# Two ways for solving a class of rational second-order difference equations

## Abstract

We consider a class of second-order rational difference equations with two parameters, and we show, in two ways, that the class of equations is solvable in closed form. One way is based on using an invariant for the class of equations, whereas the second one is based on the method of substitution. The main results extend and complement some previous ones in the literature.

## Keywords

Difference equation Solvable equation Invariant Method of substitution## MSC

39A20 39A06 39A45## 1 Introduction

We use the following standard notations \({\mathbb {N}}\), \({\mathbb {N}}_{0}\), \({\mathbb {Z}}\), \({\mathbb {R}}\), \({\mathbb {C}}\), for natural, nonnegative, whole, real and complex numbers, respectively.

Methods and ideas by de Moivre were later developed by Euler ([3]). The study was continued by Lagrange (see, e.g., [4, 5]), Laplace (see, e.g., [6]), and many other scientists. Several methods for solving some classes of nonlinear difference equations in closed form had been known to Laplace yet ([6]).

*α*,

*β*,

*γ*,

*δ*are given numbers. Let us mention that it is usually assumed that \(\alpha \delta -\beta \gamma \ne 0\ne \gamma \) to avoid dealing with the trivial and linear equations obtained from Eq. (3) (the equation with \(\gamma =0\) was solved by Lagrange in [4]; moreover, the first-order linear difference equation with nonconstant coefficients was also solved therein). For more historical details on some classical difference equations including bilinear one, see [7]. Hence, Eq. (3) is regarded as one of the basic solvable nonlinear difference equations, and can be found in many books ([8, 9, 10, 11, 12, 13, 14]; see also [15]). The equation frequently appears in papers on the solvability of difference equations (see, e.g., the recent papers [7, 16, 17, 18, 19]).

Nowadays, a part of the scientific community seems uses only computers and symbolic algebra for getting some results on solvability of various equations, including difference ones. This causes several problems, including some on originality of such obtained results. In some of our papers, among other things, we have discussed such problems (see, e.g., [7, 16, 17, 18, 19, 20, 21, 22]). Many of the known solvable difference equations and systems can be transformed to linear ones, which are solvable (for solvability of linear difference equations see, e.g., [8, 9, 14]). This motivated some authors to apply and develop the method of transformation/substitution (see, e.g., [20, 21, 22, 23, 24, 25, 26, 27, 28, 29]). Many applications of solvable difference equations, can be found in the literature which shows their importance (see, e.g., [2, 5, 6, 9, 10, 11, 12, 13, 14, 30, 31, 32, 33, 34, 35, 36]). For some related results see also [37, 38, 39, 40, 41, 42].

Now we give a formal definition of solvability of a difference equation.

### Definition 1

We say that a difference equation is *solvable in closed form* if there is a finite number of closed-form formulas from which any well-defined solution to the equation can be obtained.

### Theorem 1

*Consider the difference equation*

*where*\(d\in {\mathbb {C}}\setminus \{0\}\),

*and*\(x_{-1}, x_{0}\in {\mathbb {C}}\).

*Then the general solution to Eq*. (6)

*is given by the following formula*:

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

Invariants for difference equations can be useful in their investigations. But it is usually difficult to find those which can be of any use. Invariants for several classes of difference equations and systems of difference equations can be found, e.g., in [45, 46, 47, 48, 49, 50] (see also the related references therein). They can be useful in finding general solutions to some classes of difference equations, although it is not quite a common situation. Some examples of difference equations which are solved by using invariants can be found in [44, 51] and [52].

Now we define notion of being an invariant for a difference equation.

### Definition 2

*c*, such that

*I*is an

*invariant*for Eq. (8).

*a*and

*d*, and initial values \(x_{-1}\) and \(x_{0}\) are given numbers, can also be solved in closed form, extending and complementing the main result in [44].

We prove this by using two methods. First, this is proved by using an invariant for the difference equation. Since in this way is obtained a closed-form formula for general solution to the equation, this suggests that it might be some other, more standard, methods for getting the formula. Bearing in mind some of our previous investigations on solvability of difference equations and systems of difference equations (see, e.g., [20, 21, 22, 24, 25, 26, 27, 28, 29] and number of related references therein), the first natural choice is the method of substitution. Employing some suitably chosen changes of variables we also show the solvability of Eq. (9), confirming the guess.

## 2 Main results

Our main results are stated and proved in this section. As we have already mentioned, here we present two methods for solving difference equation (9). The first one essentially uses an invariant for the equation together with a nice trick, while the second one is based on the method of substitution where a chain of substitutions is used to get general solution to the equation.

### 2.1 Solving Eq. (9) by using an invariant

### 2.2 Solving Eq. (9) by the method of substitution

Our first idea is to write Eq. (9) in the form of a product-type difference equations, which along with systems of product-type difference equations have been considerably studied (see, e.g., [28, 53, 54, 55] and the related references therein).

There are two cases to be considered: (1) \(a=1\); (2) \(a\ne 1\).

*Case*\(a=1\)

*.*In this case Eq. (45) becomes

It is easy to see that Eqs. (56) and (57) match with Eq. (7), which shows that the method also leads to finding the general solution to Eq. (9) in this case.

*Case*\(a\ne 1\)

*.*By using the change of variables (47) in Eq. (45) we obtain

It is easy to see that Eqs. (63) and (64) match with Eq. (24) (see also (29) and (31)), which shows that the method of substitution also leads to finding a general solution to Eq. (9) in this case.

### Remark 1

Hence, a well-defined solution to Eq. (9) satisfy Eqs. (9) and (40) simultaneously.

## 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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