Abstract
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 show that if f is a Fibonacci function then lim x → ∞ f ( x + 1 ) f ( x ) = 1 + 5 2 . MSC:11B39, 39A10.
Highlights
Fibonacci numbers have been studied in many different forms for centuries and the literature on the subject is, incredibly vast
The fact that the ratio of successive Fibonacci numbers approaches the Golden ratio rather quickly as they go to infinity probably has a good deal to do with the observation made in the previous sentence
The present authors [, ] studied a Fibonacci norm of positive integers and Fibonacci sequences in groupoids in arbitrary groupoids
Summary
Fibonacci numbers have been studied in many different forms for centuries and the literature on the subject is, incredibly vast. [ ] 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. Hyers-Ulam stability of Fibonacci functional equation was studied in [ ]. 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. In this paper we consider Fibonacci functions on the real numbers R, i.e., functions f :.
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