# Generalized Tribonacci hyperbolic spinors

Zehra İŞBİLİR  Department of Mathematics, Düzce University, Düzce, 81620,
Türki̇̇ye<EMAIL_ADDRESS>,  Bahar DOĞAN YAZICI  Department of
Mathematics, Bi̇̇leci̇̇k Şeyh Edebali̇̇ University, Bi̇̇leci̇̇k, 11000,
Türki̇̇ye<EMAIL_ADDRESS>and  Murat TOSUN  Department of
Mathematics, Sakarya University, Sakarya, 54187, Türki̇̇ye.
<EMAIL_ADDRESS>
###### Abstract.

In this study, we introduce the generalized Tribonacci hyperbolic spinors and
properties of this new special numbers system by the generalized Tribonacci
numbers, which are one of the most general form of the third-order recurrence
sequences, generalized Tribonacci quaternions, and hyperbolic spinors, which
have quite an importance and framework from mathematics to physics. This study
especially improves the relations between the hyperbolic spinors and
generalized Tribonacci numbers with the help of the generalized Tribonacci
split quaternions. Furthermore, we examine some special cases of them and
construct both new equalities and fundamental properties such as recurrence
relation, Binet formula, generating function, exponential generating function,
Poisson generating function, summation formulas, special determinant
properties, matrix formula, and special determinant equations. Also, we give
some numerical algorithms with respect to the obtained materials. In addition
to these, we give a brief introduction for further research: generalized
Tribonacci polynomial hyperbolic spinor sequence.

###### Key words and phrases:

Hyperbolic spinors, generalized Tribonacci numbers, generalized Tribonacci
split quaternions, generalized Tribonacci hyperbolic spinors, generalized
Tribonacci polynomial hyperbolic spinors

##  1\. Introduction

The theory of numbers has been the most important workframe in lots of
disciplines, such as architecture, engineering, computer sciences, geometry,
graph theory, etc., due to several beneficial applications. In the existing
literature, many and varied studies have been completed and are ongoing with
respect to number systems, especially special recurrence sequences. One of the
most fundamental concept in the number system is quaternions, which were
investigated by W. R. Hamilton to extend the complex numbers. The algebra of
quaternions is an associative, non-commutative, and 4-dimensional Clifford
algebra. The quaternion (real/Hamilton type) set is represented by
$\mathbb{H}$ and defined as
$\mathbb{H}=\\{q\,\,|\,\,q={q_{0}}+{q_{1}}i+{q_{2}}j+{q_{3}}k,\,\,q_{0},q_{1},q_{2},q_{3}\in\mathbb{R}\\}$,
where $i,j,k$ are real quaternionic units that satisfy the multiplication
rules [32, 33, 34]:

$i^{2}={j^{2}}={k^{2}}=-1,\quad ij=-ji=k,\quad jk=-kj=i,\quad ki=-ik=j.$

In addition to these, J. Cockle determined the split quaternions [17]. The
split quaternionic units hold the rules [17, 21]:

(1.1) $\begin{array}[]{cc}i^{2}=-1,\quad j^{2}=k^{2}=1,\quad ijk=1,\\\
ij=-ji=k,\quad jk=-kj=-i,\quad ki=-ik=j.\end{array}$

Then, generalized quaternions, which include both real and split quaternions
and also have some special other quaternion types, are obtained [44, 45, 54,
58, 62, 94, 43]. The set of generalized quaternions is represented by
$\mathbb{H}_{\alpha\beta}$ and generalized quaternionic units satisfy the
following rules:

(1.2)
$\begin{array}[]{cc}i^{2}=-\alpha,\quad{j^{2}}=-\beta,\quad{k^{2}}=-\alpha\beta,\\\
ij=-ji=k,\quad jk=-kj=\beta i,\quad ki=-ik=\alpha j.\end{array}$

In the following Table 1, the classifications of generalized quaternions can
be examined:

Table 1. Classification of generalized quaternions For | Type of generalized quaternions
---|---
$\alpha=\beta=1$ | Real (Hamilton) quaternion [32, 33, 34]
$\alpha=1,\quad\beta=-1$ | Split quaternion [17]
$\alpha=1,\quad\beta=0$ | Semi-quaternion [60, 55]
$\alpha=1,\quad\beta=0$ | Split semi-quaternion [60]
$\alpha=\beta=0$ |  $1/4$-quaternion [60, 34]

On the other hand, special recurrence sequences with different orders are one
of the most attractive and fundamental topics in number theory. From the
beginning to the end of this article, we are interested in the third-order
special recurrence sequences, that is, the generalized Tribonacci number (or
sequence) (for short, GTN), which is the most general form of the third-order
recurrence sequences and has several special cases. Additionally, the
generalized Tribonacci sequence $\\{V_{n}(V_{0},V_{1},V_{2};r,s,t)\\}_{n\geq
0}$ satisfied the following recurrence relation:

(1.3) $V_{n}=rV_{n-1}+sV_{n-2}+tV_{n-3},\quad n\geq 3$

with the initial conditions $V_{0}=a,V_{1}=b,V_{2}=c$ are arbitrary integers
and $r,s,t$ are real numbers [75, 9]. In addition to these, some special
subgroups and special cases of generalized Tribonacci numbers can be seen in
Table 2 and Table 3, respectively [9, 10, 11, 12, 63, 65, 77, 69, 70, 71, 80,
72, 78, 73, 74, 75, 82, 83, 56, 15, 7, 16, 64, 66, 67, 68, 76, 79, 81, 84, 85,
86, 88] (see Section 2).

On the other hand, spinors were determined by Ehrenfest in the 1920s in
quantum physics [96]. Then, Cartan examined the spinors in a geometrical sense
[6]. When Cartan was examining the representation of groups, he found the
mathematical forms of spinors in 1913. Spinors satisfy a linear representation
of the groups of rotations of a space of any dimension by Cartan, and spinors
are directly concerned with geometry in addition to their relationship with
physics [35]. The set of isotropic vectors of the vector space
$\mathbb{C}^{3}$ establishes a two-dimensional surface in the two-dimensional
complex space $\mathbb{C}^{2}$ thanks to the study [6]. Conversely, these
vectors in $\mathbb{C}^{2}$ represent the same isotropic vectors. These
vectors are obtained as complex two-dimensional in the space $\mathbb{C}^{2}$
according to Cartan [6, 23]. Further, the triads of unit vectors which are
orthogonal by twos were written in terms of a single vector that has two
complex components, which is called a spinor [6, 18, 19]. Also, Pauli
matrices, which determined the electron spin in quantum theory, were studied
by Pauli in [59]. One can say that the wave function of an electron can be
represented with the help of a vector with two complex components in 1927 by
Pauli; this vector is also called a spinor [35]. To get more information
concerning the spinors, the following studies can be examined [96, 6, 18, 19,
4, 53]. In geometric meaning, del Castillo and Barrales investigated the
spinor representations of the Serret-Frenet frame [19]. Then, spinor
representations of Bishop frame [49], Darboux frame [95] and Sabban [90] frame
were introduced. Moreover, the spinor equations of involute-evolute curves
[24], Bertrand curves [23], and successor curves [25] were obtained. In
addition to these, the hyperbolic spinors were studied, bringing together the
different frames [3, 26, 47, 48].

In another respect, Vivarelli [50] determined the relation between the spinors
and quaternions, and according to the relation between quaternions and
rotations in Euclidean 3-space, the spinor representations of these three-
dimensional rotations were studied [27]. Tarakçıoğlu et al. [91] studied the
relations between the hyperbolic spinors and split quaternions.

In addition to these, studying the quaternions (real, split, generalized and
etc.) with special recurrence sequence components have been examined several
researchers. In the existing literature, lots of studies have been completed
with respect to these concepts as follows: bicomplex quaternions with
generalized Tribonacci numbers components are introduced by Kızılateş et al.
[51]. Flaut and Shpakivskyi [29] determined the generalized Fibonacci
quaternions and Fibonacci-Narayana quaternions. Cerda-Morales introduced the
real quaternions with generalized Tribonacci numbers components [9], real
quaternions with third-order Jacobsthal numbers components [8], generalized
quaternions with third-order Jacobsthal numbers components [13], and third-
order $\bar{h}$-Jacobsthal and third-order $\bar{h}$-Jacobsthal-Lucas
sequences and related quaternions [14]. Additionally, Taşcı [93] obtained the
real quaternions with Padovan and Pell-Padovan numbers components. Günay and
Taşkara defined some properties of real quaternions with Padovan numbers
components in [31]. Also, Günay [30] studied the real quaternions with some
generalized third-order recurrence numbers components. Moreover, Dişkaya and
Menken scrutinized the real quaternions with $(s,t)$-Padovan and
$(s,t)$-Perrin numbers components [20] and split quaternions with
$(s,t)$-Padovan and $(s,t)$-Perrin numbers components [21]. Further, İşbilir
and Gürses determined generalized quaternions with Pell-Padovan numbers
components [39] and generalized quaternions with Padovan and Perrin numbers
components [38], generalized quaternions with generalized Jacobsthal numbers
components [37], generalized quaternions with generalized $3$-primes and
generalized reverse $3$-primes numbers components [36]. Recently, İşbilir and
Gürses determined the generalized quaternions with generalized Tribonacci
numbers components [42], which are quite big number sequence and include
several types numbers such as real quaternions with GTN components, split
quaternions with GTN components, semi-quaternions with GTN components, split
semi-quaternions with GTN components, 1/4 quaternions with GTN components
[42].

Lately, Erişir and Güngör [28] have determined the Fibonacci and Lucas
spinors, and also Erişir [27] has examined the Horadam spinors. Then, Dişkaya
and Menken [22] have introduced the Padovan and Perrin spinors. Also, Kumari
et al. [52] have investigated the $k$-Fibonacci and $k$-Lucas spinors. Özçevik
and Dertli [57] determined the hyperbolic Jacobsthal spinors. Moreover,
İşbilir et al. determined the generalized Tribonacci spinors [41] and Padovan
and Perrin hyperbolic spinors [40].

This study is organized into 5 sections. In Section 2, we remind some required
notions and notations concerning the hyperbolic spinors, generalized
Tribonacci numbers, quaternions, and generalized Tribonacci quaternions. In
Section 3, we define and examine the generalized Tribonacci hyperbolic spinors
and some special cases. Then, we investigate some equations such as algebraic
properties, mate, conjugates, and some fundamental equations and formulas such
as recurrence relation, Binet formula, generating function, exponential
generating function, Poisson generating function, sum formulas, and special
determinant equalities. Also, we construct some numerical algorithms. Further,
we examine a special case of generalized Tribonacci hyperbolic spinors as an
example. Also, we give a short introduction for further research: generalized
Tribonacci polynomial hyperbolic spinors in Section 4. Then, in Section 5, we
present the conclusions.

##  2\. Fundamental Terminalogy

In this part of this study, we remind some fundamental and required notions
and notations with respect to the generalized Tribonacci number and special
cases, split quaternions with generalized Tribonacci numbers, and hyperbolic
spinors.

###  2.1. Split quaternions

The split quaternion $q\in\mathbb{H}_{S}$ is written as
$q={S_{q}}+{\overrightarrow{V}_{q}}$, where ${S_{q}}={q_{0}}$ is scalar part
and ${\overrightarrow{V}_{q}}={q_{1}}i+{q_{2}}j+{q_{3}}k$ is vector part. For
the split quaternions $q,p\in\mathbb{H}_{S}$, some algebraic properties are
written as follows [1]:

  * ✽

Addition/Subtraction

$q\pm
p={q_{0}}\pm{p_{0}}+\left({{q_{1}}\pm{p_{1}}}\right)i+\left({{q_{2}}\pm{p_{2}}}\right)j+\left({{q_{3}}\pm{p_{3}}}\right)k,$

  * ✽

Multiplication by a scalar

$\omega
q=\omega{q_{0}}+\omega{q_{1}}i+\omega{q_{2}}j+\omega{q_{3}}k,\quad\omega\in\mathbb{R},$

  * ✽

Multiplication

$\begin{array}[]{rl}qp&=\left({q_{0}}+{q_{1}}i+{q_{2}}j+{q_{3}}k\right)\left({p_{0}}+{p_{1}}i+{p_{2}}j+{p_{3}}k\right)\\\
&=\left(q_{0}p_{0}-q_{1}p_{1}+q_{2}p_{2}+q_{3}p_{3}\right)+\left(q_{0}p_{1}-q_{1}p_{0}-q_{2}p_{3}+q_{3}p_{2}\right)i\\\
&\,\,\,\,+\left(q_{0}p_{2}-q_{1}p_{3}+q_{2}p_{0}+q_{3}p_{1}\right)j+\left(q_{0}p_{3}-q_{1}p_{2}-q_{2}p_{1}+q_{3}p_{0}\right)k\\\
&=S_{q}S_{p}+g\left(\overrightarrow{V}_{q},\overrightarrow{V}_{p}\right)+S_{q}\overrightarrow{V}_{p}+S_{p}\overrightarrow{V}_{q}+\overrightarrow{V}_{q}\wedge\overrightarrow{V}_{p}.\end{array}$

where

$g\left(\overrightarrow{V}_{q},\overrightarrow{V}_{p}\right)=-q_{0}p_{0}+q_{1}p_{1}+q_{2}p_{2}+q_{3}p_{3}$

and

$\overrightarrow{V}_{q}\wedge\overrightarrow{V}_{p}=\begin{vmatrix}\begin{array}[]{ccc}-i&j&k\\\
q_{1}&q_{2}&q_{3}\\\ p_{1}&p_{2}&p_{3}\end{array}\end{vmatrix}.$

  * •

Conjugate The conjugate of the split quaternion $q$ is

$q^{*}={q_{0}}-{q_{1}}i-{q_{2}}j-{q_{3}}k.$

  * •

Norm: ${N_{q}}={qq^{*}}=q_{0}^{2}+q_{1}^{2}-q_{2}^{2}-q_{3}^{2}$.

###  2.2. Hyperbolic spinors

Assume that $\Omega$ is an $n\times n$ matrix that is defined on the
hyperbolic number system $\mathbb{H}$. $\Omega^{\dagger}$ defined as
transposing and conjugating of $\Omega$,
$\Omega^{\dagger}=\overline{\Omega^{t}}$, which is an $n\times n$ matrix. If
$\Omega$ is a Hermitian matrix concerning $\mathbb{H}$, then
$\Omega^{t}=\Omega$. If $\Omega$ is an anti-Hermitian matrix with respect to
$\mathbb{H}$, then $\Omega^{t}=-\Omega$. Let $\Omega$ be a Hermitian matrix,
the equation $UU^{\dagger}=U^{\dagger}U=1$ is valid for $U=e^{j\Omega}$. The
set of all $n\times n$ type matrices on $\mathbb{H}$ which satisfy the
previous equation establishes a group named hyperbolic unitary group, is
denoted by $U(n,\mathbb{H})$. If $\det U=1$, then this type group is
represented by $SU(n,\mathbb{H})$ [2, 3, 26].

Moreover, the Lorentz group is a group of all Lorentz transformations in the
Minkowski space and it is a subgroup of the Poincaré group. Then, Poincaré
group is determined as the group of all isometries in the Minkowski space. The
term “orthochronous” is a Lorentz transformation that is kept in the direction
of time. In addition to these, the orthochronous Lorentz group is determined
as that rigid transformation of Minkowski 3-space that kept both the direction
of time and orientation. If determinant $+1$, then this subgroup is
represented as $SO(1,3)$ [5, 3, 26, 48].

Additionally, there is a homomorphism between the group $SO(1,3)$, which is
the group of the rotation along the origin, and $SU(2,\mathbb{H})$, which is
the group of the unitary $2\times 2$ type matrix. While the elements of the
group $SU(2,\mathbb{H})$ present a fillip to the hyperbolic spinors, the
elements of the group $SO(1,3)$ present a fillip to the vectors with three
real components in Minkowski space [61, 3, 26, 48].

A hyperbolic spinor with two hyperbolic components as follows:

$\psi=\begin{pmatrix}\psi_{1}\\\ \psi_{2}\end{pmatrix}$

by using the vectors $a,b,c\in\mathbb{R}_{1}^{3}$ such that

(2.1) $\begin{split}a+jb&=\psi^{t}\rho\psi,\\\
c&=-\widehat{\psi}^{t}\rho\psi,\end{split}$

where “t” denotes the transposition, $\overline{\psi}$ is the conjugate of
$\psi$, $\widehat{\psi}$ is the mate of $\psi$. Then, the followings are
satisfied:

$\widehat{\psi}=-\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}\overline{\psi}=-\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}\begin{pmatrix}\overline{\psi}_{1}\\\
\overline{\psi}_{2}\end{pmatrix}=\begin{pmatrix}-\overline{\psi}_{2}\\\
\overline{\psi}_{1}\end{pmatrix}.$

Also, $2\times 2$ hyperbolic symmetric matrices, which are cartesian
components for the vector
$\varsigma=(\varsigma_{1},\varsigma_{2},\varsigma_{3})$

(2.2) $\varsigma_{1}=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix},\quad\varsigma_{2}=\begin{pmatrix}j&0\\\
0&j\end{pmatrix},\quad\varsigma_{3}=\begin{pmatrix}0&-1\\\ -1&0\end{pmatrix}$

are written [3, 26, 91, 48, 47]. Additionally, the ordered triads
$\\{a,b,c\\},\\{b,c,a\\},\linebreak\\{c,a,b\\}$ correspond to different
hyperbolic spinors, and the hyperbolic spinors $\psi$ and $-\psi$ correspond
to the same ordered orthogonal basis. For the hyperbolic spinors $\psi$ and
$\phi$, the following equations are satisfied:

(2.3) $\displaystyle\psi^{t}\rho\phi$ $\displaystyle=\phi^{t}\varsigma\psi,$
(2.4) $\displaystyle\overline{\psi^{t}\rho\phi}$
$\displaystyle=-\widehat{\psi}^{t}\varsigma\widehat{\phi},$ (2.5)
$\displaystyle\widehat{\left(\nu_{1}\psi+\nu_{2}\phi\right)}$
$\displaystyle=\overline{\nu}_{1}\widehat{\psi}+\overline{\nu}_{2}\widehat{\phi},$

where $\nu_{1},\nu_{2}\in\mathbb{H}$ [3, 48, 26, 47]. Let
$\xi=(\xi_{1},\xi_{2},\xi_{3})\in\mathbb{H}^{3}$ be an isotropic vector
(namely, length of this vector is zero: $\langle\xi,\xi\rangle=0$, $\xi\neq
0$) in $\mathbb{R}_{1}^{3}$. According to the above notions and notations, the
following equations can be given:

$\displaystyle\xi_{1}=\psi_{1}^{2}-\psi^{2}_{2},\quad\xi_{2}=j(\psi^{2}_{1}+\psi^{2}_{2}),\quad\xi_{3}=-2\psi_{1}\psi_{2}.$

Moreover, the following equations are satisfied:

(2.6)
$\displaystyle\psi_{1}=\pm\sqrt{\frac{\xi_{1}+j\xi_{2}}{2}}\quad\text{and}\quad\psi_{2}=\pm\sqrt{\frac{-\xi_{1}+j\xi_{2}}{2}}.$

Then, $||a||=||b||=||c||=\overline{\psi}^{t}\psi$. According to the (2.1) and
(2.2), the followings

$\begin{array}[]{rl}\xi_{1}=\psi^{t}\rho_{1}\psi,\quad\xi_{2}=\psi^{t}\rho_{2}\psi,\quad\xi_{3}=\psi^{t}\rho_{3}\psi\end{array}$

and

(2.7) $\displaystyle a+jb$
$\displaystyle=\left(\psi^{2}_{1}-\psi^{2}_{2},j(\psi^{2}_{1}+\psi^{2}_{2}),-2\psi_{1}\psi_{2}\right),$
(2.8) $\displaystyle c$
$\displaystyle=\left(\psi_{1}\overline{\psi}_{2}+\overline{\psi}_{1}\psi_{2},j(\psi_{1}\overline{\psi}_{2}-\overline{\psi}_{1}\psi_{2}),\left|\psi_{1}\right|^{2}-\left|\psi_{2}\right|^{2}\right)$

can be written [3, 48, 26]. To get more information about the hyperbolic
spinor, the studies [3, 48, 91, 26] can be examined, as well.

###  2.3. Relations between the hyperbolic spinors and split quaternions

Tarakçıoğlu et al. [91] and Tarakçıoğlu [92] investigated the relations
between the hyperbolic spinors and split quaternions. Let the quaternion
$q\in\mathbb{R}$ and the hyperbolic spinor $\psi$ is written, then we get [91,
92]:

(2.9) $\begin{split}f:&\,\,\mathbb{H}\rightarrow\mathbb{S}\\\
&\,\,q\rightarrow
f\left(q_{0}+q_{1}i+q_{2}j+q_{3}k\right)=\begin{bmatrix}q_{0}+q_{3}j\\\
-q_{1}+q_{2}j\end{bmatrix}\equiv\psi_{n},\end{split}$

where the function $f$ is linear, one-to-one, and onto. Hence,
$f\left(q+p\right)=f\left(q\right)+f\left(p\right)$ and $f\left(\omega
q\right)=\omega f\left(q\right)$, where $\omega\in\mathbb{R}$ and
$kerf=\\{0\\}$. According to the conjugation of the split quaternion $q$, the
following is satisfied [91, 92]:

(2.10)
$f(q^{*})=f\left(q_{0}-q_{1}i-q_{2}j-q_{3}k\right)=\begin{bmatrix}q_{0}-jq_{3}\\\
q_{1}-jq_{2}\end{bmatrix}\equiv\psi^{*}_{n}.$

For more detailed information concerning the relations and representations
between the hyperbolic spinors and split quaternions, we refer to the studies
[91, 92].

###  2.4. Generalized Tribonacci numbers

The characteristic equation of generalized Tribonacci numbers, which is given
in equation (1.3) is:

(2.11) $x^{3}-rx^{2}-sx-t=0.$

The roots of equation (2.11) are:

$\left\\{\begin{split}\sigma_{1}&={r}/{3}+\breve{\xi}+\breve{\gamma},\\\
\sigma_{2}&={r}/{3}+\breve{\epsilon}\breve{\xi}+\breve{\epsilon}^{2}\breve{\gamma},\\\
\sigma_{3}&={r}/{3}+\breve{\epsilon}^{2}\breve{\xi}+\breve{\epsilon}\breve{\gamma},\end{split}\right.$

where

$\begin{split}\left\\{\begin{split}\breve{\xi}&=\sqrt[3]{{\frac{{{r^{3}}}}{{27}}+\frac{{rs}}{6}+\frac{t}{2}+\sqrt{\breve{\Upsilon}}}},\\\
\breve{\gamma}&=\sqrt[3]{{\frac{{{r^{3}}}}{{27}}+\frac{{rs}}{6}+\frac{t}{2}-\sqrt{\breve{\Upsilon}}}},\\\
\breve{\epsilon}&=\frac{-1+i\sqrt{3}}{2},\\\
\breve{\Upsilon}&=\frac{{{r^{3}}t}}{{27}}-\frac{{{r^{2}}{s^{2}}}}{{108}}+\frac{{rst}}{6}-\frac{{{s^{3}}}}{{27}}+\frac{{{t^{2}}}}{4},\end{split}\right.\end{split}$

and
$\sigma_{1}+\sigma_{2}+\sigma_{3}=r,\quad\sigma_{1}\sigma_{2}+\sigma_{1}\sigma_{3}+\sigma_{2}\sigma_{3}=-s,\quad\sigma_{1}\sigma_{2}\sigma_{3}=t$
[9].

If $\Upsilon>0$, the equation (1.3) has one real and two non-real solutions,
the latter being conjugate complex. The Binet formula for generalized
Tribonacci numbers is given as [9]:

(2.12)
${V}_{n}=\cfrac{\Phi_{1}\sigma^{n}_{1}}{(\sigma_{1}-\sigma_{2})(\sigma_{1}-\sigma_{3})}+\cfrac{\Phi_{2}\sigma^{n}_{2}}{(\sigma_{2}-\sigma_{1})(\sigma_{2}-\sigma_{3})}+\cfrac{\Phi_{3}\sigma^{n}_{3}}{(\sigma_{3}-\sigma_{1})(\sigma_{3}-\sigma_{2})}\,\mathpunct{\raisebox{2.15277pt}{,}}$

where

(2.13)
$\begin{split}\left\\{\begin{split}\Phi_{1}=c-(\sigma_{2}+\sigma_{3})b+\sigma_{2}\sigma_{3}a,\\\
\Phi_{2}=c-(\sigma_{1}+\sigma_{3})b+\sigma_{1}\sigma_{3}a,\\\
\Phi_{3}=c-(\sigma_{1}+\sigma_{2})b+\sigma_{1}\sigma_{2}a.\end{split}\right.\end{split}$

Moreover, a beneficial way to obtain $nth$ generalized Tribonacci number is by
implementing $S$-matrix which is examined in [65, 97] and is a generalization
of the $R$-matrix. The $S$-matrix is written as follows (see [98, 46]):

$S=\left[{\begin{array}[]{*{20}{c}}r&s&t\\\ 1&0&0\\\
0&1&0\end{array}}\right].$

Due to the generalized Tribonacci number sequence, which is the most general
form of the third-order recurrence sequences, this family contains several
special cases due to the given values related to the $r,s,t$ and initial
values $a,b,c$. Special cases and some special subgroups of this number
sequence can be seen in Table 2 and Table 3. Generalized Tribonacci numbers
also categorized as related to the $r,s,t$ values in Table 2 and are
classified both $r,s,t$ and $T_{0},T_{1},T_{2}$ in Table 3 (see [9, 10, 11,
12, 63, 65, 77, 69, 70, 71, 80, 72, 78, 73, 74, 75, 82, 83, 56, 15, 7, 16, 64,
66, 67, 68, 76, 79, 81, 84, 85, 86, 88]).

Table 2. A brief classification for generalized Tribonacci numbers

Name | ${\\{V_{n}\\}=\\{V_{n}(V_{0},V_{1},V_{2};r,s,t)\\}}$ | Recurrence Relation
---|---|---
G. Tribonacci (usual) | $\\{\mathscr{A}_{n}\\}=\\{V_{n}(V_{0},V_{1},V_{2};1,1,1)\\}$ | $\mathscr{A}_{n}=\mathscr{A}_{n-1}+\mathscr{A}_{n-2}+\mathscr{A}_{n-3}$
G. Padovan | $\\{\mathscr{G}_{n}\\}=\\{V_{n}(V_{0},V_{1},V_{2};0,1,1)\\}$ | $\mathscr{G}_{n}=\mathscr{G}_{n-2}+\mathscr{G}_{n-3}$
G. Pell-Padovan | $\\{\mathscr{M}_{n}\\}=\\{V_{n}(V_{0},V_{1},V_{2};0,2,1)\\}$ | $\mathscr{M}_{n}=2\mathcal{M}_{n-2}+\mathscr{M}_{n-3}$
G. T. Pell | $\\{\mathscr{S}_{n}\\}=\\{V_{n}(V_{0},V_{1},V_{2};2,1,1)\\}$ | $\mathscr{S}_{n}=2\mathscr{S}_{n-1}+\mathscr{S}_{n-2}+\mathscr{S}_{n-3}$
G. T. Jacobsthal | $\\{\mathscr{X}_{n}\\}=\\{V_{n}(V_{0},V_{1},V_{2};1,1,2)\\}$ | $\mathcal{X}_{n}=\mathscr{X}_{n-1}+\mathscr{X}_{n-2}+2\mathscr{X}_{n-3}$
G. Jacobsthal-Padovan | $\\{\grave{\boldsymbol{\chi}}_{n}\\}=\\{V_{n}(V_{0},V_{1},V_{2};0,1,2)\\}$ | $\grave{\boldsymbol{\chi}}_{n}=\grave{\boldsymbol{\chi}}_{n-2}+2\grave{\boldsymbol{\chi}}_{n-3}$
G. Narayana | $\\{\grave{\boldsymbol{\vartheta}}_{n}\\}=\\{V_{n}(V_{0},V_{1},V_{2};1,0,1)\\}$ | $\grave{\boldsymbol{\vartheta}}_{n}=\grave{\boldsymbol{\vartheta}}_{n-1}+\grave{\boldsymbol{\vartheta}}_{n-3}$
G. 3-primes | $\\{\grave{\boldsymbol{\kappa}}_{n}\\}=\\{V_{n}(V_{0},V_{1},V_{2};2,3,5)\\}$ | $\grave{\boldsymbol{\kappa}}_{n}=2\grave{\boldsymbol{\kappa}}_{n-1}+3\grave{\boldsymbol{\kappa}}_{n-1}+5\grave{\boldsymbol{\kappa}}_{n-3}$
G. Reverse 3-primes | $\\{\grave{\boldsymbol{\nabla}}_{n}\\}=\\{V_{n}(V_{0},V_{1},V_{2};5,3,2)\\}$ | $\grave{\boldsymbol{\nabla}}_{n}=5\grave{\boldsymbol{\nabla}}_{n-1}+3\grave{\boldsymbol{\nabla}}_{n-1}+2\grave{\boldsymbol{\nabla}}_{n-3}$

* G.: Generalized, T.: Third-order.

Table 3. Some special cases of generalized Tribonacci numbers

Name | $\\{V_{n}\\}=\\{V_{n}(V_{0},V_{1},V_{2};r,s,t)\\}$ | Recurrence Relation
---|---|---
Tribonacci | $\\{A_{n}\\}=\\{V_{n}(0,1,1;1,1,1)\\}$ | $A_{n}=A_{n-1}+A_{n-2}+A_{n-3}$
Tribonacci-Lucas | $\\{B_{n}\\}=\\{V_{n}(3,1,3;1,1,1)\\}$ | $B_{n}=B_{n-1}+B_{n-2}+B_{n-3}$
Tribonacci-Perrin | $\\{C_{n}\\}=\\{V_{n}(3,0,2;1,1,1)\\}$ | $C_{n}=C_{n-1}+C_{n-2}+C_{n-3}$
M. Tribonacci | $\\{D_{n}\\}=\\{V_{n}(1,1,1;1,1,1)\\}$ | $D_{n}=D_{n-1}+D_{n-2}+D_{n-3}$
M. Tribonacci-Lucas | $\\{E_{n}\\}=\\{V_{n}(4,4,10;1,1,1)\\}$ | $E_{n}=E_{n-1}+E_{n-2}+E_{n-3}$
A. Tribonacci-Lucas | $\\{F_{n}\\}=\\{V_{n}(4,2,0;1,1,1)\\}$ | $F_{n}=F_{n-1}+F_{n-2}+F_{n-3}$
Padovan (Cordonnier) | $\\{G_{n}\\}=\\{V_{n}(1,1,1;0,1,1)\\}$ | $G_{n}=G_{n-2}+G_{n-3}$
Perrin (Padovan-Lucas) | $\\{H_{n}\\}=\\{V_{n}(3,0,2;0,1,1)\\}$ | $H_{n}=H_{n-2}+H_{n-3}$
Van der Laan | $\\{I_{n}\\}=\\{V_{n}(1,0,1;0,1,1)\\}$ | $I_{n}=I_{n-2}+I_{n-3}$
Padovan-Perrin | $\\{J_{n}\\}=\\{V_{n}(0,0,1;0,1,1)\\}$ | $J_{n}=J_{n-2}+J_{n-3}$
M. Padovan | $\\{K_{n}\\}=\\{V_{n}(3,1,3;0,1,1)\\}$ | $K_{n}=K_{n-2}+K_{n-3}$
A. Padovan | $\\{L_{n}\\}=\\{V_{n}(0,1,0;0,1,1)\\}$ | $L_{n}=L_{n-2}+L_{n-3}$
Pell-Padovan | $\\{M_{n}\\}=\\{V_{n}(1,1,1;0,2,1)\\}$ | $M_{n}=2M_{n-2}+M_{n-3}$
Pell-Perrin | $\\{N_{n}\\}=\\{V_{n}(3,0,2;0,2,1)\\}$ | $N_{n}=2N_{n-2}+N_{n-3}$
T. Fibonacci-Pell | $\\{O_{n}\\}=\\{V_{n}(1,0,2;0,2,1)\\}$ | $O_{n}=2O_{n-2}+O_{n-3}$
T. Lucas-Pell | $\\{P_{n}\\}=\\{V_{n}(3,0,4;0,2,1)\\}$ | $P_{n}=2P_{n-2}+P_{n-3}$
A. Pell-Padovan | $\\{R_{n}\\}=\\{V_{n}(0,1,0;0,2,1)\\}$ | $R_{n}=2R_{n-2}+R_{n-3}$
T. Pell | $\\{S_{n}\\}=\\{V_{n}(0,1,2;2,1,1)\\}$ | $S_{n}=2S_{n-1}+S_{n-2}+S_{n-3}$
T. Pell-Lucas | $\\{U_{n}\\}=\\{V_{n}(3,2,6;2,1,1)\\}$ | $U_{n}=2U_{n-1}+U_{n-2}+U_{n-3}$
T. modified Pell | $\\{V_{n}\\}=\\{V_{n}(0,1,1;2,1,1)\\}$ | $V_{n}=2V_{n-1}+V_{n-2}+V_{n-3}$
T. Pell-Perrin | $\\{W_{n}\\}=\\{V_{n}(3,0,2;2,1,1)\\}$ | $W_{n}=2W_{n-1}+W_{n-2}+W_{n-3}$
T. Jacobsthal | $\\{X_{n}\\}=\\{V_{n}(0,1,1;1,1,2)\\}$ | $X_{n}=X_{n-1}+X_{n-2}+2X_{n-3}$
T. Jacobsthal-Lucas | $\\{Y_{n}\\}=\\{V_{n}(2,1,5;1,1,2)\\}$ | $Y_{n}=Y_{n-1}+Y_{n-2}+2Y_{n-3}$
M. T. Jacobsthal | $\\{Z_{n}\\}=\\{V_{n}(3,1,3;1,1,2)\\}$ | $Z_{n}=Z_{n-1}+Z_{n-2}+2Z_{n-3}$
T. Jacobsthal-Perrin | $\\{\Gamma_{n}\\}=\\{V_{n}(3,0,2;1,1,2)\\}$ | $\Gamma_{n}=\Gamma_{n-1}+\Gamma_{n-2}+2\Gamma_{n-3}$
Jacobsthal-Padovan | $\\{\chi_{n}\\}=\\{V_{n}(1,1,1;0,1,2)\\}$ | $\chi_{n}=\chi_{n-2}+2\chi_{n-3}$
Jacobsthal-Perrin | $\\{\Delta_{n}\\}=\\{V_{n}(3,0,2;0,1,2)\\}$ | $\Delta_{n}=\Delta_{n-2}+2\Delta_{n-3}$
A. Jacobsthal-Padovan | $\\{\omega_{n}\\}=\\{V_{n}(0,1,0;0,1,2)\\}$ | $\omega_{n}=\omega_{n-2}+2\omega_{n-3}$
M. Jacobsthal-Padovan | $\\{\Omega_{n}\\}=\\{V_{n}(3,1,3;0,1,2)\\}$ | $\Omega_{n}=\Omega_{n-2}+2\Omega_{n-3}$
Narayana | $\\{\vartheta_{n}\\}=\\{V_{n}(0,1,1;1,0,1)\\}$ | $\vartheta_{n}=\vartheta_{n-1}+\vartheta_{n-3}$
Narayana-Lucas | $\\{\tau_{n}\\}=\\{V_{n}(3,1,1;1,0,1)\\}$ | $\tau_{n}=\tau_{n-1}+\tau_{n-3}$
Narayana-Perrin | $\\{\rho_{n}\\}=\\{V_{n}(3,0,2;1,0,1)\\}$ | $\rho_{n}=\rho_{n-1}+\rho_{n-3}$
3-primes | $\\{\kappa_{n}\\}=\\{V_{n}(0,1,2;2,3,5)\\}$ | $\kappa_{n}=2\kappa_{n-1}+3\kappa_{n-2}+5\kappa_{n-3}$
Lucas 3-primes | $\\{\theta_{n}\\}=\\{V_{n}(3,2,10;2,3,5)\\}$ | $\theta_{n}=2\theta_{n-1}+3\theta_{n-2}+5\theta_{n-3}$
M. 3-primes | $\\{\gamma_{n}\\}=\\{V_{n}(0,1,1;2,3,5)\\}$ | $\gamma_{n}=2\gamma_{n-1}+3\gamma_{n-2}+5\gamma_{n-3}$
Reverse 3-primes | $\\{\nabla_{n}\\}=\\{V_{n}(0,1,5;5,3,2)\\}$ | $\nabla_{n}=5\nabla_{n-1}+3\nabla_{n-2}+2\nabla_{n-3}$
Reverse Lucas 3-primes | $\\{\Lambda_{n}\\}=\\{V_{n}(3,5,31;5,3,2)\\}$ | $\Lambda_{n}=5\Lambda_{n-1}+3\Lambda_{n-2}+2\Lambda_{n-3}$
Reverse M. 3-primes | $\\{\phi_{n}\\}=\\{V_{n}(0,1,4;5,3,2)\\}$ | $\phi_{n}=5\phi_{n-1}+3\phi_{n-2}+2\phi_{n-3}$

* M.: Modified, A.: Adjusted, T.: Third-order.

###  2.5. Generalized quaternions with generalized Tribonacci numbers
components

The $n^{th}$ generalized quaternion with GTN components is represented by
$\breve{V}_{n}$ and determined as:

(2.14) $\breve{V}_{n}=V_{n}+V_{n+1}i+V_{n+2}j+V_{n+3}k$

where $V_{n}$ is the $n^{th}$ generalized Tribonacci number and $i,j,k$ are
the generalized quaternionic units that satisfy the multiplication rules given
in Equation (1.2). Also, the following recurrence relation is written:

(2.15)
${\breve{V}}_{n}=r{\breve{V}}_{n-1}+s{\breve{V}}_{n-2}+t{\breve{V}}_{n-3}\quad\text{for
all}\quad n\geq 3,$

where the initial values are

$\left\\{\begin{split}\breve{V}_{0}=&a+bi+cj+(rc+sb+ta)k,\\\
\breve{V}_{1}=&b+ci+(rc+sb+ta)j+\left[\left(r^{2}+s\right)c+\left(rs+t\right)b+rta\right]k,\\\
\breve{V}_{2}=&c+\left(rc+sb+ta\right)i+\left[\left(r^{2}+s\right)c+\left(rs+t\right)b+rta\right]j\\\
&+\left[\left(r^{3}+2rs+t\right)c+\left(r^{2}s+s^{2}+rt\right)b+\left(r^{2}t+st\right)a\right]k.\end{split}\right.$

The conjugate of the $n^{th}$ generalized quaternions with GTN components is
determined as $\breve{V}^{*}_{n}=V_{n}-V_{n+1}i-V_{n+2}j-V_{n+3}k$. Also,
İşbilir and Gürses give some fundamental and new properties, formulas, and
equations concerning generalized quaternions with GTN components. One can see
that if $\alpha=1$ and $\beta=-1$, then split quaternions with GTN components
are obtained [42].

##  3\. Generalized Tribonacci Hyperbolic Spinors

In this part of this study, we introduce a new number system combining the
hyperbolic spinors and one of the most popular third-order special recurrence
sequence generalized Tribonacci numbers by the generalized Tribonacci split
quaternions. Additionally, we obtain some algebraic properties and equalities
with respect to conjugations. Also, we give some equations such as the
recurrence relation, Binet formula, generating function, exponential
generating function, Poisson generating function, summation formulas, and
matrix formulas. Then, we obtain the determinant equation for calculating the
terms of this special sequence. In addition to these, we construct some
numerical algorithms concerning this new number system.

######  Definition 3.1.

Let $\breve{V}_{n}$ be the $n{th}$ generalized Tribonacci split quaternion.
The set of $n{th}$ generalized Tribonacci split quaternion is denoted by
$\mathbb{\breve{V}}$. The following transformation, with the help of the
correspondence between the split quaternions and hyperbolic spinors, can be
written as follows:

(3.1) $\begin{split}f:\mathbb{\breve{V}}&\rightarrow\mathbb{S}\\\
\breve{V}_{n}&\rightarrow
f\left(V_{n}+V_{n+1}i+V_{n+2}j+V_{n+3}k\right)=\begin{bmatrix}V_{n}+V_{n+3}j\\\
-V_{n+1}+V_{n+2}j\end{bmatrix}\equiv\varphi_{n}\end{split}$

where the split quaternionic units $i,j,k$ are satisfied by the rules which
are written in the equation (1.1). Since this transformation is linear and
one-to-one but not onto, this new type of sequence is named the generalized
Tribonacci hyperbolic spinor sequence. It is a linear recurrence sequence and
is established with the help of this transformation.

According to the above Definition 3.1, we can give the following initial
values:

(3.2)
$\begin{split}\left\\{\begin{split}\varphi_{0}&=\begin{bmatrix}a+\left(rc+sb+ta\right)j\\\
-b+cj\end{bmatrix},\\\
\varphi_{1}&=\begin{bmatrix}b+\left(\left(r^{2}+s\right)c+\left(rs+t\right)b+rta\right)j\\\
-c+\left(rc+sb+ta\right)j\end{bmatrix},\\\
\varphi_{2}&=\begin{bmatrix}c+\left(\left(r^{3}+2rs+t\right)c+\left(r^{2}s+s^{2}+rt\right)b+\left(r^{2}t+st\right)a\right)j\\\
-\left(rc+sb+ta\right)+\left(\left(r^{2}+s\right)c+\left(rs+t\right)b+rta\right)j\end{bmatrix}.\end{split}\right.\end{split}$

Let us obtain some algebraic properties related to the generalized Tribonacci
hyperbolic spinor sequence, such as addition, subtraction, and multiplication
by a scalar, respectively. Let us consider
$\varphi_{n},\varphi_{m}\in\mathbb{S}$ for $n,m\geq 0$:

  * ✽

Addition/Subtraction:

$\displaystyle\varphi_{n}\pm\varphi_{m}=$
$\displaystyle\begin{bmatrix}V_{n}+V_{n+3}j\\\
-V_{n+1}+V_{n+2}j\end{bmatrix}\pm\begin{bmatrix}V_{m}+V_{n+3}j\\\
-V_{m+1}+V_{m+2}j\end{bmatrix}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}V_{n}\pm V_{m}+\left(V_{n+3}\pm
V_{m+3}\right)j\\\ -\left(V_{n+1}\pm V_{m+1}\right)+\left(V_{n+2}\pm
V_{m+2}\right)j\end{bmatrix}.$

  * ✽

Multiplication by a scalar:

$\displaystyle\lambda\varphi_{n}=\begin{bmatrix}\lambda V_{n}+\lambda
V_{n+3}j\\\ -\lambda V_{n+1}+\lambda
V_{n+2}j\end{bmatrix},\quad\lambda\in\mathbb{R}.$

######  Theorem 3.1 (Recurrence Relation).

For all $n\geq 0$, the following recurrence relation is satisfied for the
generalized Tribonacci hyperbolic spinors:

(3.3) $\varphi_{n}=r\varphi_{n-1}+s\varphi_{n-2}+t\varphi_{n-3}.$

###### Proof.

With the help of the equations (1.3) and (3.1), we finish the proof.

$\begin{split}r\varphi_{n-1}+s\varphi_{n-2}+t\varphi_{n-3}=&r\begin{bmatrix}V_{n-1}+V_{n+2}j\\\
-V_{n}+V_{n+1}j\end{bmatrix}+s\begin{bmatrix}V_{n-2}+V_{n+1}i\\\
-V_{n-1}+V_{n}j\end{bmatrix}\\\ &+t\begin{bmatrix}V_{n-3}+V_{n}j\\\
-V_{n-2}+V_{n-1}j\end{bmatrix}\\\
=&\begin{bmatrix}rV_{n-1}+sV_{n-2}+tV_{n-3}+\left(rV_{n+2}+sV_{n+1}+tV_{n}\right)j\\\
-\left(rV_{n}+sV_{n-1}+tV_{n-2}\right)+\left(rV_{n+1}+sV_{n}+tV_{n-1}\right)j\end{bmatrix}\\\
=&\begin{bmatrix}V_{n}+V_{n+3}j\\\ -V_{n+1}+V_{n+2}j\end{bmatrix}\\\
=&\varphi_{n}.\end{split}$

∎

Now, let us construct a numerical algorithm:

Table 4. A numerical algorithm for finding ${n{th}}$ term of the Padovan
hyperbolic spinor Numerical Algorithm
---
(1) Begin
(2) Input $\varphi_{0},\varphi_{1}$ and $\varphi_{2}$
(3) Form $\varphi_{n}$ according to the equation (3.3)
(4) Compute $\varphi_{n}$
(5) Output $\varphi_{n}\equiv\begin{bmatrix}V_{n}+V_{n+3}j\\\
-V_{n+1}+V_{n+2}j\end{bmatrix}$
(6) Finish

In the following Table 5, we give some special cases (as a group) of
generalized Tribonacci hyperbolic spinors. The other table includes the
special cases with respect to both the $r,s,t$ and initial values is
constructed according to the Table 3, Definition 3.1 and equation (3.3). That
is, recurrence relations and definitions for special cases with respect to
both $r,s,t$ and initial values can be written similarly, as in the Table 5.

Table 5. Some Classification for Generalized Tribonacci Hyperbolic Spinors

Name | Definition | Recurrence Relation
---|---|---
G. Tribonacci (usual) H.S. | $\eta_{n}=\begin{bmatrix}\mathscr{A}_{n}+\mathscr{A}_{n+3}j\\\ -\mathscr{A}_{n+1}+\mathscr{A}_{n+2}j\end{bmatrix}$ | $\eta_{n}=\eta_{n-1}+\eta_{n-2}+\eta_{n-3}$
G. Padovan H.S. | $\delta_{n}=\begin{bmatrix}\mathscr{G}_{n}+\mathscr{G}_{n+3}j\\\ -\mathscr{G}_{n+1}+\mathscr{G}_{n+2}j\end{bmatrix}$ | $\delta_{n}=\delta_{n-2}+\delta_{n-3}$
G. Pell-Padovan H.S. | $\epsilon_{n}=\begin{bmatrix}\mathscr{M}_{n}+\mathscr{M}_{n+3}j\\\ -\mathscr{M}_{n+1}+\mathscr{M}_{n+2}j\end{bmatrix}$ | $\epsilon_{n}=2\epsilon_{n-2}+\epsilon_{n-3}$
G. T. Pell H.S. | $\flat_{n}=\begin{bmatrix}\mathscr{S}_{n}+\mathscr{S}_{n+3}j\\\ -\mathscr{S}_{n+1}+\mathscr{S}_{n+2}j\end{bmatrix}$ | $\flat_{n}=2\flat_{n-1}+\flat_{n-2}+\flat_{n-3}$
G. T. Jacobsthal H.S. | $\Omega_{n}=\begin{bmatrix}\mathscr{X}_{n}+\mathscr{X}_{n+3}j\\\ -\mathscr{X}_{n+1}+\mathscr{X}_{n+2}j\end{bmatrix}$ | $\Omega_{n}=\Omega_{n-1}+\Omega_{n-2}+2\Omega_{n-3}$
G. Jacobsthal-Padovan H.S. | $\upsilon_{n}=\begin{bmatrix}\grave{\boldsymbol{\chi}}_{n}+\grave{\boldsymbol{\chi}}_{n+3}j\\\ -\grave{\boldsymbol{\chi}}_{n+1}+\grave{\boldsymbol{\chi}}_{n+2}j\end{bmatrix}$ | $\upsilon_{n}=\upsilon_{n-2}+2\upsilon_{n-3}$
G. Narayana H.S. | $\Upsilon_{n}=\begin{bmatrix}\grave{\boldsymbol{\vartheta}}_{n}+\grave{\boldsymbol{\vartheta}}_{n+3}j\\\ -\grave{\boldsymbol{\vartheta}}_{n+1}+\grave{\boldsymbol{\vartheta}}_{n+2}j\end{bmatrix}$ | $\Upsilon_{n}=\Upsilon_{n-1}+\Upsilon_{n-3}$
G. 3-primes H.S. | $\phi_{n}=\begin{bmatrix}\grave{\boldsymbol{\kappa}}_{n}+\grave{\boldsymbol{\kappa}}_{n+3}j\\\ -\grave{\boldsymbol{\kappa}}_{n+1}+\grave{\boldsymbol{\kappa}}_{n+2}j\end{bmatrix}$ | $\phi_{n}=2\varphi_{n-1}+3\phi_{n-2}+5\phi_{n-3}$
G. Reverse 3-primes H.S. | $\nu_{n}=\begin{bmatrix}\grave{\boldsymbol{\nabla}}_{n}+\grave{\boldsymbol{\nabla}}_{n+3}j\\\ -\grave{\boldsymbol{\nabla}}_{n+1}+\grave{\boldsymbol{\nabla}}_{n+2}j\end{bmatrix}$ | $\tau_{n}=5\tau_{n-1}+3\tau_{n-2}+2\tau_{n-3}$

* G.: Generalized, T.:Third-order, H.S.: Hyperbolic spinor

######  Definition 3.2.

Let the conjugate of the $n{th}$ generalized Tribonacci split quaternion is
denoted by $\breve{V}^{*}_{n}=V_{n}-V_{n+1}i-V_{n+2}j-V_{n+3}k$. The
followings hold:

  * ✽

The $n{th}$ generalized Tribonacci hyperbolic spinor $\varphi_{n}^{*}$
corresponding to the conjugate of the $n{th}$ generalized Tribonacci split
quaternion is expressed by:

$f\left(\breve{V}_{n}^{*}\right)=f\left(V_{n}-V_{n+1}i-V_{n+2}j-V_{n+3}k\right)=\left[\begin{array}[]{l}V_{n}-V_{n+3}j\\\
V_{n+1}-V_{n+2}j\end{array}\right]\equiv\varphi_{n}^{*}.$

  * ✽

The matrix $C=\left[\begin{array}[]{cc}0&1\\\ -1&0\end{array}\right]$ is
given. The ordinary hyperbolic conjugate of $n{th}$ generalized Tribonacci
hyperbolic spinor $\varphi_{n}$ is obtained as follows:

$\overline{\varphi}_{n}=\left[\begin{array}[]{c}V_{n}-V_{n+3}j\\\
-V_{n+1}-V_{n+2}j\end{array}\right]$

  * ✽

Hyperbolic conjugate of generalized Tribonacci hyperbolic spinor
$\tilde{\varphi}_{n}=jC\overline{\varphi}_{n}$ of $n{th}$ generalized
Tribonacci hyperbolic spinor $\varphi_{n}$ is written as follows:

$\tilde{\varphi}_{n}=\left[\begin{array}[]{l}-V_{n+2}-V_{n+1}j\\\
V_{n+3}-V_{n}j\end{array}\right],$

where by using the study of Cartan [6].

  * ✽

The hyperbolic mate of $n{th}$ generalized Tribonacci hyperbolic spinor
$\check{\varphi}_{n}=-C\overline{\varphi}_{n}$ is

$\check{\varphi}_{n}=\left[\begin{array}[]{l}V_{n+1}+V_{n+2}j\\\
V_{n}-V_{n+3}j\end{array}\right],$

where by using the study of del Castillo and Barrales [19].

The following Theorems 3.2-3.7 are written without proofs because the proofs
are straightforward with the help of the matrix $C$ and the conjugation
properties of generalized Tribonacci hyperbolic spinors.

######  Theorem 3.2.

The following properties are satisfied.

  * ✽

$\overline{\varphi}_{n}=C\check{\varphi}_{n}$

  * ✽

$\,\check{\varphi}_{n}=-j\tilde{\varphi}_{n}$

  * ✽

$\overline{\varphi}_{n}=-jC\tilde{\varphi}_{n}$

######  Theorem 3.3.

The following properties are satisfied.

  * ✽

$\varphi_{n}+\varphi_{n}^{*}=\begin{bmatrix}2V_{n}\\\ 0\end{bmatrix}$

  * ✽

$\varphi_{n}-\varphi_{n}^{*}=2\begin{bmatrix}V_{n+3}j\\\
-V_{n+1}+V_{n+2}j\end{bmatrix}$

  * ✽

$\,\varphi_{n}+\overline{\varphi}_{n}=2\begin{bmatrix}V_{n}\\\
-V_{n+1}\end{bmatrix}$

  * ✽

$\varphi_{n}-\overline{\varphi}_{n}=2j\begin{bmatrix}V_{n+3}\\\
V_{n+2}\end{bmatrix}$

######  Theorem 3.4.

The following properties are satisfied.

  * ✽

$\varphi_{n}+\widetilde{\varphi}_{n}=\begin{bmatrix}-V_{n-1}+V_{n}j\\\
V_{n}+V_{n-1}j\end{bmatrix}$

  * ✽

$\varphi_{n}+\widetilde{\varphi}_{n}=\begin{bmatrix}V_{n}+V_{n+2}+\left(V_{n+3}+V_{n+1}\right)j\\\
-V_{n+1}-V_{n+3}+\left(V_{n+2}+V_{n}\right)j\end{bmatrix}$

  * ✽

$\varphi_{n}+\check{\varphi}_{n}=\begin{bmatrix}V_{n+3}+V_{n+5}j\\\
-V_{n+1}+V_{n}+\left(V_{n+2}-V_{n+3}\right)j\end{bmatrix}$

  * ✽

$\varphi_{n}-\check{\varphi}_{n}=\begin{bmatrix}V_{n}-V_{n+1}+\left(V_{n+3}-V_{n+2}\right)j\\\
-V_{n+3}+V_{n+5}j\end{bmatrix}$

######  Theorem 3.5.

The following properties are satisfied.

  * ✽

$\varphi_{n}^{*}+\overline{\varphi}_{n}=2\begin{bmatrix}V_{n}-V_{n+3}j\\\
V_{n+2}j\end{bmatrix}$

  * ✽

$\varphi_{n}^{*}-\overline{\varphi}_{n}=2\begin{bmatrix}0\\\
V_{n+1}\end{bmatrix}$

######  Theorem 3.6.

The following properties are satisfied.

  * ✽

$\varphi_{n}^{*}+\widetilde{\varphi}_{n}=\begin{bmatrix}V_{n}-V_{n+2}-\left(V_{n+3}+V_{n+1}\right)j\\\
V_{n+1}+V_{n+3}-\left(V_{n+2}+V_{n}\right)j\end{bmatrix}$

  * ✽

$\varphi_{n}^{*}-\widetilde{\varphi}_{n}=\begin{bmatrix}V_{n}+V_{n+2}-\left(V_{n+3}-V_{n+1}\right)j\\\
V_{n+1}-V_{n+3}-\left(V_{n+2}-V_{n}\right)j\end{bmatrix}$

  * ✽

$\varphi_{n}^{*}+\check{\varphi}_{n}=\begin{bmatrix}V_{n+3}+V_{n+5}j\\\
V_{n+3}-V_{n+5}j\end{bmatrix}$

  * ✽

$\varphi_{n}^{*}-\check{\varphi}_{n}=\begin{bmatrix}V_{n}-V_{n+1}-\left(V_{n+3}+V_{n+2}\right)j\\\
V_{n+1}-V_{n}-\left(V_{n+3}-V_{n+2}\right)j\end{bmatrix}$

######  Theorem 3.7.

The following properties are satisfied.

  * ✽

$\overline{\varphi}_{n}+\widetilde{\varphi}_{n}=\begin{bmatrix}-V_{n-1}-\left(V_{n+3}+V_{n+1}\right)j\\\
-V_{n}-\left(V_{n+2}+V_{n}\right)j\end{bmatrix}$

  * ✽

$\overline{\varphi}_{n}-\widetilde{\varphi}_{n}=\begin{bmatrix}V_{n}+V_{n+2}-V_{n}j\\\
-V_{n+1}-V_{n+3}-V_{n-1}j\end{bmatrix}$

  * ✽

$\widetilde{\varphi}_{n}+\check{\varphi}_{n}=\begin{bmatrix}V_{n+1}-V_{n+2}+\left(V_{n+2}-V_{n+1}\right)j\\\
V_{n+3}+V_{n}-V_{n+1}j\end{bmatrix}$

  * ✽

$\widetilde{\varphi}_{n}-\check{\varphi}_{n}=\begin{bmatrix}-V_{n+4}-V_{n+4}j\\\
V_{n+1}-V_{n+1}j\end{bmatrix}$

  * ✽

$\overline{\varphi}_{n}-\check{\varphi}_{n}=\begin{bmatrix}-V_{n+4}-V_{n+4}j\\\
V_{n+1}-V_{n+1}j\end{bmatrix}$

  * ✽

$\overline{\varphi}_{n}+\check{\varphi}_{n}=\begin{bmatrix}-V_{n+2}+V_{n+1}+\left(V_{n+2}-V_{n+1}\right)j\\\
V_{n+3}+V_{n}-\left(V_{n}+V_{n+3}\right)j\end{bmatrix}$

######  Definition 3.3 (Norm).

The norm of the $n{th}$ generalized Tribonacci split quaternion
$N\left(\breve{V}_{n}\right)=\breve{V}_{n}\breve{V}_{n}^{*}$ is equal to the
norm of the associated generalized Tribonacci hyperbolic spinors:

$N(\breve{V}_{n})=\overline{\varphi}_{n}^{t}\varphi_{n},$

and with the help of the Theorem 3.2, we write the norm of the generalized
Tribonacci hyperbolic spinors as follows:

$N\left(\varphi_{n}\right)=\overline{\varphi}_{n}^{t}\varphi_{n}.$

######  Theorem 3.8 (Generating Function).

Let $\varphi_{n}$ be the $n{th}$ generalized Tribonacci hyperbolic spinor. For
all $n\geq 0$, the following generating function is satisfied:

(3.4) $\displaystyle\sum\limits_{n=0}^{\infty}{\varphi}_{n}{x^{n}}$
$\displaystyle=\cfrac{{{{{\varphi}_{0}}}+({{\varphi}_{1}}-r{\varphi}_{0})x+({{{\varphi}_{2}}}-r{\varphi}_{1}-s{\varphi}_{0}){x^{2}}}}{{1-rx-s{x^{2}}-t{x^{3}}}}\,\mathpunct{\raisebox{2.15277pt}{.}}$

where

$\left\\{\begin{split}&{{{\varphi}_{0}}}=\begin{bmatrix}a+\left(rc+sb+ta\right)j\\\
-b+cj\end{bmatrix},\\\
&{{{\varphi}_{1}}}-r{\varphi}_{0}=\begin{bmatrix}b-ar+\left(sc+tb\right)j\\\
-c+rb+\left(sb+ta\right)j\end{bmatrix},\\\
&{{{\varphi}_{2}}}-r{\varphi}_{1}-s{\varphi}_{0}=\begin{bmatrix}a+\left(rc+sb+ta\right)j\\\
-b+cj\end{bmatrix}.\end{split}\right.$

###### Proof.

Assume that the following equation (3.5) is the generating function for the
generalized Tribonacci hyperbolic spinor.

(3.5)
$\sum\limits_{n=0}^{\infty}\varphi_{n}{x^{n}}={\varphi}_{0}+{{\varphi}_{1}}x+{{\varphi}_{2}}{x^{2}}+\ldots+{{\varphi}_{n}}{x^{n}}+\ldots$

If we multiply both sides of equation (3.5) by $rx,s{x^{2}},t{x^{3}}$, we get:

$\begin{array}[]{rl}r{x}\sum\limits_{n=0}^{\infty}{\varphi}_{n}{x^{n}}&=r{{\varphi}_{0}}{x}+r{{\varphi}_{1}}{x^{2}}+r{{\varphi}_{2}}{x^{3}}+\ldots+r{{\varphi}_{n}}{x^{n+1}}+\ldots\\\
s{x^{2}}\sum\limits_{n=0}^{\infty}{\varphi}_{n}{x^{n}}&=s{{\varphi}_{0}}{x^{2}}+s{{\varphi}_{1}}{x^{3}}+s{{\varphi}_{2}}{x^{4}}+\ldots+s{{\varphi}_{n}}{x^{n+2}}+\ldots\\\
{tx^{3}}\sum\limits_{n=0}^{\infty}{\varphi}_{n}{x^{n}}&=t{{\varphi}_{0}}{x^{3}}+t{{\varphi}_{1}}{x^{4}}+t{{\varphi}_{2}}{x^{5}}+\ldots+t{{\varphi}_{n}}{x^{n+3}}+\ldots\end{array}$

With the help of the equation (3.3), we obtain:

$\left(1-rx-s{x^{2}}-t{x^{3}}\right)\sum\limits_{n=0}^{\infty}{\varphi}_{n}{x^{n}}={{\varphi}_{0}}+({{\varphi}_{1}}-r{{\varphi}_{0}})x+({{\varphi}_{2}}-r{{\varphi}_{1}}-s{{\varphi}_{0}}){x^{2}}.$

Consequently, we attain the equation (3.4). ∎

######  Theorem 3.9 (Binet Formula).

Let $\varphi_{n}$ be the $n{th}$ generalized Tribonacci hyperbolic spinor. The
following Binet formula is written:

(3.6)
$\begin{split}\varphi_{n}=\cfrac{\zeta_{1}\Phi_{1}\sigma^{n}_{1}}{(\sigma_{1}-\sigma_{2})(\sigma_{1}-\sigma_{3})}+\cfrac{\zeta_{2}\Phi_{2}\sigma^{n}_{2}}{(\sigma_{2}-\sigma_{1})(\sigma_{2}-\sigma_{3})}+\cfrac{\zeta_{3}\Phi_{3}\sigma^{n}_{3}}{(\sigma_{3}-\sigma_{1})(\sigma_{3}-\sigma_{2})}\,\mathpunct{\raisebox{2.15277pt}{,}}\end{split}$

where

$\zeta_{1}=\begin{bmatrix}1+\sigma^{3}_{1}j\\\
\left(-1+\sigma_{1}j\right)\sigma_{1}\end{bmatrix},\quad\zeta_{2}=\begin{bmatrix}1+\sigma^{3}_{2}j\\\
\left(-1+\sigma_{2}i\right)\sigma_{2}\end{bmatrix},\quad\zeta_{3}=\begin{bmatrix}1+\sigma^{3}_{3}j\\\
\left(-1+\sigma_{3}j\right)\sigma_{3}\end{bmatrix}.$

###### Proof.

By using the equations (3.1) and (2.12), we obtain:

$\begin{split}\varphi_{n}=&\begin{bmatrix}\left\\{\begin{split}&\frac{\Phi_{1}\sigma^{n}_{1}}{(\sigma_{1}-\sigma_{2})(\sigma_{1}-\sigma_{3})}+\frac{\Phi_{2}\sigma^{n}_{2}}{(\sigma_{2}-\sigma_{1})(\sigma_{2}-\sigma_{3})}+\frac{\Phi_{3}\sigma^{n}_{3}}{(\sigma_{3}-\sigma_{1})(\sigma_{3}-\sigma_{2})}\\\
&+\left(\frac{\Phi_{1}\sigma^{n+3}_{1}}{(\sigma_{1}-\sigma_{2})(\sigma_{1}-\sigma_{3})}+\frac{\Phi_{2}\sigma^{n+3}_{2}}{(\sigma_{2}-\sigma_{1})(\sigma_{2}-\sigma_{3})}+\frac{\Phi_{3}\sigma^{n+3}_{3}}{(\sigma_{3}-\sigma_{1})(\sigma_{3}-\sigma_{2})}\right)j\end{split}\right\\}\\\
\left\\{\begin{split}&-\left(\frac{\Phi_{1}\sigma^{n+1}_{1}}{(\sigma_{1}-\sigma_{2})(\sigma_{1}-\sigma_{3})}+\frac{\Phi_{2}\sigma^{n+1}_{2}}{(\sigma_{2}-\sigma_{1})(\sigma_{2}-\sigma_{3})}+\frac{\Phi_{3}\sigma^{n+1}_{3}}{(\sigma_{3}-\sigma_{1})(\sigma_{3}-\sigma_{2})}\right)\\\
&+\left(\frac{\Phi_{1}\sigma^{n+2}_{1}}{(\sigma_{1}-\sigma_{2})(\sigma_{1}-\sigma_{3})}+\frac{\Phi_{2}\sigma^{n+2}_{2}}{(\sigma_{2}-\sigma_{1})(\sigma_{2}-\sigma_{3})}+\frac{\Phi_{3}\sigma^{n+2}_{3}}{(\sigma_{3}-\sigma_{1})(\sigma_{3}-\sigma_{2})}\right)j\end{split}\right\\}\end{bmatrix}\vspace{1mm}\\\
=&\begin{bmatrix}\cfrac{\left(1+\sigma^{3}_{1}j\right)\Phi_{1}\sigma^{n}_{1}}{(\sigma_{1}-\sigma_{2})(\sigma_{1}-\sigma_{3})}+\cfrac{\left(1+\sigma^{3}_{2}j\right)\Phi_{2}\sigma^{n}_{2}}{(\sigma_{2}-\sigma_{1})(\sigma_{2}-\sigma_{3})}+\cfrac{\left(1+\sigma^{3}_{3}j\right)\Phi_{3}\sigma^{n}_{3}}{(\sigma_{3}-\sigma_{1})(\sigma_{3}-\sigma_{2})}\\\
\cfrac{\left(1+\sigma_{1}j\right)\Phi_{1}\sigma^{n+1}_{1}}{(\sigma_{1}-\sigma_{2})(\sigma_{1}-\sigma_{3})}+\cfrac{\left(1+\sigma_{2}j\right)\Phi_{2}\sigma^{n+1}_{1}}{(\sigma_{2}-\sigma_{1})(\sigma_{2}-\sigma_{3})}+\cfrac{\left(1+\sigma_{3}j\right)\Phi_{3}\sigma^{n+1}_{3}}{(\sigma_{3}-\sigma_{1})(\sigma_{3}-\sigma_{2})}\end{bmatrix}\vspace{1mm}\\\
=&\begin{bmatrix}1+\sigma^{3}_{1}j\\\
\left(-1+\sigma_{1}j\right)\sigma_{1}\end{bmatrix}\frac{\Phi_{1}\sigma^{n}_{1}}{(\sigma_{1}-\sigma_{2})(\sigma_{1}-\sigma_{3})}+\begin{bmatrix}1+\sigma^{3}_{2}j\\\
\left(-1+\sigma_{2}j\right)\sigma_{2}\end{bmatrix}\frac{\Phi_{2}\sigma^{n}_{2}}{(\sigma_{2}-\sigma_{1})(\sigma_{2}-\sigma_{3})}\\\
&+\begin{bmatrix}1+\sigma^{3}_{3}j\\\
\left(-1+\sigma_{3}j\right)\sigma_{3}\end{bmatrix}\frac{\Phi_{3}\sigma^{n}_{3}}{(\sigma_{3}-\sigma_{1})(\sigma_{3}-\sigma_{2})}\vspace{1mm}\\\
=&\frac{\zeta_{1}\Phi_{1}\sigma^{n}_{1}}{(\sigma_{1}-\sigma_{2})(\sigma_{1}-\sigma_{3})}+\frac{\zeta_{2}\Phi_{2}\sigma^{n}_{2}}{(\sigma_{2}-\sigma_{1})(\sigma_{2}-\sigma_{3})}+\frac{\zeta_{3}\Phi_{3}\sigma^{n}_{3}}{(\sigma_{3}-\sigma_{1})(\sigma_{3}-\sigma_{2})}\,\mathpunct{\raisebox{2.15277pt}{.}}\end{split}$

∎

######  Theorem 3.10 (Exponential Generating Function).

Let $\varphi_{n}$ be $n{th}$ generalized Tribonacci hyperbolic spinor. For all
$n\geq 0$, the following exponential generating function is satisfied for
generalized Tribonacci hyperbolic spinor:

(3.7)
$\sum\limits_{n=0}^{\infty}{{{\varphi}_{n}}}\cfrac{y^{n}}{n!}=\cfrac{\zeta_{1}\Phi_{1}e^{\sigma_{1}y}}{(\sigma_{1}-\sigma_{2})(\sigma_{1}-\sigma_{3})}+\cfrac{\zeta_{2}\Phi_{2}e^{\sigma_{1}y}}{(\sigma_{2}-\sigma_{1})(\sigma_{2}-\sigma_{3})}+\cfrac{\zeta_{3}\Phi_{3}e^{\sigma_{1}y}}{(\sigma_{3}-\sigma_{1})(\sigma_{3}-\sigma_{2})}\,\mathpunct{\raisebox{2.15277pt}{.}}$

###### Proof.

Via the equation (3.6), we get:

$\begin{split}\sum\limits_{n=0}^{\infty}{{{\varphi}_{n}}}\cfrac{y^{n}}{n!}=&\sum\limits_{n=0}^{\infty}\left[\cfrac{\zeta_{1}\Phi_{1}\sigma^{n}_{1}}{(\sigma_{1}-\sigma_{2})(\sigma_{1}-\sigma_{3})}+\cfrac{\zeta_{2}\Phi_{2}\sigma^{n}_{2}}{(\sigma_{2}-\sigma_{1})(\sigma_{2}-\sigma_{3})}+\cfrac{\zeta_{3}\Phi_{3}\sigma^{n}_{3}}{(\sigma_{3}-\sigma_{1})(\sigma_{3}-\sigma_{2})}\right]\cfrac{y^{n}}{n!}\\\
=&\cfrac{\zeta_{1}\Phi_{1}}{(\sigma_{1}-\sigma_{2})(\sigma_{1}-\sigma_{3})}\sum\limits_{n=0}^{\infty}\cfrac{\left(\sigma_{1}y\right)^{n}}{n!}+\cfrac{\zeta_{2}\Phi_{2}}{(\sigma_{2}-\sigma_{1})(\sigma_{2}-\sigma_{3})}\sum\limits_{n=0}^{\infty}\cfrac{\left(\sigma_{2}y\right)^{n}}{n!}\\\
&+\cfrac{\zeta_{3}\Phi_{3}}{(\sigma_{3}-\sigma_{1})(\sigma_{3}-\sigma_{2})}\sum\limits_{n=0}^{\infty}\cfrac{\left(\sigma_{3}y\right)^{n}}{n!}\\\
=&\cfrac{\zeta_{1}\Phi_{1}e^{\sigma_{1}y}}{(\sigma_{1}-\sigma_{2})(\sigma_{1}-\sigma_{3})}+\cfrac{\zeta_{2}\Phi_{2}e^{\sigma_{1}y}}{(\sigma_{2}-\sigma_{1})(\sigma_{2}-\sigma_{3})}+\cfrac{\zeta_{3}\Phi_{3}e^{\sigma_{1}y}}{(\sigma_{3}-\sigma_{1})(\sigma_{3}-\sigma_{2})}\,\mathpunct{\raisebox{2.15277pt}{.}}\end{split}$

∎

######  Theorem 3.11 (Poisson Generating Function).

Let $\varphi_{n}$ be the $n{th}$ generalized Tribonacci hyperbolic spinor. The
Poisson generating function is expressed as:

$e^{-y}\sum\limits_{n=0}^{\infty}{{{\varphi}_{n}}}\cfrac{y^{n}}{n!}=\cfrac{\zeta_{1}\Phi_{1}e^{\sigma_{1}y}}{e^{y}(\sigma_{1}-\sigma_{2})(\sigma_{1}-\sigma_{3})}+\cfrac{\zeta_{2}\Phi_{2}e^{\sigma_{1}y}}{e^{y}(\sigma_{2}-\sigma_{1})(\sigma_{2}-\sigma_{3})}+\cfrac{\zeta_{3}\Phi_{3}e^{\sigma_{1}y}}{e^{y}(\sigma_{3}-\sigma_{1})(\sigma_{3}-\sigma_{2})}\,\mathpunct{\raisebox{2.15277pt}{.}}$

###### Proof.

By the equation (3.7), we have the desired equation since Poisson generating
function is expressed as multiplying the exponential generating function by
$e^{-y}$ (cf. also [89]). ∎

######  Theorem 3.12.

Let $\varphi_{n}$ be the $n{th}$ generalized Tribonacci hyperbolic spinor. For
all $n>0$, the following matrix equation holds:

$\left[{\begin{array}[]{*{20}{c}}{\varphi}_{n+2}\\\ {\varphi}_{n+1}\\\
{\varphi}_{n}\end{array}}\right]=\left[{\begin{array}[]{*{20}{c}}r&s&t\\\
1&0&0\\\
0&1&0\end{array}}\right]^{n}{\left[{\begin{array}[]{*{20}{c}}{\varphi}_{2}\\\
{\varphi}_{1}\\\ {\varphi}_{0}\end{array}}\right]}.$

###### Proof.

By mathematical induction, this proof is finished. ∎

According to Soykan [83], we obtain some summation formulas concerning the
generalized Tribonacci hyperbolic spinors in Theorem 3.13. Due to the proof
being straightforward, we skip this.

######  Theorem 3.13.

Let $\varphi_{n}$ be the $n^{th}$ generalized Tribonacci spinor. For all
$m,n>0$, the following summation formulas are written:

  * ✽

$\sum\limits_{n=0}^{m}\varphi_{n}=\cfrac{\begin{array}[]{l}\varphi_{m+3}+(1-r)\varphi_{m+2}+(1-r-s)\varphi_{m+1}-\varphi_{2}+(r-1)\varphi_{1}\\\
+(r+s-1)\varphi_{0}\end{array}}{(r+s+t-1)}\,\mathpunct{\raisebox{2.15277pt}{,}}$

  * ✽

$\sum\limits_{n=0}^{m}\varphi_{2n}=\cfrac{\begin{array}[]{l}(1-s)\varphi_{2m+2}+(t+rs)\varphi_{2m+1}+(t^{2}+rt)\varphi_{2m}+(s-1)\varphi_{2}\\\
+(-t-rs)\varphi_{1}+(r^{2}-s^{2}+rt+2s-1)\varphi_{0}\end{array}}{(r+s+t-1)(r-s+t+1)}\,\mathpunct{\raisebox{2.15277pt}{,}}$

  * ✽

$\sum\limits_{n=0}^{m}\varphi_{2n+1}=\cfrac{\begin{array}[]{l}(r+t)\varphi_{2m+2}+(s-s^{2}+t^{2}+rt)\varphi_{2m+1}+(t-st)\varphi_{2m}\\\
+(-r-t)\varphi_{2}+(-1+s+r^{2}+rt)\varphi_{1}+(-t+st)\varphi_{0}\end{array}}{(r-s+t+1)(r+s+t-1)}\,\mathpunct{\raisebox{2.15277pt}{,}}$

where denominators are not zero.

Special Cases: If $s=1$, we get the following summation formulas:

  * ✽

$\sum\limits_{n=0}^{m}\varphi_{2n}=\cfrac{\varphi_{2m+1}+t\varphi_{2m}-\varphi_{1}+r\varphi_{0}}{r+t}\,\mathpunct{\raisebox{2.15277pt}{,}}$

  * ✽

$\sum\limits_{n=0}^{m}\varphi_{2n+1}=\cfrac{\varphi_{2m+2}+t\varphi_{2m+1}-\varphi_{2}+r\varphi_{1}}{r+t}\,\mathpunct{\raisebox{2.15277pt}{,}}$

where denominators are not zero.

By inspiring Kızılateş et al. [51], we construct the following Theorem 3.14 to
compute the $n{th}$ term of generalized Tribonacci hyperbolic spinor.

######  Theorem 3.14.

Let $\varphi_{n}$ be the $n{th}$ generalized Tribonacci hyperbolic spinor. For
all $n\geq 0$, the following equation is obtained.

(3.8)
${\varphi}_{n}=\left|{\begin{array}[]{*{20}{c}}{\varphi}_{0}&-1&0&0&0&\ldots&0&0\\\
{\varphi}_{1}&0&-1&0&0&\ldots&0&0\\\ {\varphi}_{2}&0&0&-1&0&\ldots&0&0\\\
0&t&s&r&-1&\ldots&0&0\\\
\vdots&\ddots&\ddots&\ddots&\ddots&\ddots&\vdots&\vdots\\\
0&0&0&0&0&\ddots&r&-1\\\ 0&0&0&0&0&\ddots&s&r\\\
\end{array}}\right|_{(n+1)\times(n+1)}$

###### Proof.

We can finish the proof by using mathematical induction and equation (3.3). ∎

Also, we give a numerical algorithm with respect to the Theorem 3.14:

Table 6. A numerical algorithm for finding ${n{th}}$ term of the generalized
Tribonacci hyperbolic spinor Numerical Algorithm
---
(1) Begin
(2) Input $\varphi_{0},\varphi_{1}$ and $\varphi_{2}$
(3) Form $\varphi_{n}$ according to the equation (3.8)
(4) Compute $\varphi_{n}$
(5) Output $\varphi_{n}\equiv\begin{bmatrix}V_{n}+V_{n+3}j\\\
-V_{n+1}+V_{n+2}j\end{bmatrix}$
(6) Finish

Thanks to the Cereceda [15] and Cerda-Morales [7], we establish the other
method in Theorem 3.15, to calculate the $n{th}$ generalized Tribonacci
hyperbolic spinor.

######  Theorem 3.15.

For all $n\in\mathbb{N}$, the following equation holds:

(3.9)
$\varphi_{n}=\left|{\begin{array}[]{*{20}{c}}{\varphi}_{0}&1&0&0&0&0&\ldots&0&0\\\
r{\varphi}_{0}-{\varphi}_{1}&r&\cfrac{1}{{\varphi}_{0}}&0&0&0&\ldots&0&0\\\
0&r{\varphi}_{1}-{\varphi}_{2}&r&t&0&0&\ldots&0&0\\\
0&{\varphi}_{0}&-\cfrac{s}{t}&r&t&0&\ldots&0&0\\\
0&0&\cfrac{1}{t}&-\cfrac{s}{t}&r&t&\ldots&0&0\\\
\vdots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\vdots\\\
0&0&0&0&0&0&\ldots&r&t&\\\ 0&0&0&0&0&0&\ldots&-\cfrac{s}{t}&r\\\
\end{array}}\right|_{(n+1)\times(n+1)}$

where denominators are not zero.

###### Proof.

We skip this proof for the sake of brevity. ∎

Table 7. A numerical algorithm for finding ${n{th}}$ term of the generalized
Tribonacci hyperbolic spinor Numerical Algorithm
---
(1) Begin
(2) Input $\varphi_{0},\varphi_{1}$ and $\varphi_{2}$
(3) Form $\varphi_{n}$ according to the equation (3.9)
(4) Compute $\varphi_{n}$
(5) Output $\varphi_{n}\equiv\begin{bmatrix}V_{n}+V_{n+3}j\\\
-V_{n+1}+V_{n+2}j\end{bmatrix}$
(6) Finish

Let us examine a special case of generalized Tribonacci hyperbolic spinors. We
will examine the Jacobsthal-Padovan spinor sequence in the following Corollary
3.1:

######  Corollary 3.1.

Let us consider the $n^{th}$ Jacobsthal-Padovan hyperbolic spinor
$\breve{\upsilon}_{n}$ with the initial values

${\breve{\upsilon}}_{0}=\begin{bmatrix}1+3j\\\
-1+j\end{bmatrix},\quad{\breve{\upsilon}}_{1}=\begin{bmatrix}1+3j\\\
-1+3j\end{bmatrix},\quad{\breve{\upsilon}}_{2}=\begin{bmatrix}1+5j\\\
-3+3j\end{bmatrix}.$

Then the following properties are satisfied:

  * ✽

The recurrence relation of Jacobsthal-Padovan hyperbolic spinors is:

${{\breve{\upsilon}}}_{n}={{\breve{\upsilon}}}_{n-2}+2{{\breve{\upsilon}}}_{n-3},\quad
n\geq 3.$

  * ✽

The generating function of Jacobsthal-Padovan hyperbolic spinors is:

$\sum\limits_{n=0}^{\infty}{{\breve{\upsilon}}}_{n}{x^{n}}=\frac{{{{\breve{\upsilon}_{0}}}+{\breve{\upsilon}_{1}}x+({{\breve{\upsilon}_{2}}}-\breve{\upsilon}_{0}){x^{2}}}}{{1-{x^{2}}-{x^{3}}}}=\cfrac{1}{{1-x^{2}-2x^{3}}}\begin{bmatrix}1+x+\left(3+3x+2x^{2}\right)j\\\
-1-x-2x^{2}+\left(1+3x\right)j\end{bmatrix}\,\mathpunct{\raisebox{2.15277pt}{.}}$

  * ✽

The Binet formula of Jacobsthal-Padovan hyperbolic spinors is:

$\begin{split}{\breve{\upsilon}}_{n}=&\frac{(\sigma_{1}+1)\sigma_{1}^{n+1}\zeta_{1}}{(\sigma_{1}-\sigma_{2})(\sigma_{1}-\sigma_{3})}+\frac{(\sigma_{2}+1)\sigma_{2}^{n+1}\zeta_{2}}{(\sigma_{2}-\sigma_{1})(\sigma_{2}-\sigma_{3})}+\frac{(\sigma_{3}+1)\sigma_{3}^{n+1}\zeta_{3}}{(\sigma_{3}-\sigma_{1})(\sigma_{3}-\sigma_{2})}\,\mathpunct{\raisebox{2.15277pt}{,}}\end{split}$

where (see $\sigma_{1}\sigma_{2},\sigma_{3}$ in [74])

$\left\\{\begin{split}\sigma_{1}&=\left(1+\cfrac{\sqrt{78}}{9}\right)^{1/3}+\left(1-\cfrac{\sqrt{78}}{9}\right)^{1/3},\\\
\sigma_{2}&=\cfrac{-1+\sqrt{3}i}{2}\left(1+\cfrac{\sqrt{78}}{9}\right)^{1/3}+\left(\cfrac{-1+\sqrt{3}i}{2}\right)^{2}\left(1-\cfrac{\sqrt{78}}{9}\right)^{1/3},\\\
\varrho_{3}&=\left(\cfrac{-1+\sqrt{3}i}{2}\right)^{2}\left(1+\cfrac{\sqrt{78}}{9}\right)^{1/3}+\cfrac{-1+\sqrt{3}i}{2}\left(1-\cfrac{\sqrt{78}}{9}\right)^{1/3}.\end{split}\right.$

  * ✽

The exponential generating function of Jacobthal-Padovan hyperbolic spinors
is:

$\begin{split}\sum\limits_{n=0}^{\infty}{{\breve{\upsilon}}}_{n}\cfrac{y^{n}}{n!}=&\frac{(\sigma_{1}+1)\sigma_{1}\zeta_{1}e^{\sigma_{1}y}}{(\sigma_{1}-\sigma_{2})(\sigma_{1}-\sigma_{3})}+\frac{(\sigma_{2}+1)\sigma_{2}\zeta_{2}e^{\sigma_{2}y}}{(\sigma_{2}-\sigma_{1})(\sigma_{2}-\sigma_{3})}+\frac{(\sigma_{3}+1)\sigma_{2}\zeta_{3}e^{\sigma_{3}y}}{(\sigma_{3}-\sigma_{1})(\sigma_{3}-\sigma_{2})}\,\mathpunct{\raisebox{2.15277pt}{.}}\end{split}$

  * ✽

The Poisson generating function of Jacobsthal-Padovan hyperbolic spinors is:

$\begin{split}e^{-y}\sum\limits_{n=0}^{\infty}{\breve{\upsilon}}_{n}\cfrac{y^{n}}{n!}=&\frac{(\sigma_{1}+1)\sigma_{1}\zeta_{1}e^{\sigma_{1}y}}{e^{y}(\sigma_{1}-\sigma_{2})(\sigma_{1}-\sigma_{3})}+\frac{(\sigma_{2}+1)\sigma_{2}\zeta_{2}e^{\sigma_{2}y}}{e^{y}(\sigma_{2}-\sigma_{1})(\sigma_{2}-\sigma_{3})}\\\
&+\frac{(\sigma_{3}+1)\sigma_{2}\zeta_{3}e^{\sigma_{3}y}}{e^{y}(\sigma_{3}-\sigma_{1})(\sigma_{3}-\sigma_{2})}\,\mathpunct{\raisebox{2.15277pt}{.}}\end{split}$

  * ✽

For all $m\in\mathbb{N}$, the summation formulas of Jacobsthal-Padovan
hyperbolic spinors are given:

    * ❃

$\sum\limits_{n=0}^{m}{\breve{\upsilon}}_{n}=\cfrac{1}{2}\left(\breve{\upsilon}_{m+3}+\breve{\upsilon}_{m+2}-\breve{\upsilon}_{2}-\breve{\upsilon}_{1}\right),$

    * ❃

${\sum\limits_{n=0}^{m}{{\breve{\upsilon}}}_{2n}}=\cfrac{1}{2}\left({\breve{\upsilon}_{2m+1}}+2{\breve{\upsilon}_{2m}}-{\breve{\upsilon}_{1}}\right),$

    * ❃

${\sum\limits_{n=0}^{m}{{\breve{\upsilon}}}_{2n+1}}=\cfrac{1}{2}\left({\breve{\upsilon}_{2m+2}}+2{\breve{\upsilon}_{2m+1}}-{\breve{\upsilon}_{2}}\right).$

  * ✽

The following property of Jacobtshal-Padovan hyperbolic spinors is obtained:

$\left({\begin{array}[]{*{20}{c}}{\breve{\upsilon}}_{n+2}\\\
{\breve{\upsilon}}_{n+1}\\\
{\breve{\upsilon}}_{n}\end{array}}\right)=\left({\begin{array}[]{*{20}{c}}0&1&2\\\
1&0&0\\\
0&1&0\end{array}}\right)^{n}{\left({\begin{array}[]{*{20}{c}}{\breve{\upsilon}}_{2}\\\
{\breve{\upsilon}}_{1}\\\ {\breve{\upsilon}}_{0}\end{array}}\right)}.$

  * ✽

The following equations of Jacobsthal-Padovan hyperbolic spinors hold:

${\breve{\upsilon}}_{n}=\left|{\begin{array}[]{*{20}{c}}{\breve{\upsilon}}_{0}&-1&0&0&0&\ldots&0&0\\\
{\breve{\upsilon}}_{1}&0&-1&0&0&\ldots&0&0\\\
{\breve{\upsilon}}_{2}&0&0&-1&0&\ldots&0&0\\\ 0&2&1&0&-1&\ldots&0&0\\\
\vdots&\ddots&\ddots&\ddots&\ddots&\ddots&\vdots&\vdots\\\
0&0&0&0&0&\ddots&0&-1\\\ 0&0&0&0&0&\ddots&1&0\\\
\end{array}}\right|_{(n+1)\times(n+1)}.$
${\breve{\upsilon}}_{n}=\left|{\begin{array}[]{*{20}{c}}{\breve{\upsilon}}_{0}&1&0&0&\ldots&0&0\\\
-{\breve{\upsilon}}_{1}&0&\cfrac{1}{{\breve{\upsilon}}_{0}}&0&\ldots&0&0\\\
0&-{\breve{\upsilon}}_{2}&0&2&\ldots&0&0\\\
0&{\breve{\upsilon}}_{0}&-\cfrac{1}{2}&0&\ldots&0&0\\\
0&0&\cfrac{1}{2}&-\cfrac{1}{2}&\ldots&0&0\\\
\vdots&\ddots&\ddots&\ddots&\ddots&\ddots&\vdots\\\ 0&0&0&0&\ldots&0&1&\\\
0&0&0&0&\ldots&-\cfrac{1}{2}&0\\\ \end{array}}\right|_{(n+1)\times(n+1)}$

where denominators do not vanish.

##  4\. A New Further Research: Generalized Tribonacci Polynomial Hyperbolic
Spinors

In this section, we give a short introduction to some new arguments for where
to go from here. For future works, let us make a brief introduction associated
with the concept: generalized Tribonacci polynomial hyperbolic spinors, which
include the generalized Tribonacci hyperbolic spinors.

Let ${V}_{n}(x)$ represents the $nth$ generalized Tribonacci polynomial. See
for detailed information with respect to the generalized Tribonacci
polynomials (see [87]).

######  Definition 4.1.

For $n\geq 0$, the $nth$ generalized Tribonacci polynomial hyperbolic spinor
is determined as follows:

(4.1) $\widetilde{\psi}_{n}(x)=\begin{bmatrix}V_{n}(x)+V_{n+3}(x)j\\\
-V_{n+1}(x)+V_{n+2}(x)j\end{bmatrix}$

Since this transformation is linear and one-to-one but not onto, a new type
sequence, which is a linear recurrence sequence, is constructed.

######  Theorem 4.1 (Recurrence Relation).

For all $n\geq 0$, the following recurrence relations are written for the
generalized Tribonacci hyperbolic spinor sequence:

(4.2)
$\widetilde{\psi}_{n}(x)=r(x)\widetilde{\psi}_{n-1}(x)+s(x)\widetilde{\psi}_{n-2}(x)+t(x)\widetilde{\psi}_{n-3}(x).$

The other equations given in the previous section can be obtained for
generalized Tribonacci polynomial hyperbolic spinors. We intend to examine
this special sequence in a future study.

##  5\. Conclusion

In this study, we examined generalized Tribonacci hyperbolic spinors. Also, we
constructed some algebraic properties and equalities concerning the
conjugations. Then, we obtained some equations such as recurrence relation,
Binet formula, generating function, exponential generating function, Poisson
generating function, summation formulas, and matrix formulas. Further, we
constructed the determinant equalities for computing the terms of this general
sequence. Moreover, we established some numerical algorithms with respect to
this special number system. Furthermore, we gave a short introduction with
respect to the generalized Tribonacci polynomial hyperbolic spinors.

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