The sequence of the hyperbolic k-Padovan quaternions

. This work introduces the hyperbolic k-Padovan quaternion sequence, performing the process of complexification of linear and recurrent sequences, more specifically of the generalized Padovan sequence. In this sense, there is the study of some properties around this sequence, deepening the investigative mathematical study of these numbers.


Introduction and Background
Studies of recursive linear sequences have been noticed in the mathematical literature. Based on this, there is the concern to carry out an investigative study on the process of complexification of certain sequences. So soon, in this work, the hyperbolic quaternion k-Padovan sequence is introduced, presenting algebraic properties around these numbers.
The quaternions were developed by Willian Rowan Hamilton (1805-1865), arose from the attempt to generalize complex numbers in the form z = a + bi in three dimensions [10]. Thus are presented as formal sums of scalars with usual vectors of three-dimensional space, existing four dimensions. Second Halici (2012) [8], a quaternion is a hyper-complex number and is described by: where a, b, c are real numbers or scalar and i, j, k the orthogonal part at the base R 3 . The quaternionic product being i 2 = j 2 = k 2 = ijk = −1, ij = k = −ji, jk = i = −kj and ki = j = −ik.
Being q 1 = a 1 + b 1 i + c 1 j + d 1 k and q 2 = a 2 + b 2 i + c 2 j + d 2 k two distinct quaternions. The addition, equality and multiplication scalar operations between them are: And for α ∈ R, we have αq 1 = αa 1 + αb 1 i + αc 1 j + αd 1 k. The conjugate of the quaternion is denoted by q = a − bi − cj − dk.
There are also other works, such as [3,6,7,9] that address the quaternions in the scope of numerical sequences, which are also used as a basis for this research.
As for hyperbolic numbers, the set of these numbers H can be described as: The addition and multiplication of two of these hyperbolic numbers n 1 e n 2 ,are given by [12]: In this sense, there are works on hyperbolic numbers and the quaternion sequence, used as a basis for this investigative process [1,2,4,5,11].

The hyperbolic k-Padovan quaternions
The sequence of k-Padovan is defined by P k,n = P k,n−2 + kP k,n−3 , n ≥ 3, k ⩾ 1 with initial values P k,0 = P k,1 = P k,2 = 1. In turn, we have the characteristic polynomial of this sequence as being Definition 2.1. The hyperbolic k-Padovan quaternions are given by: According to the definitions presented, a study is carried out on the operations of addition, subtraction, and multiplication of hyperbolic k-Padovan quaternions.

Some properties
Hereinafter, some properties of the hyperbolic quaternion k-Padovan sequence are studied, based on the definitions discussed in the previous section.

I-invariant arithmetic convergence
Proof. Performing the multiplication of the function by x 2 , kx 3 in the equations below, we have: HP k,n x n = HP k,0 + HP k,1 x + HP k,2 x 2 + . . . + HP k,n x n + . . . (3.1) x 2 g(HP k,n , x) = HP k,0 x 2 + HP k,1 x 3 + HP k,2 x 4 + . . . + HP k,n−2 x n + . . . Thus: For n ∈ N, the Binet formula of the hyperbolic k-Padovan quaternions is expressed by: where C 1 , C 2 , C 3 are the coefficients of the Binet formula of the sequence and r 1 , r 2 , r 3 the roots of the characteristic polynomial ( Proof. Based on the k-Padovan sequence recurrence formula, its respective defined initial values and its characteristic polynomial whose roots are r 1 , r 2 , r 3 , it is possible to obtain, by solving the linear system of equations, the values of coefficients C 1 , C 2 , C 3 . The discriminant ∆ = (−k) 2 4 − 1 27 , referring to the 3rd degree polynomial, determines how the roots of the polynomial will be. Thus, when ∆ ̸ = 0 all roots will be distinct, concluding that k 2 ̸ = 64 27 . Note also that r 1 r 2 r 3 = k, r 1 + r 2 + r 3 = 0 and that when k ̸ = 0, there will be at least one root equal to zero, there being no Binet formula for this case. ■ Theorem 3.3. For n geqslant2 and n in mathbbN , the matrix form of the hyperbolic k-Padovan quaternions is given by: Proof. Through the finite induction principle, for n = 2, we have:

Ö mer KİŞİ
Checking the validity for any n = z, z ∈ N, one has: Therefore, it turns out to be valid for n = z + 1 = 1 + z:

Conclusion
The study allowed for a mathematical analysis of the k-Padovan sequence and its complex form. Thus, the hyperbolic k-Padovan quaternion sequence was introduced, addressing some mathematical properties and theorems. It is noteworthy that for the particular case of k = 1, it is possible to notice that we have the hyperbolic quaternionic Padovan sequence.

Acknowledgement
The part of research development in Brazil had the financial support of the National Council for Scientific and Technological Development -CNPq and the Ceará Foundation for Support to Scientific and Technological Development (Funcap).
The research development aspect in Portugal is financed by National Funds through FCT -Fundação para a Ciência e Tecnologia. I.P, within the scope of the UID / CED / 00194/2020 project.