Version 1
: Received: 8 September 2021 / Approved: 8 September 2021 / Online: 8 September 2021 (14:23:33 CEST)

How to cite:
Soykan, Y.; Göcen, M.; Okumuş, İ. On Tribonacci Functions and Gaussian Tribonacci Functions. Preprints2021, 2021090155 (doi: 10.20944/preprints202109.0155.v1).
Soykan, Y.; Göcen, M.; Okumuş, İ. On Tribonacci Functions and Gaussian Tribonacci Functions. Preprints 2021, 2021090155 (doi: 10.20944/preprints202109.0155.v1).

Cite as:

Soykan, Y.; Göcen, M.; Okumuş, İ. On Tribonacci Functions and Gaussian Tribonacci Functions. Preprints2021, 2021090155 (doi: 10.20944/preprints202109.0155.v1).
Soykan, Y.; Göcen, M.; Okumuş, İ. On Tribonacci Functions and Gaussian Tribonacci Functions. Preprints 2021, 2021090155 (doi: 10.20944/preprints202109.0155.v1).

Abstract

In this work, Gaussian Tribonacci functions are defined and investigated on the set of real numbers $\mathbb{R},$ \textit{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 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\limits_{x\rightarrow \infty }\frac{f_{G}(x+1)}{f_{G}(x)}=\alpha \ $and\ $\lim\limits_{x\rightarrow \infty }\frac{f_{G}(x)}{f(x)}=\alpha +i,$ 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.

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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.