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

- Durhasan T Tollu
^{1}, - Yasin Yazlik
^{2}Email author and - Necati Taskara
^{3}

**2013**:174

https://doi.org/10.1186/1687-1847-2013-174

© Tollu et al.; licensee Springer 2013

**Received:**26 April 2013**Accepted:**28 May 2013**Published:**18 June 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.

## Keywords

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

## 1 Introduction

*et al.*obtained the Fibonacci sequence in solutions of some special cases of the following difference equation

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

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*).

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.

*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

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.

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).

(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

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

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.

## Declarations

## Authors’ Affiliations

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## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.