- Open Access
On the solutions of two special types of Riccati difference equation via Fibonacci numbers
© Tollu et al.; licensee Springer 2013
- Received: 26 April 2013
- Accepted: 28 May 2013
- Published: 18 June 2013
In this study, we investigate the solutions of two special types of the Riccati difference equation and such that their solutions are associated with Fibonacci numbers.
MSC: 11B39, 39A10, 39A13.
- Riccati difference equation
- equilibrium point
- Fibonacci numbers
- golden ratio
Also, he gives specific forms of the solutions of four special cases of this equation. These specific forms also contain Fibonacci numbers.
Let us consider the following lemma which will be needed for the results in this study.
Lemma 1 
The following equalities hold:
(i) For , and , .
(ii) For , and .
(iii) For , (Cassini’s formula).
where initial conditions are and , respectively, and 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).
Firstly, it is not difficult to prove that equilibrium points of equations (7) and (8) are , and , , respectively, where is the golden ratio and is the conjugate of α. Note that one of the equilibrium points of equation (8) is the golden ratio.
Theorem 1 For , the solutions of equations (7) and (8) are as follows:
(i) For , .
(ii) For , .
Proof Firstly, in here we will just prove (ii) since (i) can be thought in the same manner.
which ends the induction and the proof. □
Theorem 2 Let the solutions of equations (7) and (8) be and , respectively and . Therefore, is satisfied if and only if the initial conditions are .
which is desired. □
Theorem 3 The following statements hold:
(i) For the initial condition (or ), equation (7) has the fixed solution (or ).
(ii) For the initial condition (or ), equation (8) has the fixed solution (or ).
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 , all the solutions of equation (7) converge to −β, where . That is, .
(ii) For , all the solutions of equation (8) converge to β, where . That is, .
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). □
Consequently, the proof is completed. □
The following theorem establishes that the Fibonacci numbers can be obtained by using the solutions of (7).
from which the result follows. □
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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