Research

# On Fibonacci functions with Fibonacci numbers

Jeong Soon Han1, Hee Sik Kim2* and Joseph Neggers3

Author Affiliations

1 Department of Applied Mathematics, Hanyang University, Ahnsan, 426-791, Korea

2 Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul, 133-791, Korea

3 Department of Mathematics, University of Alabama, Tuscaloosa, AL, 35487-0350, USA

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Advances in Difference Equations 2012, 2012:126  doi:10.1186/1687-1847-2012-126

 Received: 16 May 2012 Accepted: 11 July 2012 Published: 25 July 2012

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.

### Abstract

In this paper we consider Fibonacci functions on the real numbers R, i.e., functions such that for all , . We develop the notion of Fibonacci functions using the concept of f-even and f-odd functions. Moreover, we show that if f is a Fibonacci function then .

MSC: 11B39, 39A10.

##### Keywords:
Fibonacci function; f-even (f-odd) function; Golden ratio

### 1 Introduction

Fibonacci numbers have been studied in many different forms for centuries and the literature on the subject is, consequently, incredibly vast. One of the amazing qualities of these numbers is the variety of mathematical models where they play some sort of role and where their properties are of importance in elucidating the ability of the model under discussion to explain whatever implications are inherent in it. The fact that the ratio of successive Fibonacci numbers approaches the Golden ratio (section) rather quickly as they go to infinity probably has a good deal to do with the observation made in the previous sentence. Surveys and connections of the type just mentioned are provided in [1] and [2] for a very minimal set of examples of such texts, while in [6] an application (observation) concerns itself with a theory of a particular class of means which has apparently not been studied in the fashion done there by two of the authors of the present paper. Recently, Hyers-Ulam stability of Fibonacci functional equation was studied in [5]. Surprisingly novel perspectives are still available and will presumably continue to be so for the future as long as mathematical investigations continue to be made. In the following, the authors of the present paper are making another small offering at the same spot many previous contributors have visited in both recent and more distance pasts. The present authors [3,4] studied a Fibonacci norm of positive integers and Fibonacci sequences in groupoids in arbitrary groupoids.

In this paper we consider Fibonacci functions on the real numbers R, i.e., functions such that for all , . We develop the notion of Fibonacci functions using the concept of f-even and f-odd functions. Moreover, we show that if f is a Fibonacci function then .

### 2 Fibonacci functions

A function f defined on the real numbers is said to be a Fibonacci function if it satisfies the formula

(2.1)

for any , where R (as usual) is the set of real numbers.

Example 2.1 Let be a Fibonacci function on R where . Then . Since , we have and . Hence is a Fibonacci function, and the unique Fibonacci function of this type on R.

If we let , , then we consider the full Fibonacci sequence: ,  , i.e., for , and , the nth Fibonacci number.

Example 2.2 Let and be full Fibonacci sequences. We define a function by , where . Then for any . This proves that f is a Fibonacci function.

Note that if a Fibonacci function is differentiable on R, then its derivative is also a Fibonacci function.

Proposition 2.3Letfbe a Fibonacci function. If we definewherefor any, thengis also a Fibonacci function.

Proof Given , we have , proving the proposition. □

For example, since is a Fibonacci function, is also a Fibonacci function where .

Example 2.4 In Example 2.2, we discussed the function , where . If we let , then is a Fibonacci function. We compute and as follows: and .

Theorem 2.5Letbe a Fibonacci function and letbe a sequence of Fibonacci numbers with, . Thenfor anyandan integer.

Proof If , then . If , then we have

If we assume that it holds for the cases of n and , then

proving the theorem. □

Corollary 2.6Ifis the sequence of Fibonacci numbers with, then

(2.2)

Proof As we have seen in Example 2.1, is a Fibonacci function. Let . By applying Theorem 2.5, we have , proving that . □

Theorem 2.7Letbe the full Fibonacci sequence. Then

(2.3)
and

(2.4)

Proof The map discussed in Example 2.4 is a Fibonacci function. If we apply Theorem 2.5, then we obtain

proving the theorem. □

Corollary 2.8If, then

(2.5)
and

(2.6)

Corollary 2.9.

Proof Let in (2.5) or in (2.6). □

### 3 f-even and f-odd functions

In this section, we develop the notion of Fibonacci functions using the concept of f-even and f-odd functions.

Definition 3.1 Let be a real-valued function of a real variable such that if and is continuous then . The map is said to be an f-even function (resp., f-odd function) if (resp., ) for any .

Example 3.2 If , then implies if . By continuity of , it follows that for any integer n, and hence . Since , we see that is an f-even function.

Example 3.3 If , then implies if for any integer n. By continuity of it follows that for any integer n, and hence . Since , we see that is an f-odd function.

Theorem 3.4Letbe a function, whereis anf-even function andis a continuous function. Thenis a Fibonacci function if and only ifis a Fibonacci function.

Proof Suppose that is a Fibonacci function. Then . Hence and , i.e., and is a Fibonacci function. On the other hand, if is any Fibonacci function, then implies that is also a Fibonacci function. □

Example 3.5 It follows from Example 2.1 that is a Fibonacci function. Since is an f-even function, by Theorem 3.4, is a Fibonacci function.

Example 3.6 If we define if x is rational and if x is irrational, then for any . Also, if , then whether or not is continuous. Thus is an f-even function. In Example 3.5, we have seen that is a Fibonacci function. By applying Theorem 3.4, the map defined by

is also a Fibonacci function.

Now, we discuss f-odd functions with Fibonacci functions. Let be an f-odd function and be a continuous function. Let be a Fibonacci function such that . Then . In this situation, the characteristic equation yields solutions of the type , and thus for , the solution type is , whereas is not a real number except for special values of x.

A function f defined on R satisfying for all is said to be an odd Fibonacci function. Similarly, a sequence with is said to be an odd Fibonacci sequence.

Example 3.7 A sequence is an odd Fibonacci sequence.

Corollary 3.8Letbe a function, whereis anf-odd function andis a continuous function. Thenis a Fibonacci function if and only ifis an odd Fibonacci function.

Proof Similar to the proof of Theorem 3.4. □

Example 3.9 The function is an odd Fibonacci function. Since is an f-odd function, by Corollary 3.8, we can see that the function is a Fibonacci function.

### 4 Quotients of Fibonacci functions

In this section, we discuss the limit of the quotient of a Fibonacci function.

Theorem 4.1Ifis a Fibonacci function, then the limit of quotientexists.

Proof If we consider a quotient of a Fibonacci function , we have 4 cases: (i) , ; (ii) , ; (iii) , ; (iv) , . Consider (iii). If we let , , then , , and . In this fashion, we obtain for any natural number . Given , there exist and such that . Hence

where . Thus . Case (ii) is similar to the case (iii). Consider the case (i): , . We may change by , since any real number x (>0) can be written for some and . Consider a sequence .

since . We claim that is monotonically increasing. Since , we show that the numerator part of the quotient is positive.

which shows that the sequence is monotonically increasing. By the Monotone Convergence Theorem, there exists . The case (iv) is similar to the case (i). This proves the theorem. □

Corollary 4.2Ifis a Fibonacci function, then

Proof If we let , , then

It is shown already in the proof of Theorem 4.1 for the case of , that the limit of the quotient converges to Φ, proving the corollary. □

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

### Acknowledgement

The authors are grateful to the referee’s valuable suggestions and help.

### References

1. Atanasov, K: New Visual Perspectives on Fibonacci Numbers, World Scientific, Hackensack (2002)

2. Dunlap, RA: The Golden Ratio and Fibonacci Numbers, World Scientific, Hackensack (1997)

3. Han, JS, Kim, HS, Neggers, J: The Fibonacci norm of a positive integer n-observations and conjectures. Int. J. Number Theory. 6, 371–385 (2010). Publisher Full Text

4. Han, JS, Kim, HS, Neggers, J: Fibonacci sequences in groupoids. Adv. Differ. Equ.. 2012, (2012)

5. Jung, SM: Hyers-Ulam stability of Fibonacci functional equation. Bull. Iran. Math. Soc.. 35, 217–227 (2009)

6. Kim, HS, Neggers, J: Fibonacci means and golden section mean. Comput. Math. Appl.. 56, 228–232 (2008). Publisher Full Text