A Study on Fibonacci Functions and Gaussian Fibonacci Functions

Abstract

In this paper, we define Gaussian Fibonacci functions and investigate them
on the set of real numbers $\mathbb{R},$ i.e., functions $f_{G}$ $:$
$\mathbb{R}\rightarrow \mathbb{C}$ such that for all $x\in \mathbb{R},$ $%
n\in \mathbb{Z},$ $f_{G}(x+n)=f(x+n)+if(x+n-1)$ where $f$ $:$ $\mathbb{R}%
\rightarrow \mathbb{R}$ is a Fibonacci function which is given as $%
f(x+2)=f(x+1)+f(x)$ for all $x\in \mathbb{R}$. Then the concept of Gaussian
Fibonacci functions by using the concept of $f$-even and $f$-odd functions
is developed. Also, we present linear sum formulas of Gaussian Fibonacci
functions. Moreover, it is shown that if $f_{G}$ is a Gaussian Fibonacci
function with Fibonacci function $f$, then $\lim_{x\rightarrow \infty }\frac{%
f_{G}(x+1)}{f_{G}(x)}=\alpha $ and\ $\lim_{x\rightarrow \infty }\frac{%
f_{G}(x)}{f(x)}=1+\left( \alpha -1\right) i,$ where $\alpha $ is the
positive real root of equation $x^{2}-x-1=0$ for which $\alpha >1$.
Furthermore, matrix formulations of Fibonacci functions and Gaussian Fibonacci functions are given. We also present linear sum formulas and matrix formulations of Fibonacci functions which have not been studied in the literature.