A Study on Fibonacci Functions and Gaussian Fibonacci Functions - Soykan, Yüksel; Göcen, Melih; Taşdemir, Erkan; Murat Dikmen, Can (Sarajevo Journal of Mathematics, 2024)
In this paper, we define Gaussian Fibonacci functions and investigate them on the set of real numbers $\mathbb{R},$ i.e., functions $f_{G}$ $:$ $\mathbb{R}→ \mathbb{C}$ such that for all $x\in \mathbb{R},$ $% n\in \mathbb{Z},$ $f_{G}(x+n)=f(x+n)+if(x+n-1)$ where $f$ $:$ $\mathbb{R}% → \mathbb{R}$ 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→ ∞ }\frac{% f_{G}(x+1)}{f_{G}(x)}=α $ and\ $\lim_{x→ ∞ }\frac{% f_{G}(x)}{f(x)}=1+\left( α -1\right) i,$ where $α $ is the positive real root of equation $x^{2}-x-1=0$ for which $α >1$. 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.