On Tricomplex Horadam Numbers: A New Class of Horadam Sequences

Abstract

This study introduces an innovative approach to Horadam sequences. The aim of this paper is to investigate the Tricomplex Horadam sequence and its properties. It begins with the tricomplex ring $\mathbb{T}$ and key results related to Horadam-type sequences. The Tricomplex Horadam sequence is then defined, with a discussion of its properties, the Binet formula, and the generating function in vector form. Next, several fundamental identities, including the Tagiuri–Vajda and d’Ocagne identities, are examined, along with their implications and examples from previous Tricomplex Horadam-type sequences. Finally, the sum of terms associated with the Tricomplex Horadam sequence is presented. The research problem consists in determining properties symmetrical or analogous to the Horadam-type sequence for the Tricomplex Horadam sequence.

1. Introduction

In this work, we consider the ordinary Horadam sequence ${\left\{h_n\right\}}_{n\ge 0}$ recursively defined by a homogeneous second order linear recurrence relation of the type $h_{n+2}=p·h_{n+1}-q·h_n$, with initial terms $h_0=a$, $h_1=b$, p and q being arbitrary integers.

This sequence type was introduced in 1961 by Horadam [1], and it generalizes many sequences with a characteristic equation of recurrence relation of the form $x^2-px-q=0$. Recursive relations define several families of integers that are studied in the literature. These sequences of numbers are at the origin of many interesting identities. Horadam sequences and their generalizations have been the subject of intense research by numerous authors in recent decades. Horadam, in [2,3], established the fundamental properties of these sequences. Since then, many researchers have dedicated themselves to generalizing and expanding the concept of the Horadam sequence. For instance, a study of Horadam vectors and their geometric properties is conducted in [4], in [5,6], the Horadam sequence was extended to the domain of quaternions, while refs. [7,8] investigated Horadam spinors, opening up new perspectives for the study of these sequences. Additionally, refs. [9,10] explored the polynomial representation of Horadam sequences, and ref. [11] presented a matrix approach for studying these sequences. These various generalizations have significantly contributed to a better understanding of the properties and applications of Horadam-type sequences (see, refs. [12,13,14,15,16], among others).

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 $x_i,y_i,z_i\in \mathbb{R}$ for $i=1,2$.

Now, in $\mathbb{T}$, we present a novel type of sequence referred to as the Tricomplex Horadam sequence.

Definition 1.

For all non-negative integers $n\ge 0$, the Tricomplex Horadam sequence ${\left\{T{H}_n\right\}}_{n\ge 0}$ is defined as $T{H}_n=(h_n,h_{n+1},h_{n+2})$, where $(h_n,h_{n+1},h_{n+2})$ belongs to the tricomplex ring $\mathbb{T}$, and $h_n$ is the n-Horadam number.

To illustrate this point, consider the following examples: $T{H}_0=(h_0,h_1,h_2)$ and $T{H}_1=(h_1,h_2,h_3)$. Additionally, the value of $T{H}_2$ is given by $(h_2,h_3,h_4)$. It is important to note that

$$\begin{array}{ccc}\hfill p·T{H}_1-q·T{H}_0& =& p·(h_1,h_2,h_3)-q·(h_0,h_1,h_2)\hfill \\ & =& (p·h_1-q·h_0,p·h_2-q·h_1,p·h_3-q·h_2)\hfill \\ & =& (h_2,h_3,h_4),\hfill \end{array}$$

and so $p·T{H}_1+q·T{H}_0=T{H}_2$.

In this paper, we will look at the symmetry between the two sequences: the common Horadam sequence and the Tricomplex Horadam sequence. The structure of this work is divided into four additional sections, as outlined below. In Section 2, we briefly present the tricomplex ring $\mathbb{T}$ and list some key results of Horadam-type sequences. In Section 3, we define the Tricomplex Horadam sequence, explaining some of its characteristics and properties, and exhibit the Binet formula, which provides an explicit expression for the terms of the sequence. Furthermore, we explore the generating function, expressed in its vector form. In Section 4, we explore several fundamental identities for the Tricomplex Horadam sequence, such as those of Tagiuri–Vajda and d’Ocagne, and their consequences. The classical identities are illustrated through previously known Tricomplex Horadam-type sequences. Finally, in Section 5, we present the sum of the terms involving the terms of the Tricomplex Horadam sequence.

The Tricomplex Horadam sequence is defined using the Tricomplex ring, which has three real coordinates and real components. This construction constitutes a second-order vector generalization that unifies and extends several well-known sequences in the literature. In [17,18,19], the authors investigate the properties of three specific Tricomplex Horadam sequences, respectively, the tricomplex Fibonacci sequence ${\left\{t{f}_n\right\}}_{n\ge 0}$, the tricomplex repunit sequence ${\left\{t{r}_n\right\}}_{n\ge 0}$, and the tricomplex Pell sequence ${\left\{t{p}_n\right\}}_{n\ge 0}$. In these works, the properties, classical identities, the Binet formula, and generating functions for these particular sequences are presented. In this paper, we generalize these results for all Tricomplex Horadam sequences with $\lambda =p^2-4q>0$.

2. Background and Auxiliary Results

Since the time of Gauss, numerous researchers have explored rings with arithmetic properties similar to those of the integers. Notable examples include the Gaussian ring of integers, composed of complex numbers with integer real and imaginary parts, and the Eisenstein ring of integers, which holds significant importance in this context. In this paper, we make use of the tricomplex ring defined in three real coordinates with real components, specifically, the terms of the Horadam sequence.

2.1. The Tricomplex Ring

The tricomplex number is an element of a number system that extends the complex numbers. Just as complex numbers have a real part and an imaginary part, tricomplex numbers have two imaginary parts along with a real part. Olariu, in [20,21], introduced the concept of tricomplex numbers that are expressed in the form $(a,b,c)=a+bi+cj$, where a, b, and c are real numbers and i and j are imaginary units. According to [20], the multiplication rules for the tricomplex units $1,i$, and j are given in

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, according to 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 by 0, and the tricomplex unity is $(1,0,0)$, denoted simply by 1. Moreover, it can be checked that $\mathbb{T}$ is the commutative and unit ring, see [20,21,22,23,24]. Additionally, in [25,26], a study is presented on the three and four dimensional complex algebraic structure, with applications in constructing directional probability distributions in four-dimensional complex space in [26].

2.2. The Horadam Sequence

We recall that, given arbitrary initial values $h_0=a$ and $h_1=b$, a Horadam sequence ${\left\{h_n\right\}}_{n\ge 0}={\left\{h_n(a,b;p,q)\right\}}_{n\ge 0}$ is defined by the second-order linear recurrence relation

$$h_n=p{h}_{n-1}-q{h}_{n-2},n\ge 2,$$

where p and q are fixed constants. Particular cases of the Horadam sequence $h_n(a,b,p,q)$ are Fibonacci sequence ${\left\{f_n\right\}}_{n\ge 0}$, Lucas sequence ${\left\{l_n\right\}}_{n\ge 0}$, Pell sequence ${\left\{P_n\right\}}_{n\ge 0}$, Pell–Lucas sequence ${\left\{p_n\right\}}_{n\ge 0}$, Jacobsthal sequence $J_n$, Jacobsthal–Lucas sequence ${\left\{j_n\right\}}_{n\ge 0}$, Mersenne sequence ${\left\{M_n\right\}}_{n\ge 0}$, Mersenne–Lucas sequence ${\left\{m_n\right\}}_{n\ge 0}$, repunit sequence ${\left\{r_n\right\}}_{n\ge 0}$, and one-zero sequence ${\left\{U_n\right\}}_{n\ge 0}$, among others.

Table 2. Special sequences associated with the choice of $a,b,p\mathrm{and}q$.

Seq.	a	b	p	q	${\mathbf{h}}_{\mathbf{n}+2}={\mathbf{ph}}_{\mathbf{n}+1}-{\mathbf{qh}}_{\mathbf{n}}$	id Num [27]	Name
$f_n$	0	1	1	−1	$f_{n+2}=f_{n+1}+f_n$	A000045	Fibonacci
$l_n$	2	1	1	−1	$l_{n+2}=l_{n+1}+l_n$	A000032	Fibonacci–Lucas
$P_n$	0	1	2	−1	$P_{n+2}=2P_{n+1}+P_n$	A000129	Pell
$p_n$	1	1	2	−1	$p_{n+2}=2p_{n+1}+p_n$	A001333	Pell–Lucas
$J_n$	0	1	1	−2	$J_{n+2}=J_{n+1}+2J_n$	A001045	Jacobsthal
$j_n$	2	1	1	−2	$j_{n+2}=j_{n+1}+2j_n$	A014551	Jacobsthal–Lucas
$M_n$	0	1	3	2	$M_{n+2}=3M_{n+1}-2M_n$	A000225	Mersenne
$m_n$	2	3	3	2	$m_{n+2}=3m_{n+1}-2m_n$	A000051	Mersenne–Lucas
$r_n$	0	1	11	10	$r_{n+2}=11r_{n+1}-10r_n$	A002275	Repunit
$U_n$	0	1	101	100	$U_{n+2}=101U_{n+1}-100U_n$	A094028	One-Zero

presents the recurrence relations for some specific cases of Horadam-type sequences.

Remember that the characteristic equation for the Horadam sequence is

$$r^2-pr+q=0,$$

(3)

where $p,q$ are arbitrary integers. Solving it for r gives the roots of the characteristic equation. So, $\alpha =b+\sqrt{p^2-4q})/2$ and $\beta =b-\sqrt{p^2-4q})/2$ are the distinct roots of Equation (3); furthermore, $\alpha$ and $\beta$ are conjugate numbers. We use these to determine the Binet formula, which provides a direct way to compute the n-th Horadam number without iterating through the sequence. Recall that we considered $\lambda =p^2-4q>0$.

The next result is a special case: [2] (Equation (1.6)) or [28] (Equation (2)).

Lemma 1

(Binet-like formula). Let $c=h_1-h_0\beta$ and $d=h_0\alpha -h_1$. Then,

$$h_n={\displaystyle \frac{c{\alpha }^n+d{\beta }^n}{\alpha -\beta }},$$

(4)

where α and β are the distinct roots of Equation (3) and $h_0=a$ and $h_1=b$ are the initial terms from the Horadam sequence ${\left\{h_n\right\}}_{n\ge 0}$.

According to Equation (1.4) in [2], we have

$$\alpha +\beta =p,\alpha \beta =q,\alpha -\beta =\sqrt{p^2-4q}\mathrm{and}\lambda =p^2-4q.$$

(5)

Let ${\left\{h_n\right\}}_{n\ge 0}$ be the Horadam sequence. According to [2], considering the initial value $h_0=1$ and $h_1=p$ in the Horadam recurrence, we obtain the Fibonacci-type sequence (or the first kind of Lucas sequence), denoted by $u_n(0,p,p,q)$. On the other hand, the initial conditions $h_0=2$ and $h_1=p$ yield the Lucas-type sequence (or the second kind of Lucas sequence), denoted by $v_n(2,p,p,q)$. As a consequence of Equation (4), the Binet formulas associated with the sequences ${\left\{u_n\right\}}_{n\ge 0}$ and ${\left\{v_n\right\}}_{n\ge 0}$ are

$$u_n={\displaystyle \frac{{\alpha }^n-{\beta }^n}{\alpha -\beta }},$$

(6)

$$v_n={\alpha }^n+{\beta }^n,$$

(7)

where $\alpha$ and $\beta$ are the distinct roots of Equation (3).

Next, we present a compilation of identities concerning the Horadam sequence ${\left\{h_n\right\}}_{n\ge 0}$, collected from some of the existing literature. These will prove to be required in establishing analogous identities for the Tricomplex Horadam sequence.

Lemma 2.

Let $m,k,s$ be any non-negative integers, ${\left\{h_n\right\}}_{n\ge 0}$ be the Horadam sequence, and ${\left\{u_n\right\}}_{n\ge 0}$ and ${\left\{v_n\right\}}_{n\ge 0}$ be the Horadam-type sequence, respectively, given by Equations (6) and (7). We have the following identities

(a) 

ref. [2] (Equation (3.14)) $h_{m+n}=h_{m+1}{u}_{n+1}-q{h}_m{u}_n$;

(b) 

ref. [29] (Equation (28)) $q^n{h}_{m-n}=h_m{u}_{n+1}-h_{m+1}{u}_n$;

(c) 

ref. [29] (Equation (14)) $h_{n+1}^2-q{h}_n^2={\lambda }^2\left((h_1^2-h_0^{2}q)u_{2n+1}-h_{0}q(2h_1-h_{0}p)u_{2n}\right)$;

(d) 

ref. [2] (Equation (4.17)) $h_{n+1}^2-q^2{h}_{n-1}^2=h_1{h}_{2n+1}+(h_1-p{h}_0)h_{2n-1}$,

(e) 

ref. [30] (Equation (4)) $h_{m+n}^2-q^{3n}{h}_{m-2n}^2=(v_n^2-q^n)(h_m^2-q^n{h}_{m-n}^2)$,

(f) 

ref. [29] (Equation (16)) $h_{m+s}·h_{m-k}-h_m·h_{m+s-k}=e{q}^{n-r}(v_{s+k}-q^s{v}_{k-s})$,

where $e=p{h}_0{h}_1-q{h}_0^2-h_1^2$ and $\lambda =p^2-4q$.

In the mathematical literature, the function $G{F}_{a_n}\left(x\right)$ is referred to as the generating function for the sequence ${\left\{a_n\right\}}_{n\ge 0}$. It is defined as

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

Furthermore, the following result presents the ordinary generating function for the Horadam sequence, and it can be found in [29] (Theorem 3.1).

Lemma 3.

The ordinary generating function for the Horadam sequence ${\left\{h_n\right\}}_{n\ge 0}$, denoted by $G{F}_{h_n}\left(x\right)$, is given by

$$G{F}_{h_n}\left(x\right)={\displaystyle \frac{h_0+(h_1-p{h}_0)x}{1-px+q{x}^2}},$$

were $h_0=a$ and $h_1=b$ are the initial terms.

In the same way, the exponential generating function $E_{a_n}\left(x\right)$ of a sequence ${\left\{a_n\right\}}_{n\ge 0}$ 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 $a_n=h_n$ and, making use of Equation (4), we express the classical exponential generating function for the Horadam sequence ${\left\{h_n\right\}}_{n\ge 0}$.

Proposition 1.

Let ${\left\{h_n\right\}}_{n\ge 0}$ be a Horadam sequence. For all non-negative integers n, the exponential generating function for the Horadam sequence ${\left\{h_n\right\}}_{n\ge 0}$ is

$$E_{h_n}\left(x\right)={\displaystyle \frac{c{e}^{\alpha x}-d{e}^{\beta x}}{\alpha -\beta }},$$

where α and β are the distinct roots of Equation (3), $c=h_1-h_0\beta$, and $d=h_0\alpha -h_1$.

Proof. 

Note that

$$E_{h_n}\left(x\right)=\sum _{n=0}^{\infty }{\displaystyle \frac{h_n{x}^n}{n!}},$$

by Equation (4), we have

$$\begin{array}{ccc}\hfill E_{h_n}\left(x\right)& =& \sum _{n=0}^{\infty }({\displaystyle \frac{(c{\alpha }^n+d{\beta }^n)x^n}{(\alpha -\beta )n!}}\hfill \\ & =& {\displaystyle \frac{c}{\alpha -\beta }}\sum _{n=0}^{\infty }{\displaystyle \frac{{\left(\alpha x\right)}^n}{n!}}+{\displaystyle \frac{d}{\alpha -\beta }}\sum _{n=0}^{\infty }{\displaystyle \frac{{\left(\beta x\right)}^n}{n!}},\hfill \end{array}$$

and we have the validity of the result.

□

The Poisson generating function $P_{a_n}\left(x\right)$ for a sequence ${\left\{a_n\right\}}_{n\ge 0}$ is defined as

$$P_{a_n}\left(x\right)=\sum _{n=0}^{\infty }{a}_n{\displaystyle \frac{x^n}{n!}}{e}^{-x}=e^{-x}{E}_{a_n}\left(x\right),$$

where $E_{a_n}\left(x\right)$ denotes the exponential generating function. Consequently, the Poisson generating function for specific sequences can be derived from Proposition 1.

Corollary 1.

For all $n\ge 0$, the Poisson generating function for the Horadam sequence ${\left\{h_n\right\}}_{n\ge 0}$ is given by

$$P_{h_n}\left(x\right)={\displaystyle \frac{c{e}^{(\alpha -1)x}-d{e}^{(\beta -1)x}}{\alpha -\beta }}.$$

Another consequence of Binet’s formula is the Tagiuri–Vajda identity for the Horadam sequence. This result will prove useful in establishing a parallel result for the Tricomplex Horadam sequence.

Theorem 1

(Tagiuri–Vajda identity). Let $m,k,s$ be any non-negative integers and ${\left\{h_n\right\}}_{n\ge 0}$ be the Horadam sequence. We have the following identity

$$h_m{h}_{m+s+k}-h_{m+s}{h}_{m+k}=cd{q}^m{u}_k{u}_s,$$

(8)

where $c=h_1-h_0\beta$, $d=h_0\alpha -h_1$, α and β are the distinct roots of Equation (3), and $u_n$ are the Fibonacci numbers given in Equation (6).

Proof. 

Applying Equation (4), we obtain

$$\begin{array}{cc}\hfill & h_{m+s}{h}_{m+k}-h_m{h}_{m+s+k}\hfill \\ \hfill =& {\displaystyle \frac{c{\alpha }^{m+s}+d{\beta }^{m+s}}{\alpha -\beta }}{\displaystyle \frac{c{\alpha }^{m+k}+d{\beta }^{m+k}}{\alpha -\beta }}-{\displaystyle \frac{c{\alpha }^m+d{\beta }^m}{\alpha -\beta }}{\displaystyle \frac{c{\alpha }^{m+s+k}+d{\beta }^{m+s+k}}{\alpha -\beta }}\hfill \\ \hfill =& {\displaystyle \frac{c^2{\alpha }^{2m+s+k}+cd{\alpha }^{m+s}{\beta }^{m+k}+cd{\alpha }^{m+k}{\beta }^{m+s}+d^2{\beta }^{2m+k+s}}{{(\alpha -\beta )}^2}}\hfill \\ \hfill & -{\displaystyle \frac{c^2{\alpha }^{2m+s+k}+cd{\alpha }^m{\beta }^{m+s+k}+cd{\beta }^m{\alpha }^{m+s+k}+d^2{\beta }^{2m+s+k}}{{(\alpha -\beta )}^2}}\hfill \\ \hfill =& -cd{\left(\alpha \beta \right)}^m\left({\displaystyle \frac{{\alpha }^k-{\beta }^k}{\alpha -\beta }}\right)\left({\displaystyle \frac{{\alpha }^s-{\beta }^s}{\alpha -\beta }}\right).\hfill \end{array}$$

By Equation (5), we have $\alpha \beta =q$, and by Equation (6) $u_j={\displaystyle \frac{{\alpha }^j-{\beta }^j}{\alpha -\beta }}$ for all non-negative integers j, which verifies the result. □

To illustrate the Tagiuri–Vajda identity for the Horadam-type sequence, we present two examples.

Example 1.

Considering the Fibonacci sequence ${\left\{f_n\right\}}_{n\ge 0}$ specified when we take $p=1,q=-1,h_0=0$, and $h_1=1$ (see Table 2). So, $c=1$, $d=-1$, and we have that the Tagiuri–Vajda identity for the Fibonacci sequence is

$$f_{m+s}{f}_{m+k}-f_m{f}_{m+s+k}={(-1)}^{m+1}{f}_s{f}_k.$$

according to Equations (20a) and (20b) in [31].

Example 2.

Considering the repunit sequence ${\left\{r_n\right\}}_{n\ge 0}$ specified when we take $p=11,q=10,h_0=0$, and $h_1=1$ (see Table 2). So, $c=1$, $d=-1$, and we have that the Tagiuri–Vajda identity for the repunit sequence is

$$r_{m+s}{r}_{m+k}-r_m{r}_{m+s+k}={10}^m{r}_s{r}_k,$$

according to Theorem 3.1 in [32].

Now, to conclude this section, consider the sequence of partial sums ${\sum }_{k=0}^n{h}_k=h_0+h_1+h_2+\cdots +h_n$, for $n\ge 0$, where ${\left\{h_n\right\}}_{n\ge 0}$ is the Horadam sequence.

Lemma 4.

[2] (Equations (3.5), (3.12) and (3.13)) Let ${\left\{h_n\right\}}_{n\ge 0}$ be the Horadam sequence; then

(a) 

${\displaystyle \sum _{k=0}^n{h}_k=\frac{h_{n+2}-h_1-(p-1)(h_{n+1}-h_0)}{p-q-1}}$, with $p-q-1\ne 0$;

(b) 

${\displaystyle \sum _{k=0}^n{h}_{2k}=\frac{(1+q)(h_{2n+2}-h_0)-pq(h_{2n+1}-h_{-1})}{p^2-{(q+1)}^2}}$, with $p^2\ne {(q+1)}^2$;

(c) 

${\displaystyle \sum _{k=0}^n{h}_{2k+1}=\frac{p(h_{2n+2}-h_0)-q(1+q)(h_{2n+1}-h_{-1})}{p^2-{(q+1)}^2}}$, with $p^2\ne {(q+1)}^2$.

Finally, in this paper, we refer to an extension of the Horadam sequence with negative subscripts, as presented in [2] (Equation (2.18)). Specifically, for negative indexes, the authors define

$$h_{-n}=q^{-n}{\displaystyle \frac{h_0{u}_n-h_1{u}_{n-1}}{h_0{u}_n+(h_1-p{h}_0)u_{n-1}}}{h}_n.$$

As a consequence, we obtain the following relations for the sequences $u_n$ and $v_n$:

$$u_{-n}=-q^{-n}{u}_n,$$

(9)

and

$$v_{-n}=-{\displaystyle \frac{q^{-n}}{p}}{v}_n\mathrm{for}\mathrm{all}\mathrm{non}\text{-}\mathrm{negative}\mathrm{integers}n.$$

3. Binet’s Formula and Generating Function for Tricomplex Horadam Sequence

In this section, we explore the Tricomplex Horadam sequence ${\left\{h_n\right\}}_{n\ge 0}$, a second-order vector generalization that encompasses and unifies various well-known sequences in the literature. We demonstrate that this sequence joins to Binet-like formulas and derives both its exponential and ordinary generating functions.

First, in the ring $\mathbb{T}$, according to Definition 1, we have that the Tricomplex Horadam sequence ${\left\{T{H}_n\right\}}_{n\ge 0}$ is an element in three-dimensional space, so $T{H}_n=(h_n,h_{n+1},h_{n+2})$ for all $n\ge 0$.

Note that

$$\begin{array}{ccc}\hfill T{H}_{n+2}& =& pT{H}_n-qT{H}_{n+1}\hfill \\ \hfill (h_{n+2},h_{n+3},h_{n+4})& =& (p{h}_n-q{h}_{n+1},p{h}_{n+1}-q{h}_{n+2},p{h}_{n+2}-q{h}_{n+3}).\hfill \end{array}$$

This initial discussion leads to the following result.

Proposition 2.

The Tricomplex Horadam sequence ${\left\{T{H}_n\right\}}_{n\ge 0}$ satisfies the recurrence relation

$$T{H}_{n+1}=pT{H}_n-qT{H}_{n-1};n\ge 0,$$

(10)

with initial values $T{H}_0=(a,b,bp-aq)$ and $T{H}_1=\left(b,bp-aq,b{p}^2-abq-bq\right)$.

This follows that the vector Equation (10) is a second-order characteristic equation in each coordinate, that is the Tricomplex Horadam sequence in vector form satisfies the same recurrence relation as the Horadam-type sequence. Furthermore, vector Equation (10) is associated with the following vector characteristic equation

$$(x^2-px+q,y^2-py+q,z^2-pz+q)=(0,0,0).$$

(11)

Note that, in each coordinate, vector Equation (11) is again a second-order vector recurrence relation associated with the Tricomplex Horadam numbers ${\left\{T{H}_n\right\}}_{n\ge 0}$.

Now, we using the Binet formula for the n-th Horadam number (Equation (4)) in each coordinate in order to determine the Binet formula for the n-th Tricomplex Horadam number.

Theorem 2

(Binet-like formula). Let ${\left\{T{H}_n\right\}}_{n\ge 0}$ be the Tricomplex Horadam sequence. For all non-negative integers n, we have

$$T{H}_n=\left({\displaystyle \frac{c{\alpha }^n+d{\beta }^n}{\alpha -\beta }},{\displaystyle \frac{c{\alpha }^{n+1}+d{\beta }^{n+1}}{\alpha -\beta }},{\displaystyle \frac{c{\alpha }^{n+2}+d{\beta }^{n+2}}{\alpha -\beta }}\right),$$

(12)

where ${\left\{h_n\right\}}_{n\ge 0}$ is the Horadam sequence, $c=h_1-h_0\beta$, $d=h_0\alpha -h_1$, and α and β are the distinct roots of Equation (3).

Let ${\mathcal{GF}}_{T{H}_n}\left(x\right)=\left(G{F}_{h_n}\left(x\right),G{F}_{h_{n+1}}\left(x\right),G{F}_{h_{n+2}}\left(x\right)\right)$ be the ordinary generating function for the Tricomplex Horadam sequence ${\left\{T{H}_n\right\}}_{n\ge 0}$. Next, we will present the ordinary generating function for the Tricomplex Horadam sequence.

Proposition 3.

The ordinary generating function for the Tricomplex Horadam sequence ${\left\{T{H}_n\right\}}_{n\ge 0}$, denoted by ${\mathcal{GF}}_{T{H}_n}\left(x\right)$, is given by

$$\left({\displaystyle \frac{h_0+(h_1-p{h}_0)x}{1-px+q{x}^2}},{\displaystyle \frac{h_1-h_{0}qx}{1-px+q{x}^2}},{\displaystyle \frac{p{h}_1-q{h}_0-q{h}_{1}x}{1-px+q{x}^2}}\right),$$

(13)

where p and q are fixed constants such that $h_n=p{h}_{n-1}-q{h}_{n-2}$, for $n\ge 2$, and ${\left\{h_n\right\}}_{n\ge 0}$ is the Horadam sequence.

Proof. 

As a direct application of Lemma 3 we have $G{F}_{h_n}\left(x\right)={\displaystyle \frac{h_0+(h_1-p{h}_0)x}{1-px+q{x}^2}}$. Therefore, $G{F}_{h_{n+1}}\left(x\right)={\displaystyle \frac{h_1-h_{0}qx}{1-px+q{x}^2}}$ and $G{F}_{h_{n+2}}\left(x\right)={\displaystyle \frac{p{h}_1-q{h}_0-q{h}_{1}x}{1-px+q{x}^2}}.$

□

Let ${\mathcal{E}}_{T{H}_n}\left(x\right)=\left(E_{h_n}\left(x\right),E_{h_{n+1}}\left(x\right),E_{h_{n+1}}\left(x\right)\right)$ be the exponential generating function for the Tricomplex Horadam sequence ${\left\{T{H}_n\right\}}_{n\ge 0}$. Combining Theorem 2 and Proposition 1, we obtain the next result.

Proposition 4.

For all $n\ge 0$, the exponential generating function for the Tricomplex Horadam sequence ${\left\{T{H}_n\right\}}_{n\ge 0}$ is

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

where ${\left\{h_n\right\}}_{n\ge 0}$ is the Horadam sequence, $c=h_1-h_0\beta$, $d=h_0\alpha -h_1$, and α and β are the distinct roots of Equation (3).

To finish this section, let the Poisson generating function for the Tricomplex Horadam sequence ${\left\{T{H}_n\right\}}_{n\ge 0}$ be defined as ${\mathcal{P}}_{T{H}_n}\left(x\right)=\left(P_{h_n}\left(x\right),P_{h_{n+1}}\left(x\right),P_{h_{n+1}}\left(x\right)\right)$. In a similar way to Corollary 1, using Proposition 4 we arrive at the following result.

Corollary 2.

For all $n\ge 0$, the Poisson generating function for the Tricomplex Horadam sequence ${\left\{T{H}_n\right\}}_{n\ge 0}$ is

$${\mathcal{P}}_{T{H}_n}\left(x\right)=\left({\displaystyle \frac{c{e}^{(\alpha -1)x}-d{e}^{(\beta -1)x}}{\alpha -\beta }},{\displaystyle \frac{c\alpha e^{(\alpha -1)x}-d\beta e^{(\beta -1)x}}{\alpha -\beta }},{\displaystyle \frac{c{\alpha }^2{e}^{(\alpha -1)x}-d{\beta }^2{e}^{(\beta -1)x}}{\alpha -\beta }}\right),$$

where ${\left\{h_n\right\}}_{n\ge 0}$ is the Horadam sequence, $c=h_1-h_0\beta$, $d=h_0\alpha -h_1$, and α and β are the distinct roots of Equation (3).

4. Elementary Properties for Tricomplex Horadam Numbers

In this section, for some known identities of the Horadam sequence ${\left\{h_n\right\}}_{n\ge 0}$, we will associate a similar relation; thus, we establish some identities for the Tricomplex Horadam sequence ${\left\{T{H}_n\right\}}_{n\ge 0}$. This allows us to derive some symmetrical identities for the Tricomplex Horadam sequence. It follows from Definition 1 that $T{U}_n=(u_n,u_{n+1},u_{n+2})$, where ${\left\{u_n\right\}}_{n\ge 0}$ is the Horadam-type sequence given by Equation (6).

For the next two results for the Tricomplex Horadam sequence ${\left\{T{H}_n\right\}}_{n\ge 0}$, we make use of Lemma 2.

Proposition 5.

For any non-negative integers m and n, the Tricomplex Horadam sequence ${\left\{T{H}_n\right\}}_{n\ge 0}$ satisfies

$$T{H}_{m+1}\times T{U}_{n+1}-qT{H}_m\times T{U}_n=\left(h_{m+n}+2h_{m+n+3},h_{m+n+4}+2h_{m+n+1},3h_{m+n+2}\right),$$

(14)

where $T{U}_n=(u_n,u_{n+1},u_{n+2})$, $T{H}_n=(h_n,h_{n+1},h_{n+2})$, with ${\left\{h_n\right\}}_{n\ge 0}$ being the Horadam sequence and ${\left\{u_n\right\}}_{n\ge 0}$ being the Horadam-type sequence given by Equation (6).

Proof. 

See that

$$\begin{array}{cc}\hfill T{H}_{m+1}\times T{U}_{n+1}=& (h_{m+1},h_{m+2},h_{m+3})\times (u_{n+1},u_{n+2},u_{n+3})\hfill \\ \hfill =& (h_{m+1}{u}_{n+1}+h_{m+2}{u}_{n+3}+h_{m+3}{u}_{n+2},\hfill \\ \hfill & h_{m+3}{u}_{n+3}+h_{m+1}{u}_{n+2}+h_{m+2}{u}_{n+1},\hfill \\ \hfill & h_{m+2}{u}_{n+2}+u_{m+1}{h}_{n+3}+h_{m+3}{u}_{n+1}),\hfill \end{array}$$

(15)

and

$$\begin{array}{cc}\hfill qT{H}_m\times T{U}_n=& q\left(h_m,h_{m+1},h_{m+2}\right)\times \left(u_n,u_{n+1},u_{n+2}\right)\hfill \\ \hfill =& q(h_m{u}_n+h_{m+1}{u}_{n+2}+h_{m+2}{u}_{n+1},\hfill \\ \hfill & h_{m+2}{u}_{n+2}+h_m{u}_{n+1}+h_{m+1}{u}_n,\hfill \\ \hfill & h_{m+1}{u}_{n+1}+h_m{u}_{n+2}+h_{m+2}{u}_n).\hfill \end{array}$$

(16)

To obtain the first coordinate of Equation (14), we subtract Equation (15) from Equation (16), obtaining

$$\begin{array}{cc}\hfill & \left(h_{m+1}{u}_{n+1}-q{h}_m{u}_n\right)+\left(h_{m+2}{u}_{n+3}-q{h}_{m+1}{u}_{n+2}\right)\hfill \\ & +\left(h_{m+3}{u}_{n+2}-q{h}_{m+2}{u}_{n+1}\right)\hfill \\ \hfill =& \left(h_{m+1}{u}_{n+1}-q{h}_m{u}_n\right)+\left(h_{m+2}{u}_{n+3}-q{h}_{(m+2)-1}{u}_{(n+2)-1}\right)\hfill \\ & +\left(h_{m+3}{u}_{n+2}-q{h}_{(m+3)-1}{u}_{(n+2)-1}\right).\hfill \end{array}$$

According to item (a) of Lemma 2, we have

$$\begin{array}{cc}\hfill & \left(h_{m+1}{u}_{n+1}-q{h}_m{u}_n\right)+\left(h_{m+2}{u}_{n+3}-q{h}_{m+1}{u}_{n+2}\right)\hfill \\ & +\left(h_{m+3}{u}_{n+2}-q{h}_{m+2}{u}_{n+1}\right)\hfill \\ \hfill =& h_{m+n}+h_{(m+1)+(n+2)}+h_{(m+2)+(n+1)}\hfill \\ \hfill =& h_{m+n}+h_{m+n+3}+h_{m+n+3}\hfill \\ \hfill =& h_{m+n}+2h_{m+n+3}.\hfill \end{array}$$

This establishes the proof for the first coordinate of Equation (14). The proofs for the second and third coordinates follow in a similar manner, again using Lemma 2. □

Proposition 6.

For any non-negative integers m and n, the Tricomplex Horadam sequence ${\left\{T{H}_n\right\}}_{n\ge 0}$ satisfies the following identities

(a) 

$T{H}_{m+n}=h_{m+1}T{U}_{n+1}-q{h}_{m}T{U}_n;$

(b) 

$T{H}_{m+n}=u_{n+1}T{H}_{m+1}-q{u}_{n}T{H}_m;$

where $T{U}_n=(u_n,u_{n+1},u_{n+2})$, $T{H}_n=(h_n,h_{n+1},h_{n+2})$, with ${\left\{h_n\right\}}_{n\ge 0}$ is the Horadam sequence and ${\left\{u_n\right\}}_{n\ge 0}$ is the Horadam-type sequence given by Equation (6).

Proof. 

(a) Making use of the item (a) of Lemma 2, we obtain

$$\begin{array}{ccc}\hfill T{H}_{m+n}& =& \left(h_{m+n},h_{(m+n)+1},h_{(m+n)+2}\right)\hfill \\ & =& h_{m+1}(u_{n+1},u_{n+2},u_{n+3})-q{h}_m(u_n,u_{n+1},u_{n+2}).\hfill \end{array}$$

Thus, we achieve the desired outcome.

(b) The proof exhibits similarities to that of item (a). □

This follows directly from Proposition 6.

Corollary 3.

For any non-negative integers n, the Tricomplex Horadam sequence ${\left\{T{H}_n\right\}}_{n\ge 0}$ satisfies the following identities

(i) 

$T{H}_{2n-1}=h_{n}T{U}_{n+1}-q{h}_{n-1}T{U}_{n-1};$

(ii) 

$T{H}_{2n}=h_{n+1}T{U}_{n+1}-q{h}_{n}T{U}_n;$

(iii) 

$T{H}_{2n-1}=u_{n+1}T{H}_n-q{u}_{n}T{H}_{n-1};$

(iv) 

$T{H}_{2n}=u_{n+1}T{H}_{n+1}-q{u}_{n}T{H}_n;$

where $T{U}_n=(u_n,u_{n+1},u_{n+2})$, $T{H}_n=(h_n,h_{n+1},h_{n+2})$, with ${\left\{h_n\right\}}_{n\ge 0}$ being the Horadam sequence and ${\left\{u_n\right\}}_{n\ge 0}$ being the Horadam-type sequence given by Equation (6).

Proof. 

By setting $m=n-1$ in item (a) of Proposition 6, we obtain item (i), and by setting $m=n$, we obtain item (ii). However, in part (b) of Proposition 6, setting $m=n-1$ yields item (iii), while setting $m=n$ gives item (iv). □

For the next result for Tricomplex Horadam numbers ${\left\{T{H}_n\right\}}_{n\ge 0}$, we make use of Lemma 2, part (b). Due to the similarity with Proposition 6, we omit the proof of the next result in the interest of brevity.

Proposition 7.

For any non-negative integers $m,n$, with $m\ge n$, the following identities are satisfied by the Tricomplex Horadam sequence ${\left\{T{H}_n\right\}}_{n\ge 0}$

$$q^{n}T{H}_{m-n}=u_{n+1}T{H}_m-u_{n}T{H}_{m+1},$$

where ${\left\{u_n\right\}}_{n\ge 0}$ is the Horadam-type sequence given by Equation (6).

This is a direct consequence of Proposition 7, which makes $m=n+1$.

Corollary 4.

For any non-negative integer n, the Tricomplex Horadam sequence ${\left\{T{H}_n\right\}}_{n\ge 0}$ satisfies the following identity

$$q^{n}T{H}_1=u_{n+1}T{H}_{n+1}-u_{n}T{H}_{n+2},$$

where $T{U}_n=(u_n,u_{n+1},u_{n+2})$ and ${\left\{u_n\right\}}_{n\ge 0}$ is the Horadam-type sequence defined by Equation (6).

The next result shows the multiplication between the Tricomplex Horadam sequence ${\left\{T{H}_n\right\}}_{n\ge 0}$ and the Tricomplex Horadam-type sequence ${\left\{T{U}_n\right\}}_{n\ge 0}$.

Proposition 8.

Consider the Tricomplex Horadam sequence ${\left\{T{H}_n\right\}}_{n\ge 0}$ and the Tricomplex Horadam-type sequence ${\left\{T{U}_n\right\}}_{n\ge 0}$. For any non-negative integers m and n, the following identity is verified

$$\begin{array}{c}\hfill T{H}_m\times T{U}_{n+1}-T{H}_{m+1}\times T{U}_n=q^n(q{h}_{m-n+1}+h_{m-n}+q^2{h}_{m-n-1},\\ \hfill q{h}_{m-n-1}+q^2{h}_{m-n}+h_{m-n+1},q^2{h}_{m-n-2}+q{h}_{m-n}+h_{m-n+2}).\end{array}$$

The proof follows a similar approach to the one used in Proposition 5. In a similar way, we obtain the next three results.

The difference of the square of two consecutive terms is given below.

Proposition 9.

For arbitrary non-negative integers m, the Tricomplex Horadam sequence ${\left\{T{H}_m\right\}}_{m\ge 0}$ satisfies the following identity

$$\begin{array}{ccc}\hfill & & {\left(T{H}_{m+1}\right)}^2-q{\left(T{H}_m\right)}^2\hfill \\ \hfill & =& ({\lambda }^2\left((h_1^2-h_0^{2}q)u_{2m+1}-h_{0}q(2h_1-h_{0}p)u_{2m}\right)+2h_{m+2}(p{h}_{m+2}-2q{h}_{m+1}),\hfill \\ \hfill & & {\lambda }^2\left((h_1^2-h_0^{2}q)u_{2m+5}-h_{0}q(2h_1-h_{0}p)u_{2m+4}\right)+2h_{m+1}[(p-q)h_{m+1}-q{h}_m],\hfill \\ \hfill & & {\lambda }^2\left((h_1^2-h_0^{2}q)u_{2m+3}-h_{0}q(2h_1-h_{0}p)u_{2m+2}\right)+2h_{m+1}[p{h}_{m+2}-q(h_{m+1}-h_m)]),\hfill \end{array}$$

(17)

where ${\left\{u_m\right\}}_{m\ge 0}$ is the Horadam-type sequence given by Equation (6).

The following result gives the difference of the squares of the terms of $m+1$ and $m-1$.

Proposition 10.

The Tricomplex Horadam sequence ${\left\{T{H}_m\right\}}_{m\ge 0}$ satisfies the following identity for any non-negative integer m

$$\begin{array}{cc}\hfill & {\left(T{H}_{m+1}\right)}^2-q^2{\left(T{H}_{m-1}\right)}^2\hfill \\ \hfill =& (h_1{h}_{2m+1}+(h_1-p{h}_0)h_{2m-1}+2h_{m+1}(p{h}_{m+3}-q^2{h}_m)-2q{h}_m(h_{m+3}+q{h}_{m+1}),\hfill \\ \hfill & h_1{h}_{2m+5}+(h_1-p{h}_0)h_{2m+3}+2h_m(p{h}_{m+2}-q^2{h}_{m-1})-2q{h}_{m-1}(h_{m+2}+q{h}_m),\hfill \\ \hfill & h_1{h}_{2m+3}+(h_1-p{h}_0)h_{2m+1}+2h_{m+1}(h_{m+3}-q^2{h}_{m-1})).\hfill \end{array}$$

(18)

Proposition 11.

Let m and n be non-negative integers. The Tricomplex Horadam sequence ${\left\{T{H}_n\right\}}_{n\ge 0}$ satisfies the following identity

$$\begin{array}{cc}\hfill & {\left(T{H}_{m-n}\right)}^2-q^{3n}{\left(T{H}_{m-2n}\right)}^2\hfill \\ \hfill =& ((v_n^2-q^n)(h_m^2-q^n{h}_{m-n}^2)+2(h_{m-n+1}{h}_{m-n+2}-q^{3n}{h}_{m-2n+1}{h}_{m-2n+2}),\hfill \\ \hfill & (v_n^2-q^n)(h_{m+2}^2-q^n{h}_{m-n+2}^2)+(h_{m-n}{h}_{m-n+1}-q^{3n}{h}_{m-2n}{h}_{m-2n+1}),\hfill \\ \hfill & (v_n^2-q^n)(h_{m+1}^2-q^n{h}_{m-n+1}^2)+(h_{m-n}{h}_{m-n+2}-q^{3n}{h}_{m-2n}{h}_{m-2n+2})),\hfill \end{array}$$

(19)

where ${\left\{u_n\right\}}_{n\ge 0}$ and ${\left\{v_n\right\}}_{n\ge 0}$ are the Horadam-type sequences given by Equations (6) and (7), respectively.

5. Classical Identities for Tricomplex Horadam Numbers

In this section, we establish several fundamental identities for the Tricomplex Horadam sequence ${\left\{T{H}_n\right\}}_{n\ge 0}$. These include the Tagiuri–Vajda identity, the Catalan identity, the Cassini identity, and d’Ocagne’s identity, each of which highlights unique and elegant properties of this sequence.

To begin, we prove the Tagiuri–Vajda identity for the Tricomplex Horadam sequence ${\left\{T{H}_n\right\}}_{n\ge 0}$.

Theorem 3.

The Tricomplex Horadam sequence ${\left\{T{H}_n\right\}}_{n\ge 0}$ satisfies the following identity for non-negative integers m, s, and k,

$$T{H}_m\times T{H}_{m+s+k}-T{H}_{m+s}\times T{H}_{m+k}=cd·q^m·u_s·u_k\left((1+pq),(q^2+p),(p^2-q)\right),$$

(20)

where ${\left\{u_n\right\}}_{n\ge 0}$ is the Horadam-type sequence given by Equation (6).

Proof. 

See that

$$\begin{array}{ccc}\hfill T{H}_m\times T{H}_{m+s+k}& =& (h_m,h_{m+1},h_{m+2})\times (h_{m+s+k},h_{m+s+k+1},h_{m+s+k+2})\hfill \\ \hfill & =& (h_m·h_{m+s+k}+h_{m+1}·h_{m+s+k+2}+h_{m+2}·h_{m+s+k+1},\hfill \\ \hfill & & h_{m+2}·h_{m+s+k+2}+h_m·h_{m+s+k+1}+h_{m+1}·h_{m+s+k},\hfill \\ \hfill & & h_{m+1}·h_{m+s+k+1}+h_m·h_{m+s+k+2}+h_{m+2}·h_{m+s+k}).\hfill \end{array}$$

(21)

and

$$\begin{array}{ccc}\hfill T{H}_{m+s}\times T{H}_{m+k}& =& (h_{m+s},h_{m+s+1},h_{m+s+2})\times (h_{m+k},h_{m+k+1},h_{m+k+2})\hfill \\ \hfill & =& (h_{m+s}·h_{m+k}+h_{m+s+1}·h_{m+k+2}+h_{m+s+2}·h_{m+k+1},\hfill \\ \hfill & & h_{m+s+2}·h_{m+k+2}+h_{m+s}·h_{m+k+1}+h_{m+s+1}·h_{m+k},\hfill \\ \hfill & & h_{m+s+1}·h_{m+k+1}+h_{m+s}·h_{m+k+2}+h_{m+s+2}·h_{m+k}),\hfill \end{array}$$

(22)

To derive the first coordinate of Equation (20), we subtract Equation (22) from Equation (21), resulting in

$$\begin{array}{ccc}& & (h_m·h_{m+s+k}-h_{m+s}·h_{m+k})+(h_{m+1}·h_{m+s+k+2}-h_{m+s+1}·h_{m+k+2})\hfill \\ & & +(h_{m+2}·h_{m+s+k+1}-h_{m+s+2}·h_{m+k+1})\hfill \\ & =& \left(h_m·h_{m+s+k}-h_{m+s}·h_{m+k}\right)+\left(h_{(m+1)}·h_{(m+1)+s+(k+1)}-h_{(m+1)+s}·h_{(m+1)+(k+1)}\right)\hfill \\ & & +\left(h_{(m+2)}·h_{(m+2)+s+(k-1)}-h_{(m+2)+s}·h_{(m+2)+(k-1)}\right).\hfill \end{array}$$

According to Equation (8), we have

$$\begin{array}{ccc}& & (h_m·h_{m+s+k}-h_{m+s}·h_{m+k})+(h_{m+1}·h_{m+s+k+2}-h_{m+s+1}·h_{m+k+2})\hfill \\ & & +(h_{m+2}·h_{m+s+k+1}-h_{m+s+2}·h_{m+k+1})\hfill \\ & =& (cd·q^m·u_k·u_s)+(cd·q^{m+1}·u_{k+1}·u_s)+(cd·q^{m+2}·u_{k-1}·u_s)\hfill \\ & =& cd·q^m·u_s\left(u_k+q·u_{k+1}+q^2·u_{k-1}\right)\hfill \\ & =& cd·q^m·u_s(u_k+q(p·u_k-q·u_{k-1})+q^2·u_{k-1})\hfill \\ & =& cd·q^m·u_s·u_k(1+pq).\hfill \end{array}$$

This completes the proof of the first coordinate of Equation (20). The proofs for the second and third coordinates follow in a similar manner, utilizing Theorem 1, which presents the Tagiuri–Vajda identity for the Horadam sequence. □

To exemplify the result, we provide two examples showcasing the identities for the tricomplex Fibonacci and tricomplex Repunit sequences.

Example 3.

In [17,18], the authors study the tricomplex Fibonacci sequence ${\left\{t{f}_n\right\}}_{n\ge 0}$ and Lucas sequence ${\left\{l_n\right\}}_{n\ge 0}$, and the repunit sequence ${\left\{t{r}_n\right\}}_{n\ge 0}$, respectively, and show that the Tagiuri-Vajda identity holds for both. Namely,

(a) 

Consider $p=1,q=-1,h_0=0,h_1=1$ and $\mu =1$, we have

$$f_{m+s}^{*}{f}_{m+k}^{*}-f_m^{*}{f}_{m+s+k}^{*}={(-1)}^m·(f_1^{*}-f_0^{*})(0,2f_s,2f_k).$$

(b) 

Consider $p=1,q=-1,h_0=0,h_1=1$ and $\mu =5$, we have

$$l_{m+s}^{*}{l}_{m+k}^{*}-l_m^{*}{l}_{m+s+k}^{*}={(-1)}^m·5·(f_1^{*}-f_0^{*})(0,2f_s,2f_k).$$

(c) 

Consider $p=10,q=11,h_0=0$ and $h_1=1$, we have

$$t{r}_{m+s}t{r}_{m+k}-t{r}_{m}t{r}_{m+s+k}={10}^m·t_s{t}_k(1,1,1).$$

The remaining identities will follow as a direct consequences of the Tagiuri–Vajda identity (Theorem 3), as demonstrated below.

Proposition 12.

(d’Ocagne’s identity) The Tricomplex Horadam sequence ${\left\{T{H}_n\right\}}_{n\ge 0}$ satisfies the following identity for non-negative integers m and n. Specifically, when $m\ge n$, the following holds

$$T{H}_m\times T{H}_{n+1}-T{H}_{m+1}\times T{H}_n=-cd·q^n·u_{m-n}\left((1+pq),(q^2+p),(p^2-q)\right),$$

where ${\left\{u_n\right\}}_{n\ge 0}$ is the Horadam-type sequence given by Equation (6).

Similar to Proposition 12, we have the Catalan identity.

Proposition 13.

The Tricomplex Horadam sequence ${\left\{T{H}_n\right\}}_{n\ge 0}$ satisfies the following identity for non-negative integers m and n. In particular, for $m\ge n$, we have

$${\left(T{H}_m\right)}^2-T{H}_{m-n}\times T{H}_{m+n}=-cd{q}^{m-n}·{\left(u_n\right)}^2\left((1+pq),(q^2+p),(p^2-q)\right),$$

where ${\left\{u_n\right\}}_{n\ge 0}$ is the Horadam-type sequence given by Equation (6).

The result for $n=1$ can be derived with minimal effort from Proposition 13, which states that

Corollary 5.

(Cassini’s identity) The Tricomplex Horadam sequence ${\left\{T{H}_n\right\}}_{n\ge 0}$ satisfies the following identity for all integers $m\ge 1$,

$${\left(T{H}_m\right)}^2-T{H}_{m-1}\times T{H}_{m+1}=-cd{q}^{m-1}\left((1+pq),(q^2+p),(p^2-q)\right).$$

By making the substitution of $m=2n$ in Corollary 5, we obtain

Corollary 6.

The Tricomplex Horadam sequence ${\left\{T{H}_n\right\}}_{n\ge 0}$ satisfies the following identity for all integers $n\ge 1$,

$$T{H}_{2n-1}\times T{H}_{2n+1}-{\left(T{H}_{2n}\right)}^2=-cd{q}^{2n-1}\left((1+pq),(q^2+p),(p^2-q)\right).$$

To illustrate the above results, we present two examples with the identities for the tricomplex Fibonacci and tricomplex repunit sequences.

Example 4.

Let $m,n,k$ be any non-negative integers. According to [17], when considering the tricomplex Fibonacci sequence ${\left\{t{f}_n\right\}}_{n\ge 0}$, we have that the d’Ocagne, Catalan, and Cassini identities are

$$\begin{array}{ccc}\hfill t{f}_m^{*}\times t{f}_{n+1}^{*}-t{f}_{m+1}^{*}\times t{f}_n^{*}& =& {(-1)}^{1-n}\mu ·(f_1^{*}-f_0^{*})(0,2f_{(m-n)}^{*},2f_{(m-n)}^{*});\hfill \\ \hfill {\left(t{f}_m^{*}\right)}^2-t{f}_{m-n}^{*}\times t{f}_{m+n}^{*}& =& {(-1)}^{m+n+1}·(0,\mu ·2·{\left(f_n^{*}\right)}^2,\mu ·2·(f_1^{*}-f_0^{*})·{\left(f_n^{*}\right)}^2);\hfill \\ \hfill {\left(t{f}_m^{*}\right)}^2-t{f}_{m-1}^{*}\times t{f}_{m+1}^{*}& =& {(-1)}^m(f_1^{*}-f_0^{*})(0,2\mu ,2\mu );\hfill \end{array}$$

where ${\left\{f_n^{*}\right\}}_{n\in {\mathbb{N}}_0}$ to indicate ${\left\{f_n\right\}}_{n\ge 0}$, the Fibonacci sequence, or ${\left\{l_n\right\}}_{n\ge 0}$, the Fibonacci–Lucas sequence, $c=f_1^{*}-f_0^{*}\beta$ and $d=f_1^{*}-f_0^{*}\alpha$ and $\mu =cd$.

Example 5.

Let $m,n,k$ be any non-negative integers. According to [18], when considering the tricomplex repunit sequence ${\left\{t{r}_n\right\}}_{n\ge 0}$, we have that the d’Ocagne, Catalan, and Cassini identities are

$$\begin{array}{ccc}\hfill t{r}_m\times t{r}_{n+1}-t{r}_{m+1}\times t{r}_n& =& -{10}^n·r_3·r_{m-n}\left(1,1,1\right);\hfill \\ \hfill t{r}_{m-n}\times t{r}_{m+n}-{\left(t{r}_m\right)}^2& =& -{10}^{m-n}·r_3·{\left(r_n\right)}^2\left(1,1,1\right);\hfill \\ \hfill t{r}_{m-1}\times t{r}_{m+1}-{\left(t{r}_m\right)}^2& =& -{10}^{m-1}·r_3\left(1,1,1\right);\hfill \end{array}$$

where ${\left\{r_n\right\}}_{n\ge 0}$ is the repunit sequence.

So far, the results we have obtained have systematically used the ordinary Horadam sequence ${\left\{h_n\right\}}_{n\ge 0}$ and the Fibonacci sequence ${\left\{u_n\right\}}_{n\ge 0}$ given in Equation (6). In the following results, in addition to the ${\left\{h_n\right\}}_{n\ge 0}$ sequence, we will use the Lucas sequence ${\left\{v_n\right\}}_{n\ge 0}$ given in Equation (7).

Theorem 4

(Vajda’s Identity). The Tricomplex Horadam sequence ${\left\{T{H}_n\right\}}_{n\ge 0}$ satisfies the following identity, where m, s, and k are non-negative integers,

$$\begin{array}{cc}\hfill T{H}_{m+s}\times T{H}_{m-k}-T{H}_m\times T{H}_{m+s-k}=& e{q}^{m-k}(v_{k+s}-q^s{v}_{k-s})\left((1+pq),(q^2+p),(p^2-q)\right),\hfill \end{array}$$

(23)

where $e=p{h}_0{h}_1-q{h}_0^2-h_1^2$ and ${\left\{v_n\right\}}_{n\ge 0}$ is the Lucas sequence given in the Equation (7).

Proof. 

Note that

$$\begin{array}{cc}\hfill T{H}_{m+s}\times T{H}_{m-k}=& \left(h_{m+s},h_{m+s+1},h_{m+s+2})\times (h_{m-k},h_{m-k+1},h_{m-k+2}\right)\hfill \\ \hfill =& (h_{m+s}{h}_{m-k}+h_{m+s+1}{h}_{m-k+2}+h_{m+s+2}{h}_{m-k+1},\hfill \\ \hfill & h_{m+s+2}{h}_{m-k+2}+h_{m+s}{h}_{m-k+1}+h_{m+s+1}{h}_{m-k},\hfill \\ \hfill & h_{m+s+1}{h}_{m-k+1}+h_{m+s}{h}_{m-k+2}+h_{m+s+2}{h}_{m-k}),\hfill \end{array}$$

(24)

while

$$\begin{array}{cc}\hfill T{H}_m\times T{H}_{m+s-k}=& \left(h_m,h_{m+1},h_{m+2})\times (h_{m+s-k},h_{m+s-k+1},h_{m+s-k+2}\right)\hfill \\ \hfill =& (h_m{h}_{m+s-k}+h_{m+1}{h}_{m+s-k+2}+h_{m+2}{h}_{m+s-k+1},\hfill \\ \hfill & h_{m+2}{h}_{m+s-k+2}+h_m{h}_{m+s-k+1}+h_{m+1}{h}_{m+s-k},\hfill \\ \hfill & h_{m+1}{h}_{m+s-k+1}+h_m{h}_{m+s-k+2}+h_{m+2}{h}_{m+s-k}).\hfill \end{array}$$

(25)

To compute the first coordinate of Equation (23), we take the difference between Equations (24) and (25), resulting in

$$\begin{array}{cc}\hfill & (h_{m+s}{h}_{m-k}-h_m{h}_{m+s-k})+(h_{m+s+1}{h}_{m-k+2}-h_{m+1}{h}_{m+s-k+2})\hfill \\ & +(h_{m+s+2}{h}_{m-k+1}-h_{m+2}{h}_{m+s-k+1})\hfill \\ \hfill =& (h_{m+s}{h}_{m-k}-h_m{h}_{m+s-k})+(h_{(m+1)+s}{h}_{(m+1)-(k-1)}-h_{(m+1)}{h}_{(m+1)+s-(k-1)})\hfill \\ \hfill & +(h_{(m+2)+s}{h}_{(m+2)-(k+1)}-h_{(m+2)}{h}_{(m+2)+s-(k+1)}).\hfill \end{array}$$

According to item (f) of Lemma 2, we have

$$\begin{array}{cc}\hfill & (h_{m+s}{h}_{m-k}-h_m{h}_{m+s-k})+(h_{m+s+1}{h}_{m-k+2}-h_{m+1}{h}_{m+s-k+2})\hfill \\ & +(h_{m+s+2}{h}_{m-k+1}-h_{m+2}{h}_{m+s-k+1})\hfill \\ \hfill =& e{q}^{m-k}\left(v_{k+s}-q^s{v}_{k-s}\right)+e{q}^{m-k+2}\left(v_{k+s-1}-q^s{v}_{k-s-1}\right)+e{q}^{m-k+1}\left(v_{k+s+1}-q^s{v}_{k-s+1}\right)\hfill \\ \hfill =& e{q}^{m-k}\left[\left(v_{k+s}+q^2{v}_{k+s-1}+q{v}_{s+k+1}\right)-q^s\left(v_{k-s}+q^2{v}_{k-s-1}+q{v}_{k-s+1}\right)\right]\hfill \\ \hfill =& e{q}^{m-k}(1+pq)(v_{k+s}-q^s{v}_{k-s}).\hfill \end{array}$$

This concludes the proof for the first coordinate of Equation (23).

The proofs for the second and third coordinates are derived in a similar manner. □

The d’Ocganes, Catalan, and Cassini identities can also be obtained from the identity of Vajda. The following is a second version of these identities derived from Theorem 4 and therefore related to the Lucas sequence ${\left\{v_n\right\}}_{n\ge 0}$ given in Equation (7). Due to the similarity with the previous results in this section and the procedure for verifying them, we omit the proof of the next result in the interest of brevity.

Proposition 14

(Second d’Ocagne identiy). The Tricomplex Horadam sequence ${\left\{T{H}_n\right\}}_{n\ge 0}$ satisfies the following identity for non-negative integers m and n. Specifically, when $m\ge n$, the following holds

$$T{H}_{m+1}\times T{H}_n-T{H}_m\times T{H}_{n+1}=e·q^n(v_{m-n+1}-q·v_{m-n-1})\left((1+pq),(q^2+p),(p^2-q)\right),$$

where $e=p{h}_0{h}_1-q{h}_0^2-h_1^2$ and ${\left\{v_n\right\}}_{n\ge 0}$ is the Lucas sequence given in Equation (7).

Proposition 15

(Second Catalan identity). The Tricomplex Horadam sequence ${\left\{T{H}_n\right\}}_{n\ge 0}$ satisfies the following identity for non-negative integers m and n. In particular, for $m\ge n$, we have

$$T{H}_{m+n}\times T{H}_{m-n}-{\left(T{H}_m\right)}^2=e·q^{m-n}(v_{2n}-2·q^n)\left((1+pq),(q^2+p),(p^2-q)\right),$$

where $e=p{h}_0{h}_1-q{h}_0^2-h_1^2$ and ${\left\{v_n\right\}}_{n\ge 0}$ is the Lucas sequence given in Equation (7).

Corollary 7.

(Second Cassini’s identity) The Tricomplex Horadam sequence ${\left\{T{H}_n\right\}}_{n\ge 0}$ satisfies the following identity for all integers $m\ge 1$,

$$T{H}_{m-1}\times T{H}_{m+1}-{\left(T{H}_m\right)}^2=q^{m-1}{\lambda }^2\left((1+pq),(q^2+p),(p^2-q)\right),$$

where $\lambda =p^2-4q$.

6. Sum of Terms Involving the Tricomplex Horadam Numbers

In this section, we present results on the partial sums of terms of the Tricomplex Horadam numbers with n integers. So, we consider the sequence of partial sums ${\sum }_{k=0}^{n}T{H}_k=T{H}_0+T{H}_1+T{H}_2+\cdots +T{H}_n$, for $n\ge 0$, where ${\left\{T{H}_n\right\}}_{n\ge 0}$ is the Tricomplex Horadam sequence.

Proposition 16.

Let ${\left\{T{H}_n\right\}}_{n\ge 0}$ be the Tricomplex Horadam sequence; we now have the following formulas

$$\begin{array}{ccc}\hfill \left(a\right)\sum _{k=0}^{n}T{H}_k& =& \left({\displaystyle \frac{h_{n+2}-(p-1)(h_{n+1}-h_0)-h_1}{p-q-1}},{\displaystyle \frac{h_{n+3}-(p-1)h_{n+2}-h_1+q{h}_0}{p-q-1}},\right.\hfill \\ & & \left.{\displaystyle \frac{h_{n+4}-(p-1)h_{n+3}-(p-q)h_1+q{h}_0}{p-q-1}}\right),\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \left(b\right)\sum _{k=0}^{n}T{H}_{2k}& =& ({\displaystyle \frac{(1+q)(h_{2n+2}-h_0)-pq(h_{2n+1}-h_{-1})}{p^2-{(q+1)}^2}},\hfill \\ & & {\displaystyle \frac{p(h_{2n+2}-h_0)-q(1+q)(h_{2n+1}-h_{-1})}{p^2-{(q+1)}^2}},\hfill \\ & & {\displaystyle \frac{(1+q)h_{2n+4}-(p^2-q^2+q)h_0-pq(h_{2n+3}-h_{-1})}{p^2-{(q+1)}^2}}),\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \left(c\right)\sum _{k=0}^{n}T{H}_{2k+1}& =& \left({\displaystyle \frac{p(h_{2n+2}-h_0)-q(1+q)(h_{2n+1}-h_{-1})}{p^2-{(q+1)}^2}}\right.,\hfill \\ & & {\displaystyle \frac{(1+q)h_{2n+2}-(p^2-q^2-q)h_0-pq(h_{2n+1}-h_{-1})}{p^2-{(q+1)}^2}},\hfill \\ & & \left.{\displaystyle \frac{p(h_{2n+4}-h_0)-q(1+q)(h_{2n+3}-h_{-1})-(p^2-{(q+1)}^2)h_1}{p^2-{(q+1)}^2}}\right),\hfill \end{array}$$

where ${\left\{h_n\right\}}_{n\ge 0}$ is the ordinary Horadam sequence.

Proof. 

(a) It follows from the definition of the sum in terms of the Tricomplex Horadam numbers that

$$\begin{array}{ccc}\hfill \sum _{k=0}^{n}T{H}_k& =& T{H}_0+T{H}_1+\cdots +T{H}_n\hfill \\ & =& (h_0,h_1,h_2)+(h_1,h_2,h_3)+\cdots +(h_n,h_{n+1},h_{n+2})\hfill \\ & =& (h_0+h_1+\cdots +h_n,h_1+h_2+\cdots +h_{n+1},h_2+h_3+\cdots +h_{n+2})\hfill \\ & =& \left(\sum _{k=0}^n{h}_k,\sum _{k=1}^{n+1}{h}_k,\sum _{k=2}^{n+2}{h}_k\right)\hfill \\ & =& \left(\sum _{k=1}^n{h}_k,h_0+\sum _{k=1}^{n+1}{h}_k-h_0,h_0+h_1+\sum _{k=2}^{n+2}{h}_k-(h_0+h_1)\right)\hfill \\ & =& \left(\sum _{k=0}^n{h}_k,\sum _{k=0}^{n+1}{h}_k,\sum _{k=0}^{n+2}{h}_k\right)-(0,h_0,h_0+h_1).\hfill \end{array}$$

According to Lemma 4, item (a), we have

$$\begin{array}{ccc}& & \sum _{k=0}^{n}T{H}_k\hfill \\ & =& \left({\displaystyle \frac{h_{n+2}-(p-1)(h_{n+1}-h_0)-h_1}{p-q-1}},{\displaystyle \frac{h_{n+3}-(p-1)(h_{n+2}-h_0)-h_1}{p-q-1}},\right.\hfill \\ & & \left.{\displaystyle \frac{h_{n+4}-(p-1)(h_{n+3}-h_0)-h_1-h_0}{p-q-1}}-h_1-h_0\right),\hfill \end{array}$$

which is the result.

(b) See that

$$\begin{array}{ccc}\hfill \sum _{k=0}^{n}T{H}_{2k}& =& T{H}_0+T{H}_2+\cdots +T{H}_{2n}\hfill \\ & =& (h_0,h_1,h_2)+(h_2,h_3,h_4)+\cdots +(h_{2n},h_{2n+1},h_{2(n+1)})\hfill \\ & =& (h_0+h_2+\cdots +h_{2n},h_1+h_3+\cdots +h_{2n+1},h_2+h_3+\cdots +h_{2(n+1)})\hfill \\ & =& \left(\sum _{k=0}^n{h}_{2k},\sum _{k=0}^n{h}_{2k+1},\sum _{k=1}^{n+1}{h}_{2k}\right)\hfill \\ & =& \left(\sum _{k=0}^n{h}_{2k},\sum _{k=0}^n{h}_{2k+1},\sum _{k=0}^{n+1}{h}_{2k}-h_0\right).\hfill \end{array}$$

We get this result making use of Lemma 4, items (b) and (c).

(c) Similarly, see that

$$\begin{array}{ccc}\hfill \sum _{k=0}^{n}T{H}_{2k-1}& =& T{H}_1+T{H}_3+\cdots +T{H}_{2n+1}\hfill \\ & =& (h_1,h_2,h_3)+(h_3,h_4,h_5)+\cdots +(h_{2n+1},h_{2n+2},h_{2n+3})\hfill \\ & =& (h_1+h_3+\cdots +h_{2n+1},h_2+h_4+\cdots +h_{2n},h_3+h_5+\cdots +h_{2(n+1)-1})\hfill \\ & =& \left(\sum _{k=0}^n{h}_{2k+1},\sum _{k=1}^n{h}_{2k},\sum _{k=1}^{n+1}{h}_{2k+1}\right)=\left(\sum _{k=0}^n{h}_{2k+1},\sum _{k=0}^n{h}_{2k}-h_0,\sum _{k=0}^{n+1}{h}_{2k+1}-h_1\right).\hfill \end{array}$$

We get this result making use of Lemma 4, items (b) and (c). □

The next result is a direct consequence of Proposition 16.

Proposition 17.

Let ${\left\{T{H}_n\right\}}_{n\ge 0}$ be the Tricomplex Horadam sequence; we now have the following identities

$${\displaystyle \sum _{k=0}^n{(-1)}^{k}T{H}_k=(A_1,A_2,A_3),}$$

if n is odd and

$${\displaystyle \sum _{k=0}^n{(-1)}^{k}T{H}_k=(B_1,B_2,B_3),}$$

if n is even, where

$$\begin{array}{ccc}\hfill A_1& =& {\displaystyle \frac{(1-p+q)(h_{2n+2}-h_0)-(p-q+q^2)(h_{2n+1}-h_{-1})}{p-{(q+1)}^2}},\hfill \\ \hfill A_2& =& {\displaystyle \frac{(p-q-1)h_{2n+2}-q(1+q-q)(h_{2n+1}-h_{-1})-(p^2+p-q^2-q)h_0}{p-{(q+1)}^2}},\hfill \\ \hfill A_3& =& {\displaystyle \frac{(1+q-p)h_{2n+4}-(p^2-q^2+q+p)h_0-(pq-q-q^2)(h_{2n+3}-h_{-1})+(p^2-{(q+1)}^2)h_1}{p-{(q+1)}^2}},\hfill \\ \hfill B_1& =& {\displaystyle \frac{(1+q)h_{2n+4}-(1+q-p)h_0-pq{h}_{2n+3}-q(1+p+q)h_{-1}}{p-{(q+1)}^2}},\hfill \\ \hfill B_2& =& {\displaystyle \frac{p·h_{2n+4}-(1+q)(q·h_{2n+3}-h_{2n+2})+(p^2-p-q^2-q)h_0+q(1+q-p)h_{-1}}{p-{(q+1)}^2}},\hfill \\ \hfill B_3& =& {\displaystyle \frac{(1+q)(h_{2n+6}+q{h}_{2n+3})-pq·h_{2n+5}+\left(p^2-{(q+1)}^2\right)h_1-\left(p^2-p-q^2+q\right)h_0-(1+q-p)h_{-1}}{p-{(q+1)}^2}},\hfill \end{array}$$

and ${\left\{h_n\right\}}_{n\ge 0}$ is the ordinary Horadam sequence.

7. Conclusions

Horadam sequences and their generalizations have been extensively studied by numerous researchers. This has happened in recent decades. In this study, we introduced the Tricomplex Horadam sequence, presenting a new class of Horadam-type sequences that contribute significantly to the theoretical understanding of such sequences. We defined Tricomplex Horadam numbers and conducted a detailed analysis of their properties, as well as their relationships with traditional Horadam-type sequences. Several key identities were established to describe the behavior of this new family, and we derived both the generating function and a Binet-type formula for these sequences. This study also examined the symmetry of the Tricomplex Horadam sequence in relation to the ordinary Horadam sequence. Additionally, we explored and verified various classical identities related to Tricomplex Horadam numbers, including those proposed by Tagiuri–Vajda, d’Ocagne, Catalan, and Cassini. Our work not only expanded the understanding of this distinctive sequence family but also provided a solid foundation for future research in this area. Through the exploration of these identities and the development of a comprehensive framework, we aim to encourage further academic inquiry and progress in related mathematical fields.