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# On the solutions of two special types of Riccati difference equation via Fibonacci numbers

*Advances in Difference Equations*
**volume 2013**, Article number: 174 (2013)

## Abstract

In this study, we investigate the solutions of two special types of the Riccati difference equation ${x}_{n+1}=\frac{1}{1+{x}_{n}}$ and ${y}_{n+1}=\frac{1}{-1+{y}_{n}}$ such that their solutions are associated with Fibonacci numbers.

**MSC:** 11B39, 39A10, 39A13.

## 1 Introduction

Nonlinear difference equations have long interested researchers in the field of mathematics as well as in other sciences. They play a key role in many applications such as the natural model of a discrete process. There have been many recent investigations and interest in the field of nonlinear difference equations by several authors [1–14]. For example, in [1], Brand defined a sequence which stems from the Riccati difference equation

In [6], Cinar studied the solution of the difference equation

In [7], Papaschinopoulos and Papadopoulos studied the fuzzy difference equation

which is a special case of the Riccati difference equation. In [8], Elabbasy *et al.* obtained the Fibonacci sequence in solutions of some special cases of the following difference equation

In [9], the author deals with behavior of the solution of the nonlinear difference equation

Also, he gives specific forms of the solutions of four special cases of this equation. These specific forms also contain Fibonacci numbers.

Fibonacci numbers have been interesting to the researchers for a long time to get the main theory and applications of these numbers. For instance, the ratio of two consecutive Fibonacci numbers converges to the golden section $\alpha =\frac{1+\sqrt{5}}{2}$. The applications of the golden ratio appear in many research areas, particularly in physics, engineering, architecture, nature and art. Physicists Naschie and Marek-Crnjac gave some examples of the golden ratio in theoretical physics and physics of high energy particles [15–19]. We should recall that the Fibonacci sequence ${\{{F}_{n}\}}_{n=0}^{\mathrm{\infty}}$ has been defined by the recursive equation

with initial conditions ${F}_{0}=0$, ${F}_{1}=1$. Also, it is obtained to extend the Fibonacci sequence backward as

One can clearly obtain the characteristic equation of (1) as the form ${x}^{2}-x-1=0$ such that the roots

Hence the Binet formula for Fibonacci numbers

can be thought of as a solution of the recursive equation in (1). Also, the following ratio is satisfied:

where $r\in \mathbb{Z}$.

Let us consider the following lemma which will be needed for the results in this study.

**Lemma 1** [15]

*The following equalities hold*:

(i) *For* $n>k+1$, $n\in {\mathbb{N}}^{+}$ *and* $k\in \mathbb{N}$, ${F}_{n}={F}_{k+1}{F}_{n-k}+{F}_{k}{F}_{n-(k+1)}$.

(ii) *For* $n>0$, ${\alpha}^{n}=\alpha {F}_{n}+{F}_{n-1}$ *and* ${\beta}^{n}=\beta {F}_{n}+{F}_{n-1}$.

(iii) *For* $n>0$, ${F}_{n-1}{F}_{n+1}-{F}_{n}^{2}={(-1)}^{n}$ (*Cassini’s formula*).

In this study, we consider the Riccati difference equation

Obviously, by taking $a=c=d=1$, $b=0$ and $a=d=1$, $c=-1$, $b=0$, equation (6), respectively, is transformed into the following equations:

where initial conditions are ${x}_{0}\in \mathbb{R}-{\{-\frac{{F}_{m+1}}{{F}_{m}}\}}_{m=1}^{\mathrm{\infty}}$ and ${y}_{0}\in \mathbb{R}-{\{\frac{{F}_{m+1}}{{F}_{m}}\}}_{m=1}^{\mathrm{\infty}}$, respectively, and ${F}_{m}$ is the *m* th Fibonacci number.

The aim of this study is to investigate some relationships both between Fibonacci numbers and solutions of equations (7) and (8) and between the golden ratio and equilibrium points of equations (7) and (8).

## 2 Main results

Firstly, it is not difficult to prove that equilibrium points of equations (7) and (8) are ${\overline{x}}_{1}=-\beta $, ${\overline{x}}_{2}=-\alpha $ and ${\overline{y}}_{1}=\alpha $, ${\overline{y}}_{2}=\beta $, respectively, where $\alpha =\frac{\sqrt{5}+1}{2}$ is the golden ratio and $\beta =\frac{1-\sqrt{5}}{2}$ is the conjugate of *α*. Note that one of the equilibrium points of equation (8) is the golden ratio.

**Theorem 1** *For* $n=0,1,2,\dots $ , *the solutions of equations* (7) *and* (8) *are as follows*:

(i) *For* ${x}_{0}\in \mathbb{R}-(\{\frac{1}{\alpha},\frac{1}{\beta}\}\cup {\{-\frac{{F}_{m+1}}{{F}_{m}}\}}_{m=1}^{\mathrm{\infty}})$, ${x}_{n}=\frac{{F}_{n}+{F}_{n-1}{x}_{0}}{{F}_{n+1}+{F}_{n}{x}_{0}}$.

(ii) *For* ${y}_{0}\in \mathbb{R}-(\{\alpha ,\beta \}\cup {\{\frac{{F}_{m+1}}{{F}_{m}}\}}_{m=1}^{\mathrm{\infty}})$, ${y}_{n}=\frac{{F}_{-n}+{F}_{-(n-1)}{y}_{0}}{{F}_{-(n+1)}+{F}_{-n}{y}_{0}}$.

*Proof* Firstly, in here we will just prove (ii) since (i) can be thought in the same manner.

(ii) We will prove this theorem by induction. For $k=0$,

Now assume that

is true for all positive integers *k*. Therefore, we have to show that it is true for $k+1$. Taking into account (2) and (9), we write

which ends the induction and the proof. □

**Theorem 2** *Let the solutions of equations* (7) *and* (8) *be* ${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}$ *and* ${\{{y}_{n}\}}_{n=0}^{\mathrm{\infty}}$, *respectively and* ${x}_{0}\in \mathbb{R}-{\{-\frac{{F}_{m+1}}{{F}_{m}}\}}_{m=1}^{\mathrm{\infty}}$. *Therefore*, ${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}={\{-{y}_{n}\}}_{n=0}^{\mathrm{\infty}}$ *is satisfied if and only if the initial conditions are* ${x}_{0}=-{y}_{0}$.

*Proof* First, assume that ${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}={\{-{y}_{n}\}}_{n=0}^{\mathrm{\infty}}$. Taking into account (2), we can write

By using simple mathematical operations and the well-known Cassini’s formula for Fibonacci numbers, we have

Second, assume that ${x}_{0}=-{y}_{0}$. By considering the solutions of equation (7), we get

which is desired. □

**Theorem 3** *The following statements hold*:

(i) *For the initial condition* ${x}_{0}=\frac{1}{\alpha}$ (*or* ${x}_{0}=\frac{1}{\beta}$), *equation* (7) *has the fixed solution* ${x}_{n}=\frac{1}{\alpha}$ (*or* ${x}_{n}=\frac{1}{\beta}$).

(ii) *For the initial condition* ${y}_{0}=\alpha $ (*or* ${y}_{0}=\beta $), *equation* (8) *has the fixed solution* ${y}_{n}=\alpha $ (*or* ${y}_{n}=\beta $).

*Proof* Here we will just prove (i) since the proof of (ii) can be done quite similarly.

(i) Firstly, let ${x}_{0}=\frac{1}{\alpha}=\frac{\sqrt{5}-1}{2}$ be the initial condition of equation (7). Then, by using Lemma 1(ii), we have

Secondly, let ${x}_{0}=\frac{1}{\beta}=-\frac{\sqrt{5}+1}{2}$ be the initial condition of equation (7). Then, by considering Lemma 1(ii), we obtain

which is desired. □

**Theorem 4** *The following statements hold*:

(i) *For* ${x}_{0}\in \mathbb{R}-(\{\frac{1}{\beta}\}\cup {\{-\frac{{F}_{m+1}}{{F}_{m}}\}}_{m=1}^{\mathrm{\infty}})$, *all the solutions of equation* (7) *converge to* −*β*, *where* $\beta =\frac{1-\sqrt{5}}{2}$. *That is*, $\underset{n\to \mathrm{\infty}}{lim}{x}_{n}=-\beta $.

(ii) *For* ${y}_{0}\in \mathbb{R}-(\{\alpha \}\cup {\{\frac{{F}_{m+1}}{{F}_{m}}\}}_{m=1}^{\mathrm{\infty}})$, *all the solutions of equation* (8) *converge to* *β*, *where* $\beta =\frac{1-\sqrt{5}}{2}$. *That is*, $\underset{n\to \mathrm{\infty}}{lim}{y}_{n}=\beta $.

*Proof* To prove, we use the solutions of (7) and (8).

(i) By using Theorem 1(i), we can write

Thus, from (5), we have

(ii) The proof can be seen easily in a similar manner to Theorem 4(i). □

**Theorem 5** *Let* ${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}$ *be the solution of* (7). *Then*, *we have*

*Proof* For ${x}_{0}={F}_{0}$, the result is trivial. If ${x}_{0}\ne {F}_{0}$, by Theorem 1, then we can write

By multiplying both sides of the above equalities, we obtain

Letting $n\to \mathrm{\infty}$, the last equality gives the following result

Consequently, the proof is completed. □

The following theorem establishes that the Fibonacci numbers can be obtained by using the solutions of (7).

**Theorem 6** *Let the initial condition of equation* (7) *be* ${x}_{0}=\frac{{F}_{k}}{{F}_{k+1}}$, *where* ${F}_{k}$ *is the* *kth Fibonacci number*. *For* $n>k+1$ *and* $k,n\in {\mathbb{Z}}^{+}$, *we have*

*Proof* Firstly, taking $n-(k+1)$ instead of *n* in (10), we obtain

Secondly, dividing both sides in (11) with ${x}_{0}$, we get

Finally, by considering Lemma 1(i), we obtain

from which the result follows. □

## 3 Conclusion

In this study, we mainly obtained the relationship between the solutions of Riccati difference equations (given in (7), (8)) and Fibonacci numbers. We also presented that the nontrivial solutions of equations in (7) and (8) actually converge to −*β* and *β*, respectively, so that *β* is conjugate to the golden ratio. We finally note that the results in this paper are given in terms of Fibonacci numbers.

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Tollu, D.T., Yazlik, Y. & Taskara, N. On the solutions of two special types of Riccati difference equation via Fibonacci numbers.
*Adv Differ Equ* **2013, **174 (2013). https://doi.org/10.1186/1687-1847-2013-174

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### Keywords

- Riccati difference equation
- equilibrium point
- Fibonacci numbers
- golden ratio