Tricomplex Fibonacci Numbers: A New Family of Fibonacci-Type Sequences

Abstract

In this paper, we define a novel family of arithmetic sequences associated with the Fibonacci numbers. Consider the ordinary Fibonacci sequence ${\left\{f_n\right\}}_{n\in {\mathbb{N}}_0}$ having initial terms $f_0=0$, and $f_1=1$ and recurrence relation $f_n=f_{n-1}+f_{n-2}(n\ge 2)$. In many studies, authors worked on the generalizations of integer sequences in different ways, some by preserving the initial terms, others by preserving the recurrence relation, and some for numeric sets other than positive integers. Here, we will follow the third path. So, in this article, we study a new extension $t{f}_n^{\ast }$, with initial terms $t{f}_0^{\ast }=(f_0^{\ast },f_1^{\ast },f_2^{\ast })$ and $t{f}_1^{\ast }=(f_1^{\ast },f_2^{\ast },f_3^{\ast })$, which is generated by the recurrence relation $t{f}_n^{\ast }=t{f}_{n-1}^{\ast }+t{f}_{n-2}^{\ast }(n\ge 2)$, the Fibonacci-type sequence. The aim of this paper is to define Tricomplex Fibonacci numbers as an extension of the Fibonacci sequence and to examine some of their properties such as the recurrence relation, summation formula and generating function, and some classical identities.

1. Introduction

Recently, there has been increasing interest in various number systems. Many researchers have focused on exploring the properties and applications of these systems, uncovering new insights and potential uses. Leonardo of Pisa, commonly known as Fibonacci (12th–13th centuries), was a mathematician from the city-state of Pisa who made significant contributions to mathematics. In mathematical literature, following some historians, one of his most notable achievements was the introduction of the decimal number system to Europe from a practical standpoint. He is particularly renowned for his work on the Fibonacci sequence, $\{0,1,1,2,3,5,8,\dots \}$, where each number is the sum of the two preceding ones, defined by the recurrence relation $f_n=f_{n-1}+f_{n-2}$.

The well-known integer Fibonacci sequence ${\left\{f_n\right\}}_{n\in {\mathbb{N}}_0}$ is defined as $f_{n+2}=f_n+f_{n+1},n\ge 0$, with $f_0=0$ and $f_1=1$, the sequence $A000045$ in the OEIS [1]. The first seven terms of the Fibonacci sequence are: 0, 1, 1, 2, 3, 5, 8, 13, …. The Fibonacci recurrence, coupled with different initial terms, can be used to construct new number sequences or other similar sequences. For instance, let ${\left\{l_n\right\}}_{n\in {\mathbb{N}}_0}$ be the n-th term of a sequence with $l_0=2,l_1=1$ and $l_{n+2}=l_n+l_{n+1},n\ge 0$. The resulting sequence 2, 1, 3, 4, 7, 11, 18, …is the Fibonacci–Lucas sequence, the sequence $A000032$ in the OEIS [1]. We have both $f_n$ and $l_n$ satisfy the same recurrence. Both Fibonacci and Fibonacci–Lucas numbers satisfy numerous identities, many were discovered centuries ago. As usual, we denote by ${\mathbb{N}}_0={\mathbb{Z}}^{+}\cup \left\{0\right\}$, the set of all non-negative integers.

Several papers have been published discussing new sequences, their generalizations, extensions, and properties. These generalizations adhere to identities similar to those satisfied by the ordinary Fibonacci sequence. In Section 2.2, we provide a descriptive overview of various Fibonacci sequences (including extensions and generalizations) explored by different researchers.

Now, following [2,3,4,5], we consider $\mathbb{T}$ the ring of tricomplex numbers, that is, the set of ordered triples of real numbers $(x,y,z)$, with the operations of addition $(+)$, and multiplication $(\times )$ given by:

$$(x_1,y_1,z_1)+(x_2,y_2,z_2)=(x_1+x_2,y_1+y_2,z_1+z_2),$$

(1)

and

$$(x_1,y_1,z_1)\times (x_2,y_2,z_2)=(x_1{x}_2+y_1{z}_2+z_1{y}_2,z_1{z}_2+x_1{y}_2+y_1{x}_2,y_1{y}_2+x_1{z}_2+z_1{x}_2),$$

(2)

where $x_1,x_2,y_1,y_2,z_1$ and $z_2$ are real numbers.

By demonstrating that $\mathbb{T}$ possesses the arithmetic properties of a commutative ring with unity, ref. [2] inspired us to define numerical sequences within this domain, situated in a three-dimensional Euclidean space. This approach involves observing their behavior and exploring properties analogous to those of the ring of integers. Accordingly, in this paper, we introduce a new extension or family of the Fibonacci sequence, which we refer to as the Tricomplex Fibonacci sequence. Unlike other variations, this new extension depends on two real vectors with three coordinates in a vector linear recurrence relation.

The structure of the present work is divided into four additional sections, as outlined below. In Section 2, we briefly present the tricomplex ring $\mathbb{T}$ and describe various Fibonacci sequences studied by different researchers, also list key results on the Fibonacci-type sequence used in this work. In Section 3, we define the Tricomplex Fibonacci sequence, explaining some characteristics and properties. What the elements of this sequence look like in three-dimensional space is shown in an illustration. Additionally, we present the Binet formula, which provides an explicit expression for the terms of the sequence. Furthermore, we explore the generating function, which we express in its vectorial form. In Section 4, we will explore several fundamental identities for Tricomplex Fibonacci such as Tagiuri-Vajada, d’Ocagene, and consequences. Finally, in Section 5, we present the sum of terms involving the Tricomplex Fibonacci numbers. Our research problem is the determination of properties analogous to the Fibonacci sequence for the Tricomplex Fibonacci sequence.

2. Background and Auxiliary Results

In this section, we will make a brief presentation of the tricomplex ring $\mathbb{T}$ and a list of some results about the Fibonacci-type sequence that we will use throughout this work.

2.1. The Tricomplex Ring

Many researchers have been interested in the study of rings that are analogous to the ring of integers, in which arithmetic concepts can also be developed. Among them, we highlight the ring of Gaussian integers, whose study originates from Gauss’s investigations regarding cubic and biquadratic reciprocity. What makes this ring interesting is the fact that many results of arithmetic in Gaussian integers are analogous to the results of arithmetic in integers and, furthermore, it can be illustrated geometrically. The construction or historical facts about the structure of the complex number field, and the tricomplex ring, can be consulted in [6,7,8], among others.

A tricomplex number is an element of a number system that extends the complex numbers. Whereas complex numbers have a real part and an imaginary part, tricomplex numbers have two imaginary parts along with a real part. Olariu in [2] introduced the concept of tricomplex numbers which are expressed in the form $(x,y,z)=x+yi+zj$, where x, y, and z are real numbers and i and j are imaginary units. According to [2] the multiplication rules for the tricomplex units $1,i$ and j are given by

Table 1. The multiplication table for tricomplex units.

×	1	i	j
1	1	i	j
i	i	j	1
j	j	1	i

:

According to Equation (1), the sum of the tricomplex numbers $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ is the tricomplex number $(x_1+x_2,y_1+y_2,z_1+z_2)$. And, following Equation (2), the product of the tricomplex numbers $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ is the tricomplex number $(x_1{x}_2+y_1{z}_2+z_1{y}_2,z_1{z}_2+x_1{y}_2+y_1{x}_2,y_1{y}_2+x_1{z}_2+z_1{x}_2)$. So, tricomplex numbers and their operations can be represented by $\mathbb{T}=(\mathbb{T},+,\times )$. It can be checked through direct calculation that the tricomplex zero is $(0,0,0)$, denoted simply 0, and the tricomplex unity is $(1,0,0)$, denoted simply by 1. And furthermore, it can be checked that $\mathbb{T}$ is the commutative and unit ring, see [2,4,5] and references.

Tricomplex numbers can be useful in certain mathematical contexts, particularly in studying 3-dimensional systems. They can also have applications in physics and engineering, where three-dimensional systems are common.

2.2. Fibonacci-Type Sequence Results

A study of the history and some applications of the Fibonacci sequence, as well as some generalizations or extensions, can be found in [9,10]. In particular, we have an approach to the generating functions of this sequence by [11]. Many sequences have been defined by generalizing these number sequences in the mathematical literature. In some cases, the second-order recurrence sequences have been generalized by maintaining the recurrence relation while modifying the initial two terms of the sequence; for example, ref. [12] introduced and studied properties of a generalized Fibonacci sequence. Refs. [13] observes that several other known sequences are also of the Horadam sequence type, choosing appropriate initial terms and constants of the recurrence relation. Again, in [14], the concept of Fibonacci complex numbers and Fibonacci quaternion numbers, and extended several important identities allowing a better understanding of the relationship between the Fibonacci sequences in these complex or quaternion number systems. In particular, what Horadam studied in [12] was the generalized Fibonacci sequence. It was in [15] that Horadam studied what is now known as the Horadam sequence. In the same way, ref. [16] contributed to this process of extending the Fibonacci sequence to the domain of complex numbers, which made it possible to extend other sequences as well. The second-order recurrence sequences have been generalized by maintaining the recurrence relation while modifying the initial two terms of the sequence; for example, ref. [17] studied the generalized k-Fibonacci numbers, for a real number k, and observed that these sequences were found by studying the recursive application of two geometric transformations. Ref. [18] went on to present the k-Fibonacci generating matrices and an extension of the type $(k,t)$-Fibonacci numbers in [19] or [20], for real numbers $k,t$. Similarly, refs. [21,22,23], did some research on the sequences of numbers that arise from these sequences. Other researchers have generalized the Fibonacci sequence by preserving the first two terms of the sequence but altering the recurrence relation slightly. Refs. [24,25] define the bi-periodic Fibonacci sequences as a generalization the some Fibonacci sequences; while [26] studied the sequence ${\left\{F_n^k\right\}}_{n\in {\mathbb{N}}_0}$, the k-generalized Fibonacci number, as a generalization of the tribonacci, tetranacci, …, k-nacci sequences; giving general expressions for this type of sequence. The Fibonacci sequence has many applications, whether in nature or not, as can be seen in [9] or [10]. Refs. [27,28] in a recent study has reported the use of a Fibonacci-type sequence in the analysis of the mathematical structure of the genetic code.

Here, for easy notation, we will write ${\left\{f_n^{\ast }\right\}}_{n\in {\mathbb{N}}_0}$ to indicate $f_n$ or $l_n$. Thus, the sequence ${\left\{f_n^{\ast }\right\}}_{n\in {\mathbb{N}}_0}$ is of Fibonacci-type satisfying $f_{n+2}^{\ast }=f_{n+1}^{\ast }+f_n^{\ast },n\ge 0$, with initial terms $f_0^{\ast }$ and $f_1^{\ast }$. When $f_0^{\ast }=f_0=0$ and $f_1^{\ast }=f_1=1$, we have the classic Fibonacci numbers, and when $f_0^{\ast }=l_0=2$ and $f_1^{\ast }=l_1=1$, we have the Fibonacci–Lucas numbers. We emphasize that we do not use ${\left\{f_n^{\ast }\right\}}_{n\ge 0}$ to denote the generalized Fibonacci sequence, also called the Fibonacci sequence, usually noted as ${\left\{G_n\right\}}_{n\in {\mathbb{N}}_0}$, as can be seen in ([9], Chapter 7), among others.

The characteristic equation for the Fibonacci-type sequence is $r^2=r+1$. Solving it for r gives the roots of the characteristic equation. The Binet formula provides a direct way to compute the n-th Fibonacci-type number without iterating through the sequence.

The next result is a special case of ([9], Theorem 7.4).

Lemma 1. 

(Binet-like formula). Let $c=f_1^{\ast }-f_0^{\ast }\beta$ and $d=f_1^{\ast }-f_0^{\ast }\alpha$. Then

$$f_n^{\ast }={\displaystyle \frac{c{\alpha }^n-d{\beta }^n}{\alpha -\beta }},$$

(3)

where α is the golden ratio $(1+\sqrt{5})/2$, β is its conjugate $(1-\sqrt{5})/2$, $f_0^{\ast }$ and $f_1^{\ast }$ are the initial terms from the Fibonacci-type sequence ${\left\{f_n^{\ast }\right\}}_{n\in {\mathbb{N}}_0}$.

Thus, if ${\left\{f_n^{\ast }\right\}}_{n\in {\mathbb{N}}_0}$ is the Fibonacci sequence, ${\left\{f_n\right\}}_{n\in {\mathbb{N}}_0}$, with $f_0=0$ and $f_1=1$, then $c=1=d$ and

$$f_n={\displaystyle \frac{{\alpha }^n-{\beta }^n}{\alpha -\beta }};$$

while if ${\left\{f_n^{\ast }\right\}}_{n\in {\mathbb{N}}_0}$ is the Fibonacci–Lucas sequence, ${\left\{l_n\right\}}_{n\in {\mathbb{N}}_0}$, with $l_0=2$ and $l_1=1$, then $c=1-2\beta =\sqrt{5},d=1-2\alpha =-\sqrt{5}$ and

$$l_n={\alpha }^n+{\beta }^n.$$

Furthermore,

$$\begin{array}{ccc}\hfill cd& =& (f_1^{\ast }-f_0^{\ast }\beta )(f_1^{\ast }-f_0^{\ast }\alpha )\hfill \\ & =& {\left(f_1^{\ast }\right)}^2-f_1^{\ast }{f}_0^{\ast }-{\left(f_0^{\ast }\right)}^2.\hfill \end{array}$$

According to ([9], p. 139), this constant occurs in many of the formulas for Fibonacci-type numbers. So, the constant $\mu =cd$ is called the characteristic of the Fibonacci-type sequence. We will denote it by the Greek letter $\mu$, so $\mu ={\left(f_1^{\ast }\right)}^2-f_1^{\ast }{f}_0^{\ast }-{\left(f_0^{\ast }\right)}^2$. The characteristic of the Fibonacci sequence is $\mu =1$ and that of the Fibonacci–Lucas sequence is $\mu =-5$.

In the literature, the exponential generating function $E_{a_n}\left(x\right)$ of a sequence ${\left\{a_n\right\}}_{n\in {\mathbb{N}}_0}$ is a power series of the form

$$E_{a_n}\left(x\right)=a_0+a_{1}x+{\displaystyle \frac{a_2{x}^2}{2!}}+\cdots +{\displaystyle \frac{a_n{x}^n}{n!}}+\cdots =\sum _{n=0}^{\infty }{\displaystyle \frac{a_n{x}^n}{n!}}.$$

In the next result, we consider $a_n=f_n^{\ast }$ and making use of the Equation (3), the Binet formula for Fibonacci sequence, then we obtain the classical exponential generating function for the Fibonacci-type sequence ${\left\{f_n^{\ast }\right\}}_{n\in {\mathbb{N}}_0}$.

From Lemma 3, we obtain the following proposition directly, a special case of ([9], Equation 13.14).

Lemma 2. 

For all $n\in {\mathbb{N}}_0$, the exponential generating function for the Fibonacci sequence ${\left\{f_n^{\ast }\right\}}_{n\in {\mathbb{N}}_0}$ is

$$E_{f_n^{\ast }}\left(x\right)=\sum _{n=0}^{\infty }{\displaystyle \frac{f_n^{\ast }{x}^n}{n!}}={\displaystyle \frac{c{e}^{\alpha x}-d{e}^{\beta x}}{\alpha -\beta }},$$

where $\alpha =(1+\sqrt{5})/2$, $\beta =(1-\sqrt{5})/2$, $c=f_1^{\ast }-f_0^{\ast }\beta$ and $d=f_1^{\ast }-f_0^{\ast }\alpha$.

We have in the literature that the function $G{F}_{a_n}\left(x\right)$ is called the generating function for the sequence ${\left\{a_n\right\}}_{n\in {\mathbb{N}}_0}$, with

$${\displaystyle G{F}_{a_n}\left(x\right)=\sum _{n=0}^{\infty }{a}_n{x}^n=a_0+a_{1}x+a_2{x}^2+a_3{x}^3+\dots +a_n{x}^n+\dots }$$

In addition, our next result presents a generating function for the Fibonacci sequence as presented in ([9], Equations 13.6.1 and 13.6.3).

Lemma 3. 

The generating function for the Fibonacci sequence ${\left\{f_n^{\ast }\right\}}_{n\in {\mathbb{N}}_0}$, denoted by $G{F}_{f_n^{\ast }}\left(x\right)$, is

$$G{F}_{f_n^{\ast }}\left(x\right)={\displaystyle \frac{f_0^{\ast }+(f_1^{\ast }-f_0^{\ast })x}{1-x-x^2}}.$$

Fibonacci-type sequences satisfy many identities. The Tagiuri-Vajda identity for Fibonacci-type sequences has already appeared in some previous work; for example ([10], Equations 20a and 20b)

Lemma 4. 

[Tagiuri-Vajda’s identity] Let $m,s,k$ be any non-negative integers. We have the following identity:

$$f_{m+s}^{\ast }{f}_{m+k}^{\ast }-f_m^{\ast }{f}_{m+s+k}^{\ast }={(-1)}^m\mu (f_1^{\ast }-f_0^{\ast })f_s{f}_k.$$

where μ is the characteristic of the Fibonacci-type sequence ${\left\{f_n^{\ast }\right\}}_{n\in {\mathbb{N}}_0}$.

In the next result, the first equation is stated in ([10], Equation 25), and the second is stated in ([9], Equations 5.12 and 5.8.85).

Lemma 5. 

Let ${\left\{f_m^{\ast }\right\}}_{m\ge 0}$ be the Fibonacci-type sequence. For all $m\ge 1$ integers, we have

$$\begin{array}{ccc}\hfill \left(a\right)& {\left(f_m^{\ast }\right)}^2+{\left(f_{m+1}^{\ast }\right)}^2& =\left|\mu \right|f_{2m+1},\hfill \\ \hfill \left(b\right)& {\left(f_{m+1}^{\ast }\right)}^2-{\left(f_{m-1}^{\ast }\right)}^2& =\left|\mu \right|f_{2m},\hfill \end{array}$$

where μ is the characteristic of the Fibonacci-type sequence ${\left\{f_n^{\ast }\right\}}_{n\in {\mathbb{N}}_0}$.

Another auxiliary result about Fibonacci-type sequence.

Lemma 6 

([9], Equation 7.44). Let ${\left\{f_n^{\ast }\right\}}_{n\in {\mathbb{N}}_0}$ be the Fibonacci-type sequence. For any non-negative integers m and n, the Fibonacci-type sequence satisfies

$$f_{m-1}^{\ast }{f}_n^{\ast }-f_m^{\ast }{f}_{n-1}^{\ast }={(-1)}^{n-1}\mu f_{m-n}.$$

where μ is the characteristic of the Fibonacci-type sequence ${\left\{f_n^{\ast }\right\}}_{n\in {\mathbb{N}}_0}$.

To conclude this section, we present a result restricted to the Fibonacci sequence ${\left\{f_n\right\}}_{n\in {\mathbb{N}}_0}$, the convolution identity.

Lemma 7 

([29], Equation 1.8). For any integers m and n, the Fibonacci sequence satisfies

$$f_{m+1}{f}_n+f_m{f}_{n-1}=f_{m+n},$$

where ${\left\{f_n\right\}}_{n\in {\mathbb{N}}_0}$ is the Fibonacci sequence.

3. The Binet Formula and the Generating Function

In this section, we will present the definition of the Tricomplex Fibonacci sequence ${\left\{t{f}_n^{\ast }\right\}}_{n\in {\mathbb{N}}_0}$ and show that it is a second-order vector recurrence sequence. In addition, the Binet formulas are provided. In this section, we will derive the exponential generating and generating function of the Tricomplex Fibonacci sequence.

First, in the ring $\mathbb{T}$, we define Tricomplex Fibonacci sequence ${\left\{t{f}_n^{\ast }\right\}}_{n\in {\mathbb{N}}_0}$ as an element in the three-dimensional space.

Definition 1. 

For all non-negative integer n, the Tricomplex Fibonacci sequence ${\left\{t{f}_n^{\ast }\right\}}_{n\in {\mathbb{N}}_0}$ is given by:

$$t{f}_n^{\ast }=(f_n^{\ast },f_{n+1}^{\ast },f_{n+2}^{\ast }),$$

where ${\left\{f_n^{\ast }\right\}}_{n\in {\mathbb{N}}_0}$ is the Fibonacci-type sequence.

For instance, $t{f}_0=(0,1,1)$, $t{f}_1=(1,1,2)$ and $t{f}_2=(1,2,3)$; while $t{l}_0=(2,1,3)$, $t{l}_1=(1,3,4)$ and $t{l}_2=(3,4,7)$.

The Figure 1 illustrates the terms of the Tricomplex Fibonacci sequence $t{f}_2,t{f}_3$ and $t{f}_4$, respectively, represented by $(x_2,y_2,z_2)$, $(x_3,y_3,z_3)$ and $(x_4,y_4,z_4)$ in three-dimensional space ${\mathbf{R}}^3$.

A direct calculation shows the result stated in the next proposition.

Proposition 1. 

The Tricomplex Fibonacci sequence ${\left\{t{f}_n^{\ast }\right\}}_{n\in {\mathbb{N}}_0}$ satisfies the recurrence relation

$$t{f}_{n+2}^{\ast }=t{f}_n^{\ast }+t{f}_{n+1}^{\ast },n\ge 0,$$

(4)

with initial terms $t{f}_0^{\ast }=(f_0^{\ast },f_1^{\ast },f_2^{\ast })$ and $t{f}_1^{\ast }=(f_1^{\ast },f_2^{\ast },f_3^{\ast })$, where ${\left\{f_n^{\ast }\right\}}_{n\ge 1}$ is the Fibonacci-type sequence.

Therefore, the vector sequence ${\left\{t{f}_n^{\ast }\right\}}_{n\in {\mathbb{N}}_0}$, derived from the Fibonacci-type sequence ${\left\{f_n^{\ast }\right\}}_{n\in {\mathbb{N}}_0}$, is also a Fibonacci-type sequence, because the sequence obeys the same recursion as the Fibonacci sequence.

It can be observed that in each coordinate of Equation (4), a second-order characteristic equation is present, that is

$$\begin{array}{ccc}\hfill t{f}_{n+2}^{\ast }& =& t{f}_n^{\ast }+t{f}_{n+1}^{\ast }\hfill \\ \hfill (f_{n+2}^{\ast },f_{n+3}^{\ast },f_{n+4}^{\ast })& =& (f_n^{\ast }+f_{n+1}^{\ast },f_{n+1}^{\ast }+f_{n+2}^{\ast },f_{n+2}^{\ast }+f_{n+3}^{\ast }).\hfill \end{array}$$

So, the vector Equation (4) is equivalent of the following vector characteristic equation

$$(x^2-x+1,y^2-y+1,z^2-z+1)=(0,0,0).$$

(5)

In each component of the vector Equation (5), the variables $x,y,$ and z are represented by quadratic equations in terms of the indeterminate $x,y,$ and z. So, the vector Equation (5) is again second-order vector recurrence relation associated with the Tricomplex Fibonacci numbers ${\left\{t{f}_n^{\ast }\right\}}_{n\in {\mathbb{N}}_0}$.

Using the Binet formula for n-th Fibonacci number, Equation (3), in each coordinate, we have just shown the Binet formula for n-th Tricomplex Fibonacci number.

Theorem 1. 

For all non-negative integers n, the Binet formula for the Tricomplex Fibonacci sequence ${\left\{t{f}_n^{\ast }\right\}}_{n\in {\mathbb{N}}_0}$ is

$$(f_n^{\ast },f_{n+1}^{\ast },f_{n+2}^{\ast })=\left({\displaystyle \frac{c{\alpha }^n-d{\beta }^n}{\alpha -\beta }},{\displaystyle \frac{c{\alpha }^{n+1}-d{\beta }^{n+1}}{\alpha -\beta }},{\displaystyle \frac{c{\alpha }^{n+2}-d{\beta }^{n+2}}{\alpha -\beta }}\right),$$

(6)

where ${\left\{f_n^{\ast }\right\}}_{n\ge 1}$ is the Fibonacci-type sequence, $\alpha =(1+\sqrt{5})/2$, $\beta =(1-\sqrt{5})/2$, $c=f_1^{\ast }-f_0^{\ast }\beta$ and $d=f_1^{\ast }-f_0^{\ast }\alpha$.

Let ${\mathcal{E}}_{t{f}_n^{\ast }}\left(x\right)=\left(E_{f_n^{\ast }}\left(x\right),E_{f_{n+1}^{\ast }}\left(x\right),E_{f_{n+1}^{\ast }}\left(x\right)\right)$ be the vector exponential generating function for the Tricomplex Fibonacci sequence ${\left\{t{f}_n^{\ast }\right\}}_{n\in {\mathbb{N}}_0}$. Combining Theorem 1 and Lemma 2 we obtain the next result.

Proposition 2. 

the exponential generating function for the Tricomplex Fibonacci sequence ${\left\{t{f}_n^{\ast }\right\}}_{n\in {\mathbb{N}}_0}$ is

$${\mathcal{E}}_{t{f}_n^{\ast }}\left(x\right)=\left({\displaystyle \frac{c{e}^{\alpha x}-d{e}^{\beta x}}{\alpha -\beta }},{\displaystyle \frac{c\alpha e^{\alpha x}-d\beta e^{\beta x}}{\alpha -\beta }},{\displaystyle \frac{c{\alpha }^2{e}^{\alpha x}-d{\beta }^2{e}^{\beta x}}{\alpha -\beta }}\right),$$

where ${\left\{f_n^{\ast }\right\}}_{n\ge 1}$ is the Fibonacci-type sequence, $\alpha =(1+\sqrt{5})/2$, $\beta =(1-\sqrt{5})/2$, $c=f_1^{\ast }-f_0^{\ast }\beta$ and $d=f_1^{\ast }-f_0^{\ast }\alpha$.

Now, let ${\mathcal{GF}}_{t{f}_n^{\ast }}\left(x\right)=\left(G{F}_{f_n^{\ast }}\left(x\right),G{F}_{f_{n+1}^{\ast }}\left(x\right),G{F}_{f_{n+1}^{\ast }}\left(x\right)\right)$ be the vector generating function for the Tricomplex Fibonacci sequence ${\left\{t{f}_n^{\ast }\right\}}_{n\in {\mathbb{N}}_0}$. Next, we will present the generating function for the Tricomplex Fibonacci sequence.

Proposition 3. 

The generating function for the Tricomplex Fibonacci sequence ${\left\{t{f}_n^{\ast }\right\}}_{n\in {\mathbb{N}}_0}$, denoted by ${\mathcal{GF}}_{t{f}_n^{\ast }}\left(x\right)$, is given by

$$\left({\displaystyle \frac{f_0^{\ast }+(f_1^{\ast }-f_0^{\ast })x}{1-x-x^2}},{\displaystyle \frac{f_1^{\ast }+f_0^{\ast }x}{1-x-x^2}},{\displaystyle \frac{(f_0^{\ast }+f_1^{\ast })x+f_1^{\ast }{x}^2}{x-x^2-x^3}}\right),$$

(7)

where ${\left\{f_n^{\ast }\right\}}_{n\in {\mathbb{N}}_0}$ is the Fibonacci-type sequence.

Proof. 

The first coordinate of the vector defined in Equation (7) is a direct application of the Lemma 3.

We have ${\displaystyle G{F}_{f_{n+1}^{\ast }}\left(x\right)=\sum _{n=0}^{\infty }{f}_{n+1}^{\ast }{x}^n}$. Then, by expanding equation $xG{F}_{f_{n+1}^{\ast }}\left(x\right)$, we obtain

$$\begin{array}{ccc}\hfill xG{F}_{f_{n+1}^{\ast }}\left(x\right)& =& f_1^{\ast }x+f_2^{\ast }{x}^2+f_3^{\ast }{x}^3+\dots \hfill \\ & =& G{F}_{f_n^{\ast }}\left(x\right)-f_0^{\ast }\hfill \\ & =& {\displaystyle \frac{f_0^{\ast }+(f_1^{\ast }-f_0^{\ast })x}{1-x-x^2}}-f_0^{\ast }\hfill \\ & =& {\displaystyle \frac{f_1^{\ast }+f_0^{\ast }x}{1-x-x^2}}.\hfill \end{array}$$

This proves the second coordinate of the vector defined in Equation (7). Proof of the third coordinate of the vector defined in Equation (7) is performed similarly. □

4. Elementary Properties of Tricomplex Fibonacci Numbers

This section is dedicated to establishing some fundamental identities for the Tricomplex Fibonacci sequence, defined by the set of terms ${\left\{t{f}_n^{\ast }\right\}}_{n\in {\mathbb{N}}_0}$. These will include the Tagiuri-Vajda, Catalan, Cassini, and d’Ocagne identities.

First, we state the Tagiuri-Vajda’s Identity for the Tricomplex Fibonacci sequence ${\left\{t{f}_n^{\ast }\right\}}_{n\in {\mathbb{N}}_0}$. Remember that the characteristic of the Fibonacci sequence is $\mu =1$ and that of the Fibonacci–Lucas sequence is $\mu =-5$.

Theorem 2. 

Let $m,s,k$ be any non-negative integers. We have

$$t{f}_{m+s}^{\ast }t{f}_{m+k}^{\ast }-t{f}_m^{\ast }t{f}_{m+s+k}^{\ast }={(-1)}^m\mu ·(f_1^{\ast }-f_0^{\ast })\left(0,2f_s{f}_k,2f_s{f}_k\right),$$

(8)

where ${\left\{t{f}_n^{\ast }\right\}}_{n\in {\mathbb{N}}_0}$ is the Tricomplex Fibonacci sequence, ${\left\{f_n\right\}}_{n\in {\mathbb{N}}_0}$ is the Fibonacci sequence, and μ is the characteristic of the Fibonacci-type sequence ${\left\{f_n^{\ast }\right\}}_{n\in {\mathbb{N}}_0}$.

Proof. 

We have

$$\begin{array}{ccc}\hfill t{f}_{m+s}^{\ast }t{f}_{m+k}^{\ast }& =& (f_{m+s}^{\ast },f_{m+s+1}^{\ast },f_{m+s+2}^{\ast })\times (f_{m+k}^{\ast },f_{m+k+1}^{\ast },f_{m+k+2}^{\ast })\hfill \\ \hfill & =& (f_{m+s}^{\ast }{f}_{m+k}^{\ast }+f_{m+s+1}^{\ast }{f}_{m+k+2}^{\ast }+f_{m+s+2}^{\ast }{f}_{m+k+1}^{\ast },\hfill \\ \hfill & & f_{m+s+2}^{\ast }{f}_{m+k+2}^{\ast }+f_{m+s}^{\ast }{f}_{m+k+1}^{\ast }+f_{m+s+1}^{\ast }{f}_{m+k}^{\ast },\hfill \\ \hfill & & f_{m+s+1}^{\ast }{f}_{m+k+1}^{\ast }+f_{m+s}^{\ast }{f}_{m+k+2}^{\ast }+f_{m+s+2}^{\ast }{f}_{m+k}^{\ast }),\hfill \end{array}$$

(9)

and,

$$\begin{array}{ccc}\hfill t{f}_m^{\ast }t{f}_{m+s+k}^{\ast }& =& (f_m^{\ast },f_{m+1}^{\ast },f_{m+2}^{\ast })\times (f_{m+s+k}^{\ast },f_{m+s+k+1}^{\ast },f_{m+s+k+2}^{\ast })\hfill \\ \hfill & =& (f_m^{\ast }{f}_{m+s+k}^{\ast }+f_{m+1}^{\ast }{f}_{m+s+k+2}^{\ast }+f_{m+2}^{\ast }{f}_{m+s+k+1}^{\ast },\hfill \\ \hfill & & f_{m+2}^{\ast }{f}_{m+s+k+2}^{\ast }+f_m^{\ast }{f}_{m+s+k+1}^{\ast }+f_{m+1}^{\ast }{f}_{m+s+k}^{\ast },\hfill \\ \hfill & & f_{m+1}^{\ast }{f}_{m+s+k+1}^{\ast }+f_m^{\ast }{f}_{m+s+k+2}^{\ast }+f_{m+2}^{\ast }{f}_{m+s+k}^{\ast }).\hfill \end{array}$$

(10)

To obtain the first coordinate of the vector defined in Equation (8), we subtract Equation (10) from Equation (9), that is,

$$\begin{array}{ccc}& & (f_{m+s}^{\ast }{f}_{m+k}^{\ast }-f_m^{\ast }{f}_{m+s+k}^{\ast })+(f_{m+s+1}^{\ast }{f}_{m+k+2}^{\ast }-f_{m+1}^{\ast }{f}_{m+s+k+2}^{\ast })\hfill \\ & & +(f_{m+s+2}^{\ast }{f}_{m+k+1}^{\ast }-f_{m+2}^{\ast }{f}_{m+s+k+1}^{\ast })\hfill \\ & =& (f_{m+s}^{\ast }{f}_{m+k}^{\ast }-f_m^{\ast }{f}_{m+s+k}^{\ast })\hfill \\ & & +(f_{(m+1)+s}^{\ast }{f}_{(m+1)+(k+1)}^{\ast }-f_{(m+1)}^{\ast }{f}_{(m+1)+s+(k+1)}^{\ast })\hfill \\ & & +(f_{(m+2)+s}^{\ast }{f}_{(m+2)+(k-1)}^{\ast }-f_{(m+2)}^{\ast }{f}_{(m+2)+s+(k-1)}^{\ast }).\hfill \end{array}$$

By Lemma 4, we have

$$\begin{array}{ccc}& & (f_{m+s}^{\ast }{f}_{m+k}^{\ast }-f_m^{\ast }{f}_{m+s+k}^{\ast })+(f_{m+s+1}^{\ast }{f}_{m+k+2}^{\ast }-f_{m+1}^{\ast }{f}_{m+s+k+2}^{\ast })\hfill \\ & & +(f_{m+s+2}^{\ast }{f}_{m+k+1}^{\ast }-f_{m+2}^{\ast }{f}_{m+s+k+1}^{\ast })\hfill \\ & =& \left({(-1)}^m\mu (f_1^{\ast }-f_0^{\ast })f_s{f}_k\right)+\left({(-1)}^{m+1}\mu (f_1^{\ast }-f_0^{\ast })f_s{f}_{k+1}\right)\hfill \\ & & +\left({(-1)}^{m+2}\mu (f_1^{\ast }-f_0^{\ast })f_s{f}_{k-1}\right)\hfill \\ & =& {(-1)}^m\mu f_s(f_1^{\ast }-f_0^{\ast })(f_k-f_{k+1}+f_{k-1})\hfill \\ & =& {(-1)}^m\mu (f_1^{\ast }-f_0^{\ast })f_s·0.\hfill \end{array}$$

This proves the first coordinate of the vector defined in Equation (8). Proofs of the second and the third coordinates of the vector defined in Equation (8) are performed similarly using the Lemma 4, the Tagiuri-Vajda identity for Fibonacci-type sequence. □

Here are two examples, one using the Fibonacci sequence and the other using the Fibonacci–Lucas sequence.

Example 1. 

Consider the Fibonacci sequence ${\left\{f_n\right\}}_{n\in {\mathbb{N}}_0}$, $m=3$, $k=4$ and $s=5$, so $t{f}_3=(2,3,5)$, $t{f}_7=(13,21,34)$, $t{f}_8=(21,34,55)$, and $t{f}_{12}=(144,233,377)$. By Equation (2) we obtain

$$t{f}_{7}t{f}_8-t{f}_{3}t{f}_{12}=(2584,2753,2143)-(2584,2783,2173)=(0,-30,-30).$$

On the other hand,

$${(-1)}^3·(0,2f_5{f}_4,2f_4{f}_5)=-(0,2·3·5,2·3·5).$$

Example 2. 

Consider the Fibonacci–Lucas sequence ${\left\{l_n\right\}}_{n\in {\mathbb{N}}_0}$, $m=1$, $k=1$ and $s=2$, so $t{l}_1=(1,3,4)$, $t{l}_2=(3,4,7)$, $t{l}_3=(4,7,11)$, and $t{l}_4=(7,11,18)$. According to Equation (2) we have

$$t{l}_{2}t{l}_3-t{1}_{3}t{4}_{12}=(105,114,89)-(105,104,79)=(0,10,10).$$

On the other hand,

$${(-1)}^{1+1}·5·(0,2f_1{f}_2,2f_1{f}_2)=5·(0,2·1·1,2·1·1).$$

The following expressions for Fibonacci and Fibonacci–Lucas numbers with negative subscripts are well-known:

$$f_{-n}={(-1)}^{n+1}{f}_{n}and{l}_{-n}={(-1)}^n{l}_n.$$

The next set of results makes use of these and the Tagiuri-Vajda identity stated in Theorem 2.

Proposition 4.(d’Ocagne’s Identity) Let $m,n$ be any non-negative integers. For $m\ge n$ we have

$$\begin{array}{c}\hfill t{f}_m^{\ast }t{f}_{n+1}^{\ast }-t{f}_{m+1}^{\ast }t{f}_n^{\ast }={(-1)}^{1-n}\mu ·(f_1^{\ast }-f_0^{\ast })(0,2·f_{(m-n)},2·f_{(m-n)}),\end{array}$$

(11)

where ${\left\{t{f}_n^{\ast }\right\}}_{n\in {\mathbb{N}}_0}$ is the Tricomplex Fibonacci sequence, ${\left\{f_n\right\}}_{n\in {\mathbb{N}}_0}$ is the Fibonacci sequence, and μ is the characteristic of the Fibonacci-type sequence ${\left\{f_n^{\ast }\right\}}_{n\in {\mathbb{N}}_0}$.

Proof. 

In the second and third coordinate of the vector defined in Equation (11) let $k=n-m$ and $s=1$ in Theorem 2,

$$\begin{array}{ccc}\hfill {(-1)}^m(f_1^{\ast }-f_0^{\ast })\mu ·2·f_s{f}_k& =& {(-1)}^m(f_1^{\ast }-f_0^{\ast })\mu ·2·f_1{f}_{n-m}\hfill \\ & =& {(-1)}^m(f_1^{\ast }-f_0^{\ast })\mu ·2·f_1{f}_{-(m-n)}\hfill \\ & =& {(-1)}^m(f_1^{\ast }-f_0^{\ast })\mu ·2·f_1{(-1)}^{m-n+1}{f}_{(m-n)}\hfill \\ & =& {(-1)}^{2m+1-n}\mu ·2·(f_1^{\ast }-f_0^{\ast })f_1{f}_{(m-n)},\hfill \end{array}$$

since $f_{-n}={(-1)}^{n+1}{f}_n$ for all n, and as $f_1=1$, we obtain the result. □

We present an example:

Example 3. 

We have

$$\begin{array}{ccc}\hfill t{f}_{4}t{f}_3-t{f}_{5}t{f}_2& =& {(-1)}^{0+1-2}(0,2·f_{(4-2)},2·f_{(4-2)})\hfill \\ & =& (0,-2,-2);\hfill \end{array}$$

and,

$$\begin{array}{ccc}\hfill t{l}_{4}t{l}_3-t{l}_{5}t{l}_2& =& {(-1)}^{1+1-2}·(-5)·(0,2·f_{(4-2)},2·f_{(4-2)})\hfill \\ & =& (0,-10,-10).\hfill \end{array}$$

Similar to Proposition 4 we have the Catalan Identity.

Proposition 5. 

Let $m,n$ be any non-negative integers. For $m\ge n$ we have

$${\left(t{f}_m^{\ast }\right)}^2-t{f}_{m-n}^{\ast }t{f}_{m+n}^{\ast }={(-1)}^{m+n+1}\left(0,\mu ·2·{\left(f_n\right)}^2,\mu ·2·(f_1^{\ast }-f_0^{\ast }){\left(f_n\right)}^2\right),$$

where ${\left\{t{f}_n^{\ast }\right\}}_{n\in {\mathbb{N}}_0}$ is the Tricomplex Fibonacci sequence, ${\left\{f_n\right\}}_{n\in {\mathbb{N}}_0}$ is the Fibonacci sequence, and μ is the characteristic of the Fibonacci-type sequence ${\left\{f_n^{\ast }\right\}}_{n\in {\mathbb{N}}_0}$

Proof. 

Take $s=n$ and $k=-n$ in Theorem 2 we have

$$\begin{array}{ccc}\hfill {(-1)}^m\mu ·(f_1^{\ast }-f_0^{\ast })·2·f_s{f}_k& =& {(-1)}^m\mu ·(f_1^{\ast }-f_0^{\ast })·2·f_n{f}_{-n}\hfill \\ & =& {(-1)}^m\mu ·(f_1^{\ast }-f_0^{\ast })·2·f_n·{(-1)}^{n+1}·f_n\hfill \\ & =& {(-1)}^{m+n+1}·\mu (f_1^{\ast }-f_0^{\ast })·2·f_n{f}_n\hfill \\ & =& {(-1)}^{m+n+1}·\mu (f_1^{\ast }-f_0^{\ast })·2·{\left(f_n\right)}^2,\hfill \end{array}$$

the result follows. □

Taking $n=1$ in Proposition 5, we obtain:

Corollary 1.(Cassini’s Identity) For all $m\ge 1$ integer, we have

$${\left(t{f}_m^{\ast }\right)}^2-t{f}_{m-1}^{\ast }t{f}_{m+1}^{\ast }={(-1)}^m(f_1^{\ast }-f_0^{\ast })\left(0,2\mu ,2\mu \right),$$

where ${\left\{t{f}_n^{\ast }\right\}}_{n\in {\mathbb{N}}_0}$ is the Tricomplex Fibonacci sequence, ${\left\{f_n\right\}}_{n\in {\mathbb{N}}_0}$ is the Fibonacci sequence, and μ is the characteristic of the Fibonacci-type sequence ${\left\{f_n^{\ast }\right\}}_{n\in {\mathbb{N}}_0}$.

Using the same technique employed in Theorem 2, and Lemma 5, we omit the proof of the next result in the interest of brevity.

Proposition 6. 

Let ${\left\{t{f}_n^{\ast }\right\}}_{n\in {\mathbb{N}}_0}$ be the Tricomplex Fibonacci sequence. For all $m\ge 1$ integer, we have

$$\begin{array}{c}\hfill \left(a\right){\left(t{f}_m^{\ast }\right)}^2+{\left(t{f}_{m+1}^{\ast }\right)}^2=\left|\mu \right|(f_{2m+1}+2f_{m+2}(f_{m+1}+f_{m+3}),\\ \hfill f_{2m+5}+2f_{m+1}(f_m+f_{m+2}),f_{2m+3}+2(f_m{f}_{m+2}+f_{m+1}{f}_{m+3}));\\ \\ \hfill \left(b\right){\left(t{f}_{m+1}^{\ast }\right)}^2-{\left(t{f}_{m-1}^{\ast }\right)}^2=\left|\mu \right|(f_{2m}+2(f_{m+2}{f}_{m+3}-f_m{f}_{m+1}),\\ \hfill f_{2(m+2)}+2(f_{m+1}{f}_{m+2}-f_{m-1}{f}_m),f_{2(m+1)}+2f_{m+1}(f_{m+3}-f_{m-1})),\end{array}$$

where μ is the characteristic of the Fibonacci-type sequence ${\left\{f_n^{\ast }\right\}}_{n\in {\mathbb{N}}_0}$, and ${\left\{f_n\right\}}_{n\in {\mathbb{N}}_0}$ is the Fibonacci sequence.

We have the following result for the Tricomplex Fibonacci sequence.

Proposition 7. 

Let ${\left\{t{f}_n^{\ast }\right\}}_{n\in {\mathbb{N}}_0}$ be the Tricomplex Fibonacci sequence. For any non-negative integers m and n, the Tricomplex Fibonacci sequence satisfies

$$t{f}_{m-1}^{\ast }t{f}_n^{\ast }-t{f}_m^{\ast }t{f}_{n-1}^{\ast }={(-1)}^{n-1}2\mu (0,f_{m-n},f_{m-n}),$$

(12)

where μ is the characteristic of the Fibonacci-type sequence ${\left\{f_n^{\ast }\right\}}_{n\in {\mathbb{N}}_0}$, and ${\left\{f_n\right\}}_{n\in {\mathbb{N}}_0}$ is the Fibonacci sequence.

Proof. 

We have

$$\begin{array}{ccc}\hfill t{f}_{m-1}^{\ast }t{f}_n^{\ast }& =& (f_{m-1}^{\ast },f_m^{\ast },f_{m+1}^{\ast })\times (f_n^{\ast },f_{n+1}^{\ast },f_{n+2}^{\ast })\hfill \\ \hfill & =& (f_{m-1}^{\ast }{f}_n^{\ast }+f_m^{\ast }{f}_{n+2}^{\ast }+f_{m+1}^{\ast }{f}_{n+1}^{\ast },\hfill \\ \hfill & & f_{m+1}^{\ast }{f}_{n+2}^{\ast }+f_{m-1}^{\ast }{f}_{n+1}^{\ast }+f_m^{\ast }{f}_n^{\ast },\hfill \\ \hfill & & f_m^{\ast }{f}_{n+1}^{\ast }+f_{m-1}^{\ast }{f}_{n+2}^{\ast }+f_{m+1}^{\ast }{f}_n^{\ast }),\hfill \end{array}$$

(13)

and

$$\begin{array}{ccc}\hfill t{f}_m^{\ast }t{f}_{n-1}^{\ast }& =& (f_m^{\ast },f_{m+1}^{\ast },f_{m+2}^{\ast })\times (f_{n-1}^{\ast },f_n^{\ast },f_{n+1}^{\ast })\hfill \\ \hfill & =& (f_m^{\ast }{f}_{n-1}^{\ast }+f_{m+1}^{\ast }{f}_{n+1}^{\ast }+f_{m+2}^{\ast }{f}_n^{\ast },\hfill \\ \hfill & & f_{m+2}^{\ast }{f}_{n+1}^{\ast }+f_m^{\ast }{f}_n^{\ast }+f_{m+1}^{\ast }{f}_{n-1}^{\ast },\hfill \\ \hfill & & f_{m+1}^{\ast }{f}_n^{\ast }+f_m^{\ast }{f}_{n+1}^{\ast }+f_{m+2}^{\ast }{f}_{n-1}^{\ast }).\hfill \end{array}$$

(14)

To obtain the second components of the vector defined in Equation (12), we make the difference between the Equations (13) and (14), that is,

$$\begin{array}{c}\hfill (f_{m+1}^{\ast }{f}_{n+2}^{\ast }-f_{m+2}^{\ast }{f}_{n+1}^{\ast })+(f_{m-1}^{\ast }{f}_{n+1}^{\ast }-f_m^{\ast }{f}_n^{\ast })+(f_m^{\ast }{f}_n^{\ast }-f_{m+1}^{\ast }{f}_{n-1}^{\ast })\end{array}$$

By Lemma 6, we obtain

$$\begin{array}{ccc}& & (f_{m+1}^{\ast }{f}_{n+2}^{\ast }-f_{m+2}^{\ast }{f}_{n+1}^{\ast })+(f_{m-1}^{\ast }{f}_{n+1}^{\ast }-f_m^{\ast }{f}_n^{\ast })+(f_m^{\ast }{f}_n^{\ast }-f_{m+1}^{\ast }{f}_{n-1}^{\ast })\hfill \\ & =& {(-1)}^{n+1}\mu f_{m+2-(n+2)}+{(-1)}^n\mu f_{m-(n+1)}+{(-1)}^{n-1}\mu f_{m+1-n}\hfill \\ & =& {(-1)}^{n-1}\mu \left({(-1)}^2{f}_{m-n}+(-1)f_{m-n-1}+f_{m-n+1}\right)\hfill \\ & =& {(-1)}^{n-1}\mu (f_{m-n}-f_{m-n-1}+f_{m-n+1})\hfill \\ & =& {(-1)}^{n-1}\mu (f_{m-n+2}-f_{m-n+1})\hfill \\ & =& {(-1)}^{n-1}\mu ·2f_{m-n}.\hfill \end{array}$$

This proves the second component of the vector defined in Equation (12). Proofs of the first and the third coordinates of the vector defined in Equation (12) are performed similarly using also Lemma 6. □

In concluding this section, we present a specific result that applies to the Tricomplex Fibonacci sequence, which is restricted to the Fibonacci sequence ${\left\{f_n\right\}}_{n\in {\mathbb{N}}_0}$. This result gives a convolution identity for the Tricomplex Fibonacci sequence.

Proposition 8. 

Let ${\left\{t{f}_n\right\}}_{n\in {\mathbb{N}}_0}$ be the Tricomplex Fibonacci sequence. For any non-negative integers m and n, the Tricomplex Fibonacci sequence satisfies

$$t{f}_{m+1}t{f}_n+t{f}_{m}t{f}_{n-1}=(f_{m+n}+2f_{m+n+3},f_{m+n+4}+2f_{m+n+1},3f_{m+n+2}),$$

(15)

where ${\left\{f_n\right\}}_{n\in {\mathbb{N}}_0}$ is the Fibonacci sequence.

Proof. 

We have

$$\begin{array}{ccc}\hfill t{f}_{m+1}t{f}_n& =& (f_{m+1},f_{m+2},f_{m+3})\times (f_n,f_{n+1},f_{n+2})\hfill \\ \hfill & =& (f_{m+1}{f}_n+f_{m+2}{f}_{n+2}+f_{m+3}{f}_{n+1},\hfill \\ \hfill & & f_{m+3}{f}_{n+2}+f_{m+1}{f}_{n+1}+f_{m+2}{f}_n,\hfill \\ \hfill & & f_{m+2}{f}_{n+1}+f_{m+1}{f}_{n+2}+f_{m+3}{f}_n),\hfill \end{array}$$

(16)

and,

$$\begin{array}{ccc}\hfill t{f}_{m}t{f}_{n-1}& =& (f_m,f_{m+1},f_{m+2})\times (f_{n-1},f_n,f_{n+1})\hfill \\ \hfill & =& (f_m{f}_{n-1}+f_{m+1}{f}_{n+1}+f_{m+2}{f}_n,\hfill \\ \hfill & & f_{m+2}{f}_{n+1}+f_m{f}_n+f_{m+1}{f}_{n-1},\hfill \\ \hfill & & f_{m+1}{f}_n+f_m{f}_{n+1}+f_{m+2}{f}_{n-1}).\hfill \end{array}$$

(17)

To extract the first components of the vector given in Equation (15), we add Equations (16) and (17), that is,

$$\begin{array}{ccc}& & (f_{m+1}{f}_n+f_m{f}_{n-1})+(f_{m+2}{f}_{n+2}+f_{m+1}{f}_{n+1})\hfill \\ & & +(f_{m+3}{f}_{n+1}+f_{m+2}{f}_n)\hfill \\ & =& (f_{m+1}{f}_n+f_m{f}_{n-1})+(f_{(m+1)+1}{f}_{n+2}+f_{m+1}{f}_{(n+2)-1})\hfill \\ & & +(f_{(m+2)+1}{f}_{n+1}+f_{m+2}{f}_{(n+1)-1}).\hfill \end{array}$$

From Lemma 7, we deduce

$$\begin{array}{ccc}& & (f_{m+1}{f}_n+f_m{f}_{n-1})+(f_{m+2}{f}_{n+2}+f_{m+1}{f}_{n+1})\hfill \\ & & +(f_{m+3}{f}_{n+1}+f_{m+2}{f}_n)\hfill \\ & =& f_{m+n}+f_{(m+1)+(n+2)}+f_{(m+2)+(n+1)}\hfill \\ & =& f_{m+n}+f_{m+n+3}+f_{m+n+3}\hfill \\ & =& f_{m+n}+2f_{m+n+3}.\hfill \end{array}$$

This proves the first coordinate of the vector defined in Equation (15). Proofs of the second and the third components of the vector defined in Equation (15) are performed similarly. □

Example 4. 

Consider the Fibonacci sequence ${\left\{f_n\right\}}_{n\in {\mathbb{N}}_0}$, $n=3$ and $m=3$, so $t{f}_2=(1,2,3)$, $t{f}_3=(2,3,5)$ and $t{f}_4=(3,5,8)$. According to Equation (2) we have

$$t{f}_{4}t{f}_2+t{f}_{3}t{f}_2=(55,59,46)+(21,22,17)=(76,81,63).$$

On the other hand,

$$(f_6+2f_9,f_{10}+2f_7,3f_8)=(8+2·13,55+2·13,3·21)=(76,81,63).$$

5. Sum of Terms Involving the Tricomplex Fibonacci Numbers

In this section, we present results on partial sums of terms of the Tricomplex Fibonacci numbers with n integers. Initially, consider the sequence of partial sums $f_n^{\ast }={\sum }_{k=0}^n{f}_k^{\ast }=f_0^{\ast }+f_1^{\ast }+f_2^{\ast }+\cdots +f_n^{\ast }$, for $n\in {\mathbb{N}}_0$, where ${\left\{f_n^{\ast }\right\}}_{n\in {\mathbb{N}}_0}$ is the Fibonacci-type sequence.

Now, we consider the sequence of partial sums $t{f}_n^{\ast }={\sum }_{k=0}^{n}t{f}_k^{\ast }=t{f}_0^{\ast }+t{f}_1^{\ast }+t{f}_2^{\ast }+\cdots +t{f}_n^{\ast }$, for $n\in {\mathbb{N}}_0$, where ${\left\{t{f}_n^{\ast }\right\}}_{n\in {\mathbb{N}}_0}$ is the Tricomplex Fibonacci sequence.

Proposition 9. 

Let ${\left\{t{f}_n^{\ast }\right\}}_{n\in {\mathbb{N}}_0}$ be the Tricomplex Fibonacci sequence. For all non-negative integers n, we have the following identities:

$$\begin{array}{ccc}\hfill \left(a\right)\sum _{k=0}^{n}t{f}_k^{\ast }& =& (f_{n+2}^{\ast },f_{n+3}^{\ast },f_{n+4}^{\ast })-(f_1^{\ast },f_2^{\ast },f_3^{\ast }),\hfill \\ \hfill \left(b\right)\sum _{k=0}^{n}t{f}_{2k}^{\ast }& =& (f_{2n+1}^{\ast },f_{2n+2}^{\ast },f_{2n+3}^{\ast })+(f_0^{\ast },f_1^{\ast },f_2^{\ast })-(f_1^{\ast },f_2^{\ast },f_3^{\ast }),\hfill \\ \hfill \left(c\right)\sum _{k=0}^{n}t{f}_{2k+1}^{\ast }& =& (f_{2n+2}^{\ast },f_{2n+3}^{\ast },f_{2n+4}^{\ast })-(f_0^{\ast },f_1^{\ast },f_2^{\ast }),\hfill \end{array}$$

where ${\left\{f_n^{\ast }\right\}}_{n\in {\mathbb{N}}_0}$ is the Fibonacci-type sequence.

Proof. (a) According to Proposition 1 we have

$$t{f}_n^{\ast }=t{f}_{n+2}^{\ast }-t{f}_{n+1}^{\ast }.$$

So

$$\begin{array}{ccc}\hfill t{f}_0^{\ast }& =& t{f}_2^{\ast }-t{f}_1^{\ast }\hfill \\ \hfill t{f}_1^{\ast }& =& t{f}_3^{\ast }-t{f}_2^{\ast }\hfill \\ \hfill t{f}_2^{\ast }& =& t{f}_4^{\ast }-t{f}_3^{\ast }\hfill \\ \hfill ⋮& & ⋮\hfill \\ \hfill t{f}_{n-1}^{\ast }& =& t{f}_{n+1}^{\ast }-t{f}_n^{\ast }\hfill \\ \hfill t{f}_n^{\ast }& =& t{f}_{n+2}^{\ast }-t{f}_{n+1}^{\ast }.\hfill \end{array}$$

From this we get

$$\begin{array}{ccc}\hfill t{f}_n^{\ast }& =& t{f}_0^{\ast }+t{f}_1^{\ast }+t{f}_2^{\ast }+\cdots +t{f}_{n-1}^{\ast }+t{f}_n^{\ast }\hfill \\ & =& (t{f}_2^{\ast }-t{f}_1^{\ast })+(t{f}_3^{\ast }-t{f}_2^{\ast })+(t{f}_4^{\ast }-t{f}_3^{\ast })+\cdots +(t{f}_{n+1}^{\ast }-t{f}_n^{\ast })+(t{f}_{n+2}^{\ast }-t{f}_{n+1}^{\ast })\hfill \\ & =& t{f}_{n+2}^{\ast }-t{f}_1^{\ast },\hfill \end{array}$$

and we obtain the result.

(b) According to Proposition 1 we also have

$$t{f}_{n+1}^{\ast }=t{f}_{n+2}^{\ast }-t{f}_n^{\ast }.$$

So

$$\begin{array}{ccc}\hfill t{f}_2^{\ast }& =& t{f}_3^{\ast }-t{f}_1^{\ast }\hfill \\ \hfill t{f}_4^{\ast }& =& t{f}_5^{\ast }-t{f}_3^{\ast }\hfill \\ \hfill ⋮& & ⋮\hfill \\ \hfill t{f}_{2n}^{\ast }& =& t{f}_{2n+1}^{\ast }-t{f}_{2n-1}^{\ast }.\hfill \end{array}$$

From this we get

$$\begin{array}{ccc}\hfill t{f}_{2n}^{\ast }& =& t{f}_0^{\ast }+t{f}_2^{\ast }+t{f}_4^{\ast }+\cdots +t{f}_{2n}^{\ast }\hfill \\ & =& t{f}_0^{\ast }+(t{f}_3^{\ast }-t{f}_1^{\ast })+(t{f}_5^{\ast }-t{f}_3^{\ast })+(t{f}_7^{\ast }-t{f}_5^{\ast })+\cdots +(t{f}_{2n+1}^{\ast }-t{f}_{2n-1}^{\ast })\hfill \\ & =& t{f}_{2n+1}^{\ast }+t{f}_0^{\ast }-t{f}_1^{\ast },\hfill \end{array}$$

and the result follows.

(c) Similarly,

$$\begin{array}{ccc}\hfill t{f}_1^{\ast }& =& t{f}_2^{\ast }-t{f}_0^{\ast }\hfill \\ \hfill t{f}_3^{\ast }& =& t{f}_4^{\ast }-t{f}_2^{\ast }\hfill \\ \hfill ⋮& & ⋮\hfill \\ \hfill t{f}_{2n+1}^{\ast }& =& t{f}_{2n+2}^{\ast }-t{f}_{2n}^{\ast }.\hfill \end{array}$$

From this we obtain

$$\begin{array}{ccc}\hfill t{f}_{2n+1}^{\ast }& =& t{f}_1^{\ast }+t{f}_3^{\ast }+t{f}_5^{\ast }+\cdots +t{f}_{n-1}^{\ast }+t{f}_n^{\ast }\hfill \\ & =& (t{f}_2^{\ast }-t{f}_0^{\ast })+(t{f}_4^{\ast }-t{f}_2^{\ast })+(t{f}_6^{\ast }-t{f}_4^{\ast })+\cdots ++(t{f}_{2n+2}^{\ast }-t{f}_{2n}^{\ast })\hfill \\ & =& t{f}_{2n+2}^{\ast }-t{f}_0^{\ast },\hfill \end{array}$$

as required. □

A direct consequence of the Proposition 9.

Proposition 10. 

Let ${\left\{t{f}_n^{\ast }\right\}}_{n\in {\mathbb{N}}_0}$ be the Tricomplex Fibonacci sequence. For all non-negative integers n, we have the following identities:

$${\displaystyle \left(a\right)\sum _{k=0}^n{(-1)}^{k}t{f}_k^{\ast }=-t{f}_{2n}^{\ast }-t{f}_1^{\ast };}$$

if the last term is negative, and

$${\displaystyle \left(b\right)\sum _{k=0}^n{(-1)}^{k}t{f}_k^{\ast }=t{f}_{2n+1}^{\ast }-t{f}_1^{\ast };}$$

if the last term is positive.

Proof. (a) First consider when the last term is negative. Then

$$\begin{array}{ccc}\hfill \sum _{k=0}^{2n+1}{(-1)}^{k}t{f}_k^{\ast }& =& t{f}_0^{\ast }-t{f}_1^{\ast }+t{f}_2^{\ast }-t{f}_3^{\ast }+\cdots +t{f}_{2n}^{\ast }-t{f}_{2n+1}^{\ast }\hfill \\ & =& (t{f}_0^{\ast }+t{f}_2^{\ast }+\cdots +t{f}_{2n}^{\ast })-(t{f}_1^{\ast }+t{f}_3^{\ast }+\cdots +t{f}_{2n+1}^{\ast })\hfill \\ & =& \sum _{k=0}^{n}t{f}_{2k}^{\ast }-\sum _{k=0}^{n}t{f}_{(2k+1)}^{\ast }\hfill \end{array}$$

According to Proposition 9, items (b) and (c), it follows that

$$\begin{array}{ccc}\hfill \sum _{k=0}^{2n+1}{(-1)}^{k}t{f}_k^{\ast }& =& t{f}_{2n+1}^{\ast }+t{f}_0^{\ast }-t{f}_1^{\ast }-(t{f}_{2n+2}^{\ast }+t{f}_0^{\ast })\hfill \\ & =& t{f}_{2n+1}^{\ast }-t{f}_{2n+2}^{\ast }-t{f}_1^{\ast }.\hfill \end{array}$$

We obtain the result, using the Proposition 1.

(b) In case the last term is positive, we have

$$\begin{array}{ccc}\hfill \sum _{k=0}^{2(n+1)}{(-1)}^{k}t{f}_k^{\ast }& =& t{f}_0^{\ast }-t{f}_1^{\ast }+t{f}_2^{\ast }-t{f}_3^{\ast }+\cdots +t{f}_{2n}^{\ast }-t{f}_{2n+1}^{\ast }+t{f}_{2n+2}^{\ast }\hfill \\ & =& \sum _{k=0}^{n+1}t{f}_{2k}^{\ast }-\sum _{k=0}^{n}t{f}_{2k+1}^{\ast }\hfill \end{array}$$

As in item (a), apply the Proposition 9 and Proposition 1, the result follows. □

6. Conclusions

In this paper, we introduced a new family of arithmetic sequences related to the Fibonacci numbers. We investigated Fibonacci-type sequences by introducing the Fibonacci sequence into the tricomplex ring $\mathbb{T}$. In doing so, we extended and explored potentially innovative mathematical concepts and results to advance the knowledge of tricomplex numbers, as evidenced by the results presented in this study. Certainly, by changing the initial terms, we will obtain new sequences that deserve study. We leave it to the interested reader to study these extensions in more detail. That is, the paper presented tricomplex Fibonacci sequences. Possible extensions to higher dimensions, such as the complex, quaternions, or other multidimensional number systems, could be investigated. Of course, further extensions would depend on verifying that the ring structure of $\mathbb{T}$ remains valid when using coefficients from more complex systems such as complex numbers, quaternions, or other multidimensional numbers. Preserving the ring properties is crucial to the consistency and applicability of these generalized Fibonacci-type sequences in higher-dimensional spaces.