On Tribonacci functions and Gaussian Tribonacci functions

Abstract

 In this work, Gaussian Tribonacci functions are defined and investigated on the set of real numbers $\mathbb{R}$, i.e., functions $f_G: \mathbb{R} \rightarrow \mathbb{C}$ such that for all $x \in \mathbb{R}, n \in \mathbb{Z}, f_G(x+n)=$ $f(x+n)+i f(x+n-1)$ where $f: \mathbb{R} \rightarrow \mathbb{R}$ is a Tribonacci function which is given as $f(x+3)=$ $f(x+2)+f(x+1)+f(x)$ for all $x \in \mathbb{R}$. Then the concept of Gaussian Tribonacci functions by using the concept of $f$-even and $f$-odd functions is developed. Also, we present linear sum formulas of Gaussian Tribonacci functions. Moreover, it is showed that if $f_G$ is a Gaussian Tribonacci function with Tribonacci function $f$, then $\lim _{x \rightarrow \infty} \frac{f_G(x+1)}{f_G(x)}=\alpha$ and $\lim _{x \rightarrow \infty} \frac{f_G(x)}{f(x)}=\alpha+i$, where $\alpha$ is the positive real root of equation $x^3-x^2-x-1=0$ for which $\alpha>1$. Finally, matrix formulations of Tribonacci functions and Gaussian Tribonacci functions are given. In the literature, there are several studies on the functions of linear recurrent sequences such as Fibonacci functions and Tribonacci functions. However, there are no study on Gaussian functions of linear recurrent sequences such as Gaussian Tribonacci and Gaussian Tetranacci functions and they are waiting for the investigating. We also present linear sum formulas and matrix formulations of Tribonacci functions which have not been studied in the literature.