Keywords:Fibonacci function; f-even (f-odd) function; Golden ratio
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  and  for a very minimal set of examples of such texts, while in  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 . 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
Note that if a Fibonacci function is differentiable on R, then its derivative is also a Fibonacci function.
proving the theorem. □
proving the theorem. □
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.
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.
Proof Similar to the proof of Theorem 3.4. □
4 Quotients of Fibonacci functions
In this section, we discuss the limit of the quotient of a Fibonacci function.
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
The authors declare that they have no competing interests.
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.
The authors are grateful to the referee’s valuable suggestions and help.
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
Kim, HS, Neggers, J: Fibonacci means and golden section mean. Comput. Math. Appl.. 56, 228–232 (2008). Publisher Full Text