I. INTRODUCTION

In this research we will address two second-order linear recursive sequences, they are: Pell and Jacobsthal. The Pell sequence came from the English mathematician John Pell (1611 - 1685), known for being one of the most enigmatic mathematicians of the century XVII. The Jacobsthal sequence is named after the mathematician Ernest Erich Jacobsthal (1882-1965), specialist in Number Theory.

These sequences have similarity in their recurrence, the Pell sequence is defined by:

being $P{e}_0=0$ $P{e}_1=1$ its initial conditions.

As for the Jacobsthal sequence, it is presented by the recurrence:

being $J_0=0$ $J_1=1$ its initial conditions.

On the other hand, we have the quaternions numbers, it is believed that the quaternions arose in an attempt to transform the complex number $z=a+bi$ in three dimensions [1]. Quaternions are presented as formal sums of scalars with usual vectors of three-dimensional space, existing four dimensions and is described by: $q=a+bi+cj+dk$, where $a$, $b$, $c$ and $d$ are real numbers and $i$, $j$, $k$ the orthogonal part at the base ${\mathbf{R}}^3$. In the work of [3] the authors present the quaternionic product as $i^2=j^2=k^2=-1$, $ij=k=-ij$, $jk=i=-kj$ and $ki=j=-ik$.

On the other hand, there are the hybrid numbers, presented initially by [4], which he studied and lists three sets of numbers, they are: the complex, hyperbolic and dual. A hybrid number is defined as:

The hybridization of linear sequences consists of relating the hybrid numbers with linear sequences, researches around this hybridization are found in the mathematical literature, as we can find in [10, 11, 12, 13, 14, 15, 16]. Regarding the study on quaternions, we find work on the Padovan and Perrin quaternions in [5], Pell-Padovan in [6] and Fibonacci and Fibonacci Complexes in [2, 7, 8, 9].

Finally, in this research, we take as a basis the works of [17, 18] to present, in the following sections, the hybrid quaternions of Pell and Jacobsthal.

II. PELL HYBRID QUATERNIONS NUMBERS

In this section, we will present Pell's hybrid quaternions based on the work of [17, 18].

Definition 2.1. Pell's hybrid number, denoted by $HP{e}_n$, is defined by:

Definition 2.2. The Recurrence Relationship for Pell's Hybrids, $n\ge 2$, is defined by:

with $HP{e}_0=i+2\epsilon +5h$ and $HP{e}_1=1+2i+5\epsilon +12h$ its initial terms.

Definition 2.3. Pell's quaternion number, denoted by $QP{e}_n$, it is given by:

Definition 2.4. The recurrence relation for Pell's quaternions, $n\ge 2$, is defined by:

with $QP{e}_0=i+2j+5k$ and $QP{e}_1=1+2i+5j+12k$ its initial terms.

Now, from what was seen above, we will approach Pell's hybrid quaternions.

Definition 2.5. Pell's hybrid quaternion number is defined as:

where $i$, $j$, $k$ are the units of the quaternions and $H{P}_n$ it's the $n$ -th Pell hybrid number. Thus, Pell's hybrid quaternions can be rewritten by:

where $i,\epsilon$ and $h$ are the imaginary units of the hybrid numbers and ${\widehat{HPe}}_n=P{e}_n+P{e}_{n+1}i+P{e}_{n+2}j+P{e}_{n+3}k$.

Definition 2.6. The recurrence relationship for Pell's hybrid quaternions, $n\ge 2$, is defined by:

with the following initial terms: $\tilde{P}{e}_0=HP{e}_0+HP{e}_{1}i+HP{e}_{2}j+HP{e}_{3}k$ and $\tilde{P}{e}_1=HP{e}_1+HP{e}_{2}i+HP{e}_{3}j+HP{e}_{4}k$.

And yet, extending these non-positive integer indices, we have:

Definition 2.7. The recurrence relation for the hybrid quaternions of non-positive Pell indices, $n\ge 0$, is defined by:

Theorem 2.1. Pell's hybrid quaternion, $\overline{F}{e}_n$, satisfies the following recurrence:

Proof.

According to the recurrence relationship, ${\widetilde{Pe}}_{n+1}=2{\widetilde{Pe}}_n+{\widetilde{Pe}}_{n-1}$, one can present its characteristic equation, defined by $x^2-2x-1=0$ where is a quadratic equation having two real roots $x_1=1+\sqrt{2}$ and $x_2=1-\sqrt{2}$.

Definition 2.8. Pell's hybrid quaternion conjugate can be defined in three different types for ${\widetilde{Pe}}_n={\widehat{HPe}}_n+{\widehat{HPe}}_{n+1}i+{\widehat{HPe}}_{n+2}\epsilon +{\widehat{HPe}}_{n+3}h$:

• $\text{Quaternion conjugate,}\overline{\stackrel{~}{P}{e}_n};\overline{\stackrel{~}{P}{e}_n}=\widehat{HP{E}_n}+\widehat{HP{E}_{n+1}}i+\widehat{HP{E}_{n+2}}\epsilon +\widehat{HP{E}_{n+3}}h$;
• Hybrid conjugate, $(\tilde{P}{e}_n{)}^C:(\tilde{P}{e}_n{)}^C={\widehat{HPe}}_n-{\widehat{HPe}}_{n+1}i-{\widehat{HPe}}_{n+2}\epsilon -{\widehat{HPe}}_{n+3}h;$
• Total conjugate, $(\tilde{P}{e}_n{)}^T:(\tilde{P}{e}_n{)}^T=\overline{(\stackrel{~}{P}{e}_n{)}^C}=\widehat{HP{e}_n}-\widehat{HP{e}_{n+1}}i-\widehat{HP{e}_{n+2}}\epsilon -$ $\widehat{HP{e}_n^{+3}}h$

Theorem 2.2. The generating function of Pell's hybrid quaternions, denoted by $G_{\tilde{P}{e}_n}(x)$, with $n\in \mathbf{N}$, is presented by:

Proof. To define the generating function of Pell's hybrid quaternions we will write a sequence in which each term of the sequence corresponds to the coefficients.

Making algebraic manipulations due to the recurrence relation we can write this sequence as:

Adding each member, we have:

Theorem 2.3. For $n\ge 0$, we have that the Binet formula for the Pell's hybrid quaternion is given by:

where $x_1$ and $x_2$ are the real roots of the characteristic equation.

Proof. The Binet formula can be represented as follows:

For $n=0$, there is: $A+B=\tilde{P}{e}_0$ a, for $n=1$, we have $A{x}_1+B{x}_2=\tilde{P}{e}_1$. Thus, we have the following linear system:

Solving the linear system, we have that the coefficients found were: $A=\frac{\tilde{P}{e}_1-\tilde{P}{e}_0{x}_2}{x_1-x_2}$ and $B=\frac{\tilde{P}{e}_0{x}_1-\tilde{P}{e}_1}{x_1-x_2}$.

Making the appropriate substitutions in the Binet formula, we have:

III. JACOBSTHAL HYBRID QUATERNIONS NUMBERS

Jacobsthal's hybrid quaternion numbers are defined from below.

Definition 3.1. Jacobsthal's hybrid number, denoted by $H{J}_n$, is defined by:

Definition 3.2. The Recurrence Relationship for Jacobsthal's hybrid, $n\ge 2$, is defined by:

with $H{J}_0=i+\epsilon +3h$ and $H{J}_1=1+i+3\epsilon +5h$ its initial terms.

Definition 3.3. Jacobsthal's quaternion number, denoted by $Q{J}_n$, it is given by:

Definition 3.4. The recurrence relation for Jacobsthal's quaternions, $n\ge 2$, is defined by:

with $Q{J}_0=i+j+3k$ and $Q{J}_1=1+i+3j+5k$ its initial terms.

Now, from what was seen above, we will approach Jacobsthal's hybrid quaternions.

Definition 3.5. Jacobsthal's hybrid quaternion number is defined as:

where $i,j,k$ are the units of the quaternions and $H{J}_n$ it's the $n$ -th Jacobsthal hybrid number. Thus, Jacobsthal's hybrid quaternions can be rewritten by: