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On Fibonacci functions with periodicity
Advances in Difference Equations volume 2014, Article number: 293 (2014)
Abstract
In this paper we discuss Fibonacci functions using the (ultimately) periodicity and we also discuss the exponential Fibonacci functions. Especially, given a non-negative real-valued function, we obtain several exponential Fibonacci functions.
MSC:11B39, 39A10.
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 [3] Kim and Neggers showed that there is a mapping on means such that if M is a Fibonacci mean so is DM, and that if M is the harmonic mean, then DM is the arithmetic mean, and if M is a Fibonacci mean, then is the golden section mean. The Hyers-Ulam stability of Fibonacci functional equation was studied in [4]. 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 as many previous contributors have visited in both recent and more distant pasts.
Han et al. [5] considered several properties of Fibonacci sequences in arbitrary groupoids. They discussed Fibonacci sequences in both several groupoids and groups. The present authors [6] introduced the notion of generalized Fibonacci sequences over a groupoid and discussed these in particular for the case where the groupoid contains idempotents and pre-idempotents. Using the notion of Smarandache-type P-algebras they obtained several relations on groupoids which are derived from generalized Fibonacci sequences.
In [7] Han et al. discussed Fibonacci functions on the real numbers R, i.e., functions such that, for all , , and developed the notion of Fibonacci functions using the concept of f-even and f-odd functions. Moreover, they showed that if f is a Fibonacci function, then .
In this paper we discuss Fibonacci functions using the (ultimately) periodicity and we also discuss the exponential Fibonacci functions. Especially, given a non-negative real-valued function, we obtain several exponential Fibonacci functions.
2 Preliminaries
A function f defined on the real numbers is said to be a Fibonacci function [7] if it satisfies the formula
for any , where R (as usual) is the set of real numbers.
Example 2.1 ([7])
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 n th Fibonacci number.
Example 2.2 ([7])
Let and be full Fibonacci sequences. We define a function by , where . Then for any . This proves that f is a Fibonacci function.
Example 2.3 Let , be any real-valued-functions defined on and let and be full Fibonacci sequences, respectively. Define a map , where . Then . Since and thus , we obtain , i.e., is a Fibonacci function.
Note that Example 2.2 is a special case of Example 2.3 with , . By choosing of the suitable functions φ and ψ, e.g., , , we obtain another example.
Remark Note that the Fibonacci function in Example 2.3 does not have the form , which is monotone in any case. In fact, let and be full Fibonacci sequences with , , . Then , , and , . Thus and . Assume where . If , then and . If , then and . If , then . Thus does not have the form , which is monotone in any case.
Using the notions of an f-even function and an f-odd function, we obtain many Fibonacci functions as discussed in [7].
Definition 2.4 ([7])
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 .
Theorem 2.5 ([7])
Let be a function, where is an f-even function and is a continuous function. Then is a Fibonacci function if and only if is a Fibonacci function.
Example 2.6 ([7])
It follows from Example 2.1 that is a Fibonacci function. Since is an f-even function, by Theorem 2.5, is a Fibonacci function.
Example 2.7 ([7])
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 2.6, we have seen that is a Fibonacci function. By applying Theorem 2.5, the map defined by
is also a Fibonacci function.
Corollary 2.8 ([7])
Let be a function, where is an f-odd function and is a continuous function. Then is a Fibonacci function if and only if is an odd Fibonacci function.
Example 2.9 ([7])
The function is an odd Fibonacci function. Since is an f-odd function, by Corollary 2.8, we can see that the function is a Fibonacci function.
Note that if a Fibonacci function is differentiable on R, then its derivative is also a Fibonacci function.
Proposition 2.10 ([7])
Let f be a Fibonacci function. If we define where for any , then g is also a Fibonacci function.
For example, since is a Fibonacci function, is also a Fibonacci function where .
Theorem 2.11 ([7])
If is a Fibonacci function, then the limit of the quotient exists.
Corollary 2.12 ([7])
If is a Fibonacci function, then
3 Fibonacci functions with periodicity
In this section, we obtain several results on Fibonacci functions using the periodicity.
Proposition 3.1 Let and be Fibonacci functions with for some . Then
Proof If and are Fibonacci functions, then by Corollary 2.12 we obtain
This proves the proposition. □
Corollary 3.2 Let and be Fibonacci functions with for some . Then
for all natural numbers k.
Proof It follows from the following equation:
 □
Corollary 3.3 Let and be Fibonacci functions with for some . If , then
Proof It follows from Corollary 3.2 that
 □
A map is said to be ultimately periodic of period if
Note that discussed in Proposition 3.1 is ultimately periodic of period 1.
Example 3.4 Let . If , then , showing that is ultimately periodic of period p for all .
Using Example 3.4, we obtain the following example.
Example 3.5 If , then is ultimately periodic of period p for all .
Example 3.6 If , then . It follows that
Since does not exist, is not ultimately periodic of period unless and , i.e., for any integer .
Proposition 3.7 If and are ultimately periodic of period , then is also ultimately periodic of period for all .
Proof Since and are ultimately periodic of period , there exist such that and where (). We know that for some . In fact, . This shows that
This proves the proposition. □
Proposition 3.8 If and are ultimately periodic of period , then is also ultimately periodic of period .
Proof It follows from the following equation:
 □
Theorem 3.9 The collection of all functions which are ultimately periodic of period forms an algebra.
Proof It follows immediately from Propositions 3.7 and 3.8. □
Proposition 3.10 If and for all for some λ, then .
Proof It follows from the following equation:
 □
Proposition 3.11 If , then for all natural numbers k.
Proof The proof is similar to that of Corollary 3.2. □
A map f defined on the set of all real numbers R is said to be periodic of period if for all . It is obvious that every map of period of periodic 1 is ultimately periodic of period 1.
Proposition 3.12 Let be a Fibonacci function and let be periodic of period 1. If , then is a Fibonacci function.
Proof Given , since is periodic of period 1, we have
 □
We ask the following question: Are there a Fibonacci function and a function which is ultimately periodic of period 1 but not periodic of period 1 such that is also a Fibonacci function?
Theorem 3.13 Let , be Fibonacci functions with . If for all , then
Proof Since for all , we have
It follows that , which implies
By Corollary 2.12, we obtain
proving the theorem. □
If we let , then . This shows that cannot be a Fibonacci function for any Fibonacci function .
Note that cannot be an increasing function on for some . In fact, we suppose that there is an such that for all . Then . It follows that , a contradiction.
Given , if we let and we let
then . It follows that
Theorem 3.14 Let be a Fibonacci function and let be a map with condition (2). If is a Fibonacci function for all , then
for all .
Proof Let be a function satisfying the condition (2). Since is a Fibonacci function, we have the following:
Since is also a Fibonacci function, we obtain
It follows that
This shows that
for all . □
4 Exponential Fibonacci functions
Consider a map with domain . If we let , then is a bijective function. If such that , then .
Theorem 4.1 Let be a map such that and let where . Then is a Fibonacci function.
Proof If we assume is a Fibonacci function, then
where . Then . Let and let . Then . It follows that
This shows that
Hence and , i.e., . A is a constant, and are also constants for all . Hence there exists such that . It follows that
If we let , a constant, for all , then . It follows that , i.e., . This shows that is a Fibonacci function where . □
The map T discussed above is useful for the following proposition.
Proposition 4.2 If is a Fibonacci function where , then there exists such that
Proof If , , then . Assume
for some . If we let , then
It follows that
proving the proposition. □
The converse of Proposition 4.2 need not be true in general. If , then it is not a Fibonacci function. If we let , then and . It follows that .
We construct Fibonacci functions directly as follows.
Theorem 4.3 Let be a constant and let f be a map defined on . Define a map by for all . Then is a Fibonacci function.
Proof If we let , then . It follows that
proving the theorem. □
Proposition 4.4 There is no Fibonacci function such that , where is differentiable and is a Fibonacci function.
Proof Assume that is a Fibonacci function. Since is differentiable, we have
Since is a Fibonacci function, we have . Since and is differentiable, , i.e., is also a Fibonacci function. It follows from that
Since , we obtain
By (4) and (5), we obtain
This shows that
It follows that
a contradiction. Hence should not be a Fibonacci function. □
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Acknowledgements
The authors are grateful for a referee’s valuable suggestions and help. This research was supported by Hallym University Research Fund 2014 (HRF-201401-008).
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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.
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Kim, H.S., Neggers, J. & So, K.S. On Fibonacci functions with periodicity. Adv Differ Equ 2014, 293 (2014). https://doi.org/10.1186/1687-1847-2014-293
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DOI: https://doi.org/10.1186/1687-1847-2014-293