# Some representations of the general solution to a difference equation of additive type

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## Abstract

## Keywords

Third-order difference equation Solvable difference equation Linear difference equation Representation of solutions## MSC

39A20 39A06 39A45## 1 Introduction

Regarding some notations which are used in this paper, say that we denote the sets of positive, nonnegative, integer, and complex numbers by \({\mathbb {N}}\), \({\mathbb {N}}_{0}\), \({\mathbb {Z}}\), and \({\mathbb {C}}\), respectively. If \(l_{1},l_{2}\in {\mathbb {Z}}\), \(l_{1}\le l_{2}\), then the notation \(l=\overline {l_{1},l_{2}}\), which is frequently used throughout the paper, denotes the set of all \(l\in {\mathbb {Z}}\) satisfying the inequalities \(l_{1}\le l\le l_{2}\). We also use the standard convention \(\prod_{j=k}^{k-1}a_{j}=1\), \(k\in {\mathbb {Z}}\).

Solvability of equation (1) of the corresponding nonhomogeneous linear difference equation, as well as of the nonhomogeneous linear first-order difference equation whose coefficients can be nonconstant, is frequently used in pure and applied mathematics as well as in other branches of science. For some recently studied classes of nonlinear difference equations and systems of nonlinear difference equations which have been solved by using one of these equations, see, e.g., [12, 13, 14, 15, 16, 17]. For some recent applications of their solvability, see, e.g., [18, 19, 20]. For example, by using solvability methods in [19] and [20], some classes of difference equations are transformed to “integral” forms, which are usually more suitable for studying the existence of bounded or periodic solutions to difference equations than the equations in their original forms.

Employing various changes of variables, many classes of nonlinear difference equations can be transformed to special cases of equation (1), from which solvability of the nonlinear equations can follow.

For a method which uses an associated two-dimensional system of difference equations in solving equation (2), see, e.g., [7] or [26]. For some other results on the difference equation and for some applications of the equation, see, e.g., [3, 12, 25, 27, 28, 29, 30].

### Remark 1

This should be the main reason why Brunacci tried to solve equation (7) by using the change of variables (8). It is also interesting to note that in this way Brunacci “solved” equation (7) in terms of a product of continuous fractions, which is, in fact, a quite complicated representation of general solution to the difference equation.

Our further investigations on solvability of difference equations and systems of difference equations have shown that these solutions, the solutions to the linear difference equations with constant coefficients of the third order such that \(x_{-2}=x_{-1}=0\) and \(x_{0}=1\), as well as the solutions to the linear difference equations with constant coefficients of the fourth order such that \(x_{-3}=x_{-2}=x_{-1}=0\) and \(x_{0}=1\), naturally appear in representations of general solutions to several other classes of difference equations and systems of difference equations. For example, they appear in papers [25, 28, 29, 30] predominately connected to the bilinear difference equations, as well as in the representations of general solutions to the product-type difference equations and systems of difference equations, e.g., in [15, 33, 34, 35, 36, 37, 38, 39] (see also the references therein).

For some other results on solvability of difference equations and systems of difference equations, related topics such as invariants for difference equations and systems, and their applications, see, e.g., [3, 5, 6, 7, 9, 10, 23, 26, 32, 40, 41, 42, 43, 44, 45, 46, 47].

Motivated, among other things, by some results in [15, 25, 33, 34, 35, 36, 37, 38, 39], recently in [48] we have proved the following theorem extending and theoretically explaining a recent result in the literature.

### Theorem 1

*Let*\(a, b\in {\mathbb {C}}\), \(c\in {\mathbb {C}}\setminus \{0\}\),

*and let*\((t_{n})_{n\ge-2}\)

*be the solution to equation*(14)

*such that*

*Then every well*-

*defined solution to equation*(12)

*has the following representation*:

*for*\(n\ge-1\).

The following cases are considered separately: (1) all the roots of the polynomial are distinct; (2) there is a unique double root of the polynomial; (3) there is a triple root of the polynomial and one simple; (4) there is a quadruple root of the polynomial; (5) there are two distinct double roots of the polynomial.

## 2 Main results

The main results in this paper are stated and proved in this section.

### 2.1 A representation of general solution to equation (17)

Our first result concerns the problem of representing the general solution to equation (17) in terms of the parameters *a*, *b*, *c*, *d*, initial values, and a specially chosen solution to a linear difference equation with constant coefficients of the fourth order associated with the equation. It complements known representations of the general solutions to the corresponding difference equation of the first order (i.e., of equation (5)), as well as of the second order (i.e., of equation (12)).

### Theorem 2

*Let*\(a, b, c\in {\mathbb {C}}\), \(d\in {\mathbb {C}}\setminus\{0\}\),

*and*\((s_{n})_{n\ge-2}\)

*be the solution to the equation*

*such that*

*Then every well*-

*defined solution to equation*(17)

*has the following representation*:

*for*\(n\ge-2\).

### Proof

We consider the following four cases: (1) \(a\ne0\), (2) \(a=0\), \(b\ne0\), (3) \(a=b=0\), \(c\ne0\), (4) \(a=b=c=0\), separately.

*Case*\(a\ne0\)*.* The first part of the proof in this case was essentially presented in [49] (see also [36, 38]). Hence, we will present only essential details for the completeness and benefit of the reader.

First note that, by employing (13) in equation (17), it is transformed to equation (19).

*n*is replaced by \(n-2\), in relation (19), where

*n*is replaced by \(n-1\), and using (22), we have

*n*is replaced by \(n-l-1\) in (25), it follows that

*Case*\(a=0\)

*,*\(b\ne0\)

*.*Since \(a=0\), we have

*n*is replaced by \(n-3\), in equality (37), where

*n*is replaced by \(n-1\), we obtain

*n*is replaced by \(n-l-2\) in (41), we have

*Case*\(a=b=0\ne c\)

*.*Since \(a=b=0\), equation (19) is

*n*is replaced by \(n-4\), in equality (52), where

*n*is replaced by \(n-1\), we obtain

*n*is replaced by \(n-l-3\) in (56), we have

From (54), (55), (58), (59), and the induction it follows that (56) and (57) hold for \(l,n\in {\mathbb {N}}\) such that \(2\le l\le n-2\).

*Case*\(a=b=c=0\ne d\)

*.*Since \(a=b=c=0\), equation (19) is

### Remark 2

The case \(a\ne0\) has been essentially treated in [49]. However, the proofs in the other cases, in all of which is \(a=0\), have been omitted there; although, as it is seen in the proof of Theorem 2, they follow the idea of the proof in the case \(a\ne0\). Hence, the proof of Theorem 2 can be regarded as a completion of the proof of Theorem 6 in [49] in the case \(a=0\).

### 2.2 About the roots of polynomial \(P_{4}(\lambda )\)

Here we present the roots of polynomial \(P_{4}\) and some facts related to them.

First, we formulate a well-known lemma, whose proofs can be found, e.g., in [26, 38, 50].

### Lemma 1

*Let*\(t_{j}\), \(j=\overline {1,k}\)

*be the roots of the polynomial*

*such that*\(t_{i}\ne t_{j}\), \(i\ne j\),

*then*

*s*so that

Multiplicity of the zeros of the polynomial \(P_{4}\) are described in the following lemma (see, e.g., [51]).

### Lemma 2

*Let polynomial*\(P_{4}\)*be defined in* (18), \(\Delta _{0}\)*in* (82), \(\Delta _{1}\)*in* (83), *Q**in* (84), *and let*\(\Delta =\frac{1}{27}(4\Delta _{0}^{3}-\Delta _{1}^{2})\), \(P=-8b-3a^{2}\), *and*\(D=-64d-16b^{2}-16a^{2}b-16ac-3a^{4}\).

*Then the following statements hold*:

- (a)
*If*\(\Delta \ne0\),*then the zeros of*\(P_{4}\)*are distinct*. - (b)
*If*\(\Delta =0\)*and*\(P<0\), \(D<0\), \(\Delta _{0}\ne0\),*or*\(D>0\),*or*\(P>0\), \(D\ne0\),*or*\(P>0\), \(Q\ne0\),*then exactly two zeros of*\(P_{4}\)*are equal*. - (c)
*If*\(\Delta=0\), \(\Delta _{0}=0\),*and*\(D\ne0\),*then there is a triple zero of*\(P_{4}\)*and one simple*. - (d)
*If*\(\Delta=0\), \(D=0\),*and if*\(P<0\),*or*\(P>0\)*and*\(Q=0\),*then*\(P_{4}\)*has two distinct double zeros*. - (e)
*If*\(\Delta=0\), \(D=0\),*and*\(\Delta _{0}=0\),*then all the zeros of*\(P_{4}\)*are equal to*\(a/4\).

### 2.3 Possible forms of the sequence \(s_{n}\)

While formula (21) gives a representation of the general solution to equation (17) in terms of the sequence \(s_{n}\), it does not tell anything about its possible forms. In this section we will describe the forms in terms of the roots \(\lambda _{j}\), \(j=\overline {1,4}\), of polynomial (18).

*Case*\(\Delta \ne0\)

*.*In this case, by Lemma 2(a), we have that the roots \(\lambda _{j}\), \(j=\overline {1,4}\), are distinct. By using Lemma 1 and the general solution to equation (19) in this case, we have proved in [38] (see also [36]) that the following formula holds:

*ε*is a complex root of the equation \(t^{3}=1\), from which along with (87) it follows that

From the above analysis, we get the following corollary.

### Corollary 1

*Consider equation*(17)

*with*\(a,b,c\in {\mathbb {C}}\)

*and*\(d\in {\mathbb {C}}\setminus\{0\}\).

*Assume that*\(\Delta\ne 0\).

*Then the following statements are true*.

- (a)
*If*\(a+b+c+d\ne1\),*then the general solution to equation*(17)*is given by*(21),*where sequence*\((s_{n})_{n\ge-3}\)*is given by*(85),*whereas*\(\lambda _{j}\), \(j=\overline {1,4}\),*are given by*(78)*–*(81). - (b)
*If*\(a+b+c+d=1\),*then the general solution to equation*(17)*is given by*(21),*where sequence*\((s_{n})_{n\ge-3}\)*is given by*(85)*where*\(\lambda _{1}=1\), \(\lambda _{j}\), \(j=\overline {2,4}\),*are given by*(89).

### Remark 3

*Case*\(\Delta =0\)

*and*\(P<0\)

*,*\(D<0\)

*,*\(\Delta _{0}\ne0\)

*, or*\(D>0\)

*, or*\(P>0\)

*,*\(D\ne0\)

*, or*\(P>0\)

*,*\(Q\ne0\)

*.*By Lemma 2(b) in this case \(P_{4}\) has exactly one double zero. We may assume \(\lambda _{1}=\lambda _{2}\ne \lambda _{j}\), \(j=3,4\). To find the solution satisfying (20), we let \(\lambda _{1}\to \lambda _{2}\) in (85) and get

### Corollary 2

*Consider equation*(17)

*with*\(a,b,c\in {\mathbb {C}}\)

*and*\(d\in {\mathbb {C}}\setminus\{0\}\).

*Assume that*\(\Delta =0\)

*and*\(P<0\), \(D<0\), \(\Delta _{0}\ne0\),

*or*\(D>0\),

*or*\(P>0\), \(D\ne0\),

*or*\(P>0\), \(Q\ne0\).

*Then the following statements hold*.

- (a)
*If*\(3a+2b+c\ne4\),*then the general solution to equation*(17)*is given by formula*(21),*where sequence*\((s_{n})_{n\ge-3}\)*is given by*(94). - (b)
*If*\(a+b+c+d-1=4-3a-2b-c=0\),*then the general solution to equation*(17)*is given by formula*(21),*where sequence*\((s_{n})_{n\ge-3}\)*is given by*(95),*whereas*\(\lambda _{3,4}\)*are given by*(97).

*Case*\(\Delta=0\)

*,*\(D=0\)

*, and*\(P<0\)

*, or*\(P>0\)

*and*\(Q=0\)

*.*By Lemma 2(d) in this case \(P_{4}\) has two pairs of double zeros. By letting \(\lambda _{3}\to \lambda _{4}\) in (94), it is obtained

### Corollary 3

*Consider equation*(17)

*with*\(a,b,c\in {\mathbb {C}}\)

*and*\(d\in {\mathbb {C}}\setminus\{0\}\).

*Assume that*\(\Delta =0\), \(D=0\),

*and*\(P<0\),

*or*\(P>0\)

*and*\(Q=0\).

*Then the following statements hold*.

- (a)
*If*\(a+b+c+d\ne1\),*then the general solution to equation*(17)*is given by formula*(21),*where sequence*\((s_{n})_{n\ge-3}\)*is given by*(98). - (b)
*If*\(a+b+c+d=1\),*then the general solution to equation*(17)*is given by formula*(21),*where sequence*\((s_{n})_{n\ge-3}\)*is given by*(99),*whereas*\(\lambda _{3,4}\)*are given by*(100).

*Case*\(\Delta=0\)

*,*\(\Delta _{0}=0\)

*, and*\(D\ne0\)

*.*By Lemma 2(c) we have that \(P_{4}\) has a triple zero and one simple. We may assume that \(\lambda _{1}=\lambda _{2}=\lambda _{3}\). In [35] we proved that in this case

### Corollary 4

*Consider equation*(17)

*with*\(a,b,c\in {\mathbb {C}}\)

*and*\(d\in {\mathbb {C}}\setminus\{0\}\).

*Assume that*\(\Delta =0\), \(\Delta _{0}=0\),

*and*\(D\ne0\).

*Then the following statements hold*.

- (a)
*If*\(3a+2b+c\ne4\),*then the general solution to equation*(17)*is given by formula*(21),*where sequence*\((s_{n})_{n\ge-3}\)*is given by*(101). - (b)
*If*\(a+b+c+d=1\)*and*\(3a+2b+c=4\),*then the general solution to equation*(17)*is given by formula*(21),*where sequence*\((s_{n})_{n\ge-3}\)*is given by*(102),*whereas*\(\lambda _{4}\)*is given by*(103).

*Case*\(\Delta=0\)*,*\(D=0\)*, and*\(\Delta _{0}=0\)*.* By Lemma 2(e) we have that all the zeros to \(P_{4}\) are equal to \(a/4\).

## Notes

### Acknowledgements

The study in the paper is a part of the investigations 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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