# 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

## 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

## References

- Brand L:
**A sequence defined by a difference equation.***Am. Math. Mon.*1955, 62: 489-492. 10.2307/2307362MathSciNetView ArticleGoogle Scholar - Agarwal RP:
*Difference Equations and Inequalities*. 1st edition. Dekker, New York; 1992. (2nd ed., (2000))MATHGoogle Scholar - Gibbons CH, Kulenović MRS, Ladas G:On the recursive sequence ${x}_{n+1}=\frac{\alpha +\beta {x}_{n-1}}{\gamma +{x}_{n}}$.
*Math. Sci. Res. Hot-Line*2000, 4: 1-11.MathSciNetGoogle Scholar - Grove EA, Kostrov Y, Ladas G, Schultz SW:
**Riccati difference equations with real period-2 coefficients.***Commun. Appl. Nonlinear Anal.*2007, 14: 33-56.MathSciNetMATHGoogle Scholar - Taskara N, Uslu K, Tollu DT:
**The periodicity and solutions of the rational difference equation with periodic coefficients.***Comput. Math. Appl.*2011, 62: 1807-1813. 10.1016/j.camwa.2011.06.024MathSciNetView ArticleMATHGoogle Scholar - Cinar C:On the positive solutions of the difference equation ${x}_{n+1}={x}_{n-1}/(1+{x}_{n}{x}_{n-1})$.
*Appl. Math. Comput.*2004, 150: 21-24. 10.1016/S0096-3003(03)00194-2MathSciNetView ArticleMATHGoogle Scholar - Papaschinopoulos G, Papadopoulos BK:On the fuzzy difference equation ${x}_{n+1}=A+\frac{B}{{x}_{n}}$.
*Soft Comput.*2002, 6: 456-461. 10.1007/s00500-001-0161-7View ArticleMathSciNetMATHGoogle Scholar - Elabbasy EM, El-Metwally HA, Elsayed EM:
**Global behavior of the solutions of some difference equations.***Adv. Differ. Equ.*2011., 2011: Article ID 28 10.1186/1687-1847-2011-28Google Scholar - Elsayed EM:
**Solution and attractivity for a rational recursive sequence.***Discrete Dyn. Nat. Soc.*2011., 2011: Article ID 982309Google Scholar - Elsayed EM:
**On the solution of some difference equations.***Eur. J. Pure Appl. Math.*2011, 4: 287-303.MathSciNetGoogle Scholar - Elsayed EM:
**Solutions of rational difference system of order two.***Math. Comput. Model.*2012, 55: 378-384. 10.1016/j.mcm.2011.08.012MathSciNetView ArticleMATHGoogle Scholar - Touafek N, Elsayed EM:
**On the solutions of systems of rational difference equations.***Math. Comput. Model.*2012, 55: 1987-1997. 10.1016/j.mcm.2011.11.058MathSciNetView ArticleMATHGoogle Scholar - Elsayed EM:
**Behavior and expression of the solutions of some rational difference equations.***J. Comput. Anal. Appl.*2013, 15(1):73-81.MathSciNetMATHGoogle Scholar - El-Metwally H, Elsayed EM:
**Solution and behavior of a third rational difference equation.***Util. Math.*2012, 88: 27-42.MathSciNetMATHGoogle Scholar - Koshy T:
*Fibonacci and Lucas Numbers with Applications*. Wiley, New York; 2001.View ArticleMATHGoogle Scholar - Vajda S:
*Fibonacci and Lucas Numbers, and the Golden Section: Theory and Applications*. Dover, New York; 2007.MATHGoogle Scholar - El Naschie MS:
**The golden mean in quantum geometry, Knot theory and related topics.***Chaos Solitons Fractals*1999, 10(8):1303-1307. 10.1016/S0960-0779(98)00167-2MathSciNetView ArticleMATHGoogle Scholar - Marek-Crnjac L:
**On the mass spectrum of the elementary particles of the standard model using El Naschie’s golden field theory.***Chaos Solitons Fractals*2003, 15(4):611-618. 10.1016/S0960-0779(02)00174-1View ArticleMATHGoogle Scholar - Falcon S, Plaza A:
**The metallic ratios as limits of complex valued transformations.***Chaos Solitons Fractals*2009, 41: 1-13. 10.1016/j.chaos.2007.11.011MathSciNetView ArticleMATHGoogle Scholar

## 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.