On Fibonacci functions with Fibonacci numbers
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RESEARCH
Open Access
On Fibonacci functions with Fibonacci numbers Jeong Soon Han1 , Hee Sik Kim2* and Joseph Neggers3 *
Correspondence: [email protected] 2 Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul, 133-791, Korea Full list of author information is available at the end of the article
Abstract In this paper we consider Fibonacci functions on the real numbers R, i.e., functions f : R → R such that for all x ∈ R, f (x + 2) = f (x + 1) + f (x). We develop the notion of Fibonacci functions using the concept of f -even and f -odd functions. Moreover, we √ 1+ 5 = . show that if f is a Fibonacci function then limx→∞ f (x+1) f (x) 2 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 [] 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 [, ] 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 f : R → R such that for all x ∈ R, f (x + ) = f (x + ) + f (x). 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 limx→∞ f (x+) f (x) © 2012 Han et al.; licensee Springer. 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.
Han et al. Advances in Difference Equations 2012, 2012:126 http://www.advancesindifferenceequ
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