̃*,

*h̃*,

*p̃*,

*q̃*,

*r̃*,

*s̃*

*are defined by*(67)

*–*(70), (72)

*–*(75), (77)

*–*(80),

*respectively*.

### Remark 2

### Remark 3

It is time consuming but not difficult to check that the above procedure employed to the two-dimensional close-to-symmetric and three-dimensional close-to-cyclic bilinear systems of difference equations can be also applied to close-to-cyclic bilinear systems of difference equations of fourth order. However, since the procedure leads to some long calculations and quite long formulas, we also leave getting the corresponding formulas for solutions to the system to the interested reader.

## 3 On the name of equation (8)

As we have already mentioned, difference equations appeared in the form of recurrent relations/series in 1718 at the latest (before that time they usually had appeared, in a way, indirectly or descriptively). The methods by de Moivre were systematized and studied further by Euler and can be found in [7]. Later in 1759 Lagrange in [34] studied the “integrability” (i.e., solvability) of linear difference equations by modifying the methods which had been used in studying differential equations and essentially laid a cornerstone for further investigations.

There has been a recent custom that the bilinear difference equation is called Riccati difference equation. During the last two decades several colleagues, some of them who used the terminology, asked me if the attribution is correct, bearing in mind that it is known that there were not so many investigations on difference equations during the life of Riccati and that nobody has seen a paper written by him on the topic. The frequent question and recent studies of solvability of difference equations motivated me to conduct a considerable, but, of course, not thorough, literature investigation, to try to give a possible answer for the “open problem” in history of difference equations.

As far as we could see, the bilinear difference equation had not been investigated by Riccati at all, which is not so strange bearing in mind that he lived until 1754 and that a serious investigation of the solvability of difference equations practically started in Lagrange’s 1759 paper [34].

Reputable sources from the end of eighteenth century, which are almost of an encyclopedic character, such us Cousin’s book [63] (1796) and famous Lacroix book [64] (1800) did not mention Riccati in chapters devoted to difference equations at all. Moreover, Lacroix, besides already mentioned mathematicians and other French ones, mentioned in [64] several Italians, such as Brunacci, Paoli, and Malfatti, who had worked on difference equations. If we follow further the nineteenth century literature on difference equations all over Europe (see, for example, Lacroix [65] (1816), Schlömilch [66] (1848), Boole [20] (1880, 1st ed. 1860), Markoff [67, 68] (1896, 2nd ed. 1910)), we see that nobody mention Riccati at all. Although, for example, Boole devoted considerable space in his book to studying bilinear difference equations. It should be also mentioned that the elementary course on differential and integral calculus [65] by Lacroix (1816) considers the bilinear difference equation and solves it by using the change of variables (14), which means that already at the time the solvability of the equation was regarded as a matter of general mathematical culture, as something which is easily understandable. Nörlund in his known book [69] did not consider equation (8), but the book contains a huge number of references not mentioning Riccati at all.

The cherry on the cake could be the classroom note [21] by Brand (1955) published in a widely read popular journal, which solves the bilinear difference equation in the way as it was suggested by old French masters [6, 65] and describes the long-term behavior of its solutions. The note starts with: “The sequences defined by a difference equation of Riccati type”, but the note is devoted only to the bilinear difference equation, that is, to equation (8) (with constant coefficients). Interestingly [21] did not cite any paper except [16] because of the Kronecker lemma. Bearing in mind that the terminology is similar to the one in [24], we can suppose that Brand borrowed it from there. As we can see, Brand also did not attribute the bilinear difference equation to Riccati, but it seems this, combined with book [24], caused the chain reaction among some experts who started calling equation (8) by a wrong name, that is, a Riccati equation.

### Remark 4

Recall that there has been a related problem with solution (2) to equation (1), which has been frequently called Binet’s solution (we personally met the attribution for the first time in a Russian issue of book [12], which could have been simply another case of a wrong chain reaction). Nowadays it is well known that, in fact, formula (2) belongs to de Moivre. The formula and the method that leads to getting it can be also regarded as a cornerstone in the study of solvability of difference equations (one of several cornerstones by de Moivre, besides the formula \((\cos \varphi +i \sin \varphi )^{n}=\cos n\varphi +i\sin n\varphi \), and de Moivre–Laplace theorem).

## Notes

### Acknowledgements

The study in the paper is a part of the investigation under the projects III 41025 and III 44006 by the Serbian Ministry of Education and Science.

### Availability of data and materials

Not applicable.

### Authors’ contributions

The author has contributed solely to the writing of this paper. He read and approved the manuscript.

### Funding

Not applicable.

### Competing interests

The author declares that he has no competing interests.

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