Abstract
A function is said to be a Fibonacci function if for all . In 2012, some properties on the Fibonacci functions were presented. In this paper, for any positive integer , a function is said to be a Fibonacci function with period if for all ; we present some properties on the Fibonacci functions with period .
Highlights
There are many research articles about Fibonacci numbers
For any positive integer k, a function f : R → R is said to be a Fibonacci function with period k if f(x + 2k) = f(x + k) + f(x) for all x ∈ R; we present some properties on the Fibonacci functions with period k using the concept of f-even and f-odd functions with period k
Let f(x) = ax/k be a Fibonacci function with period k ∈ N, where a > 0. It follows that a(x/k)+2 = a(x/k)+1 + ax/k for all x ∈ R, so a2 = a + 1
Summary
There are many research articles about Fibonacci numbers (see [1]). Fibonacci numbers are involved in the golden ratio (see [2]). In 2012, Han et al [6] studied Fibonacci sequences in groupoids They [7] gave some properties on Fibonacci functions; a function f : R → R is said to be a Fibonacci function if f(x + 2) = f(x + 1) + f(x), for all x ∈ R, using the concept of f-even and f-odd functions. For any positive integer k, a function f : R → R is said to be a Fibonacci function with period k if f(x + 2k) = f(x + k) + f(x) for all x ∈ R; we present some properties on the Fibonacci functions with period k using the concept of f-even and f-odd functions with period k. We present some properties on the odd Fibonacci functions with period k
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