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On Fibonacci functions with Fibonacci numbers
Advances in Difference Equations volume 2012, Article number: 126 (2012)
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.
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
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 n th 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 2.3 Let f be a Fibonacci function. If we definewherefor any, then g is 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 2.5 Letbe 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 2.6 Ifis the sequence of Fibonacci numbers with, then
Proof As we have seen in Example 2.1, is a Fibonacci function. Let . By applying Theorem 2.5, we have , proving that . □
Theorem 2.7 Letbe the full Fibonacci sequence. Then
and
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 2.8 If, then
and
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 3.4 Letbe a function, whereis an f-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 3.8 Letbe a function, whereis an f-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 4.1 Ifis 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 4.2 Ifis 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. □
References
Atanasov K, et al.: New Visual Perspectives on Fibonacci Numbers. World Scientific, Hackensack; 2002.
Dunlap RA: The Golden Ratio and Fibonacci Numbers. World Scientific, Hackensack; 1997.
Han JS, Kim HS, Neggers J: The Fibonacci norm of a positive integer n -observations and conjectures. Int. J. Number Theory 2010, 6: 371–385. 10.1142/S1793042110003009
Han JS, Kim HS, Neggers J: Fibonacci sequences in groupoids. Adv. Differ. Equ. 2012., 2012:
Jung SM: Hyers-Ulam stability of Fibonacci functional equation. Bull. Iran. Math. Soc. 2009, 35: 217–227.
Kim HS, Neggers J: Fibonacci means and golden section mean. Comput. Math. Appl. 2008, 56: 228–232. 10.1016/j.camwa.2007.12.003
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The authors are grateful to the referee’s valuable suggestions and help.
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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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Han, J.S., Kim, H.S. & Neggers, J. On Fibonacci functions with Fibonacci numbers. Adv Differ Equ 2012, 126 (2012). https://doi.org/10.1186/1687-1847-2012-126
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DOI: https://doi.org/10.1186/1687-1847-2012-126
Keywords
- Fibonacci function
- f-even (f-odd) function
- Golden ratio