sequence of the hyperbolic k-Padovan quaternions

MALAYA JOURNAL OF MATEMATIK Malaya J. Mat. 11(03)(2023), 324–331. http://doi.org/10.26637/mjm1103/009 The sequence of the hyperbolic k-Padovan quaternions R ENATA V IEIRA* 1 , F RANCISCO R EGIS A LVES2 AND PAULA C ATARINO3 1 2 3 Post-Graduate Program in Education of the Nordeste Education Network – Polo RENOEN-UFC, Federal University of Ceara, Brazil. Federal Institute of Science and Technology Education of the State of Ceará, Brazil. University of Trás-os-Montes and Alto Douro, Portugal. Received 01 August 2022; Accepted 16 June 2023 Abstract. 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. AMS Subject Classifications: 11B37, 11B39. Keywords: hyperbolic numbers, quaternions, k-Padovan sequence. Contents 1 Introduction and Background 324 2 The hyperbolic k-Padovan quaternions 325 3 Some properties 326 4 Conclusion 328 5 Acknowledgement 328 1. 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 Padovan sequence is a linear and recurrent third-order sequence, named after the Italian architect Richard Padovan. Thus, its recurrence is given by: Pn = Pn−2 + Pn−3 , n ≥ 3 and being P0 = P1 = P2 = 1 your initial conditions [13–16]. 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: q = a + bi + cj + dk ∗ Corresponding author. Email address: re.passosm@gmail.com (Renata Vieira) https://www.malayajournal.org/index.php/mjm/index ©2023 by the authors. I-invariant arithmetic convergence where a, b, c are real numbers or scalar and i, j, k the orthogonal part at the base R3 . The quaternionic product being i2 = j 2 = k 2 = ijk = −1, ij = k = −ji, jk = i = −kj and ki = j = −ik. Being q1 = a1 + b1 i + c1 j + d1 k and q2 = a2 + b2 i + c2 j + d2 k two distinct quaternions. The addition, equality and multiplication scalar operations between them are: q1 + q2 = (a1 + a2 ) + (b1 + b2 )i + (c1 + c2 )j + (d1 + d2 )k. q1 = q2 only if a1 = a2 , b1 = b2 , c1 = c2 , d1 = d2 . And for α ∈ R, we have αq1 = αa1 + αb1 i + αc1 j + αd1 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:  H = z = x + hy|h ∈ / R, h2 = 1, x, y ∈ R . The addition and multiplication of two of these hyperbolic numbers n1 e n2 ,are given by [12]: n1 ± n2 = (x1 + hy1 ) ± (x2 + hy2 ) = (x1 ± x2 ) + h(y1 ± y2 ) n1 n2 = (x1 + hy1 )(x2 + hy2 ) = (x1 x2 ) + h(y1 y2 ) + h(x1 y2 + x2 y1 ) 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]. 2. The hyperbolic k-Padovan quaternions The sequence of k-Padovan is defined by Pk,n = Pk,n−2 + kPk,n−3 , n ≥ 3, k ⩾ 1 with initial values Pk,0 = Pk,1 = Pk,2 = 1. In turn, we have the characteristic polynomial of this sequence as being x3 − x − k = 0. Definition 2.1. The hyperbolic k-Padovan quaternions are given by: HPk,n = Pk,n + iPk,n+1 + jPk,n+2 + kPk,n+3 , where i2 = j 2 = k 2 = −1,ij = k = −ji,jk = i = −kj,ki = j = −ik. According to the definitions presented, a study is carried out on the operations of addition, subtraction, and multiplication of hyperbolic k-Padovan quaternions. HPk,n ± HPk,m = (Pk,n ± Pk,m ) + i(Pk,n+1 ± Pk,m+1 ) + j(Pk,n+2 ± Pk,m+2 ) + k(Pk,n+3 ± Pk,m+3 ), HPk,n HPk,m = (Pk,n Pk,m + Pk,n+1 Pk,m+1 + Pk,n+2 Pk,m+2 + Pk,n+3 Pk,m+3 ) + i(Pk,n Pk,m+1 + Pk,n+1 Pk,m + Pk,n+2 Pk,m+3 − Pk,n+3 Pk,m+2 ) + j(Pk,n Pk,m+2 + Pk,n+2 Pk,m − Pk,n+1 Pk,m+3 + Pk,n+3 Pk,m+1 ) + k(Pk,n Pk,m+3 + Pk,n+3 Pk,m + Pk,n+1 Pk,m+2 − Pk,n+2 Pk,m+1 ) ̸= HPk,m HPk,n The conjugate of the hyperbolic k-Padovan quaternary numbers is represented by: HP k,n = Pk,n − iPk,n+1 − jPk,n+2 − kPk,n+3 . 325 Ömer KİŞİ Theorem 2.2. Let Pk,n be the nth term of the k-Padovan sequence and HPk,n the nth term of the quaternionic k-Padovan sequence hyperbolic, for n ⩾ 1 the following relations are given: (i)HPk,n+3 = HPk,n+1 + kHPk,n ; (ii)HPk,n − iHPk,n+1 + jHPk,n+2 − kHPk,n+3 = Pk,n + Pk,n+2 + Pk,n+4 + Pk,n+6 . Proof. (i) Based on Definition 2.1, we have: HPk,n+1 + kHPk,n = Pk,n+1 + iPk,n+2 + jPk,n+3 + kPk,n+4 + k(Pk,n + iPk,n+1 + jPk,n+2 + kPk,n+3 ) = (Pk,n+1 + kPk,n ) + i(Pk,n+2 + kPk,n+1 ) + j(Pk,n+3 + kPk,n+2 ) + k(Pk,n+4 + kPk,n+3 ) = Pk,n+3 + iPk,n+4 + jPk,n+5 + kPk,n+6 = HPk,n+3 For (ii), we have: HPk,n − iHPk,n+1 + jHPk,n+2 − kHPk,n+3 = Pk,n + iPk,n+1 + jPk,n+2 + kPk,n+3 − i(Pk,n+1 + iPk,n+2 + jPk,n+3 + kPk,n+4 ) − j(Pk,n+2 + iPk,n+3 + jPk,n+4 + kPk,n+5 ) − k(Pk,n+3 + iPk,n+4 + jPk,n+5 + kPk,n+6 ) = Pk,n + Pk,n+2 − kPk,n+3 + jPk,n+4 + kPk,n+3 + Pk,n+4 − iPk,n+5 − jPk,n+4 + iPk,n+5 + Pk,n+6 = Pk,n + Pk,n+2 + Pk,n+4 + Pk,n+6 ■ Theorem 2.3. Let HP k,n be the quaternionic conjugate of hyperbolic k-Padovan, then: HPk,n + HP k,n = 2Pk,n Proof. According to Definition 2.1, we have: HPk,n + HP k,n = Pk,n + iPk,n+1 + jPk,n+2 + kPk,n+3 + Pk,n − iPk,n+1 − jPk,n+2 − kPk,n+3 = 2Pk,n ■ 3. Some properties Hereinafter, some properties of the hyperbolic quaternion k-Padovan sequence are studied, based on the definitions discussed in the previous section. Theorem 3.1. The generating function of the hyperbolic k-Padovan quaternions is given by: g(HPk,n , x) = HPk,0 + HPk,1 x + (HPk,2 − HPk,0 )x2 . 1 − x2 − kx3 326 I-invariant arithmetic convergence Proof. Performing the multiplication of the function by x2 , kx3 in the equations below, we have: g(HPk,n , x) = ∞ X HPk,n xn = HPk,0 + HPk,1 x + HPk,2 x2 + . . . + HPk,n xn + . . . (3.1) n=0 x2 g(HPk,n , x) = HPk,0 x2 + HPk,1 x3 + HPk,2 x4 + . . . + HPk,n−2 xn + . . . 3 3 4 5 n kx g(HPk,n , x) = HPk,0 kx + HPk,1 kx + HPk,2 kx + . . . + HPk,n−3 kx + . . . (3.2) (3.3) Based on the Equation (3.1-3.2+3.3), we have: (1 − x2 − kx3 )g(HPk,n , x) = HPk,0 + HPk,1 x + (HPk,2 − HPk,0 )x2 + (HPk,3 − HPk,1 − HPk,0 )x3 + . . . + (HPk,n − HPk,n−2 − HPk,n−3 )xn + . . . Thus: (1 − x2 − kx3 )g(HPk,n , x) = HPk,0 + HPk,1 x + (HPk,2 − HPk,0 )x2 g(HPk,n , x) = HPk,0 + HPk,1 x + (HPk,2 − HPk,0 )x2 . 1 − x2 − kx3 ■ Theorem 3.2. For n ∈ N, the Binet formula of the hyperbolic k-Padovan quaternions is expressed by: (n) Qk,n = C1 r1n + C2 r2n + C3 r3n , where C1 , C2 , C3 are the coefficients of the Binet formula of the sequence and r1 , r2 , r3 the roots of the characteristic polynomial (x3 − x − k = 0). Proof. Based on the k-Padovan sequence recurrence formula, its respective defined initial values and its characteristic polynomial whose roots are r1 , r2 , r3 , it is possible to obtain, by solving the linear system of equations, the values of coefficients C1 , C2 , C3 . 2 1 The discriminant ∆ = (−k) − 27 , referring to the 3rd degree polynomial, determines how the roots of the 4 64 . Note also that polynomial will be. Thus, when ∆ ̸= 0 all roots will be distinct, concluding that k 2 ̸= 27 r1 r2 r3 = k, r1 + r2 + r3 = 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: n      01k Qk,2 Qk,1 Qk,0 Hk,n+2 Hk,n+1 Hk,n 1 0 0 Qk,1 Qk,0 Qk,−1  = Hk,n+1 Hk,n Hk,n−1  . 010 Qk,0 Qk,−1 Qk,−2 Hk,n Hk,n−1 Hk,n−2 Proof. Through the finite induction principle, for n = 2, we have:  2      01k Hk,2 Hk,1 Hk,0 1k0 Hk,2 Hk,1 Hk,0 1 0 0 Hk,1 Hk,0 Hk,−1  = 0 1 k  Hk,1 Hk,0 Hk,−1  010 100 Hk,0 Hk,−1 Hk,−2 Hk,0 Hk,−1 Hk,−2   Hk,2 + kHk,1 Hk,1 + kHk,0 Hk,0 + kHk,−1 = Hk,1 + kHk,0 Hk,0 + kHk,−1 Hk,−1 + kHk,−2  Hk,2 Hk,1 Hk,0   Hk,4 Hk,3 Hk,2 = Hk,3 Hk,2 Hk,1  . Hk,2 Hk,1 Hk,0 327 Ömer KİŞİ Checking the validity for any n = z, z ∈ N, one has:  z     01k Hk,2 Hk,1 Hk,0 Hk,z+2 Hk,z+1 Hk,z 1 0 0 Hk,1 Hk,0 Hk,−1  = Hk,z+1 Hk,z Hk,z−1  . 010 Hk,0 Hk,−1 Hk,−2 Hk,z Hk,z−1 Hk,z−2 Therefore, it turns out to be valid for n = z + 1 = 1 + z:  1+z     z   01k Hk,2 Hk,1 Hk,0 01k 01k Hk,2 Hk,1 Hk,0 1 0 0  Hk,1 Hk,0 Hk,−1  = 1 0 0 1 0 0 Hk,1 Hk,0 Hk,−1  010 Hk,0 Hk,−1 Hk,−2 010 010 Hk,0 Hk,−1 Hk,−2    01k Hk,z+2 Hk,z+1 Hk,z = 1 0 0 Hk,z+1 Hk,z Hk,z−1  010 Hk,z Hk,z−1 Hk,z−2   Hk,z+1 + kHk,z Hk,z + kHk,z−1 Hk,z−1 + kHk,z−2  = Hk,z+2 Hk,z+1 Hk,z Hk,z+1 Hk,z Hk,z−1   Hk,z+3 Hk,z+2 Hk,z+1 = Hk,z+2 Hk,z Hk,z  . Hk,z+1 Hk,z Hk,z−1 ■ 4. 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. 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