The Third Order Jacobsthal Octonions: Some Combinatorial Properties

Transcription

1 DOI: /auom An. Şt. Univ. Ovidius Constanţa Vol. 26),2018, The Third Order Jacobsthal Octonions: Some Combinatorial Properties Gamaliel Cerda-Morales Abstract Various families of octonion number sequences such as Fibonacci octonion, Pell octonion and Jacobsthal octonion) have been established by a number of authors in many different ways. In addition, formulas and identities involving these number sequences have been presented. In this paper, we aim at establishing new classes of octonion numbers associated with the third order Jacobsthal and third order Jacobsthal- Lucas numbers. We introduce the third order Jacobsthal octonions and the third order Jacobsthal-Lucas octonions and give some of their properties. We derive the relations between third order Jacobsthal octonions and third order Jacobsthal-Lucas octonions. 1 Introduction Recently, the topic of number sequences in real normed division algebras has attracted the attention of several researchers. It is worth noticing that there are exactly four real normed division algebras: real numbers R), complex numbers C), quaternions H) and octonions O). In [4] Baez gives a comprehensive discussion of these algebras. The real quaternion algebra H = {q = q r + q i i + q j j + q k k : q r, q s R, s = i, j, k} Key Words: Third order Jacobsthal numbers, third order Jacobsthal-Lucas numbers, third order Jacobsthal octonions, third order Jacobsthal-Lucas octonions, octonion algebra Mathematics Subject Classification: Primary 11B9; Secondary 11R52, 05A15. Received: Accepted:

2 PROPERTIES 58 is a 4-dimensional R-vector space with basis {1 e 0, i e 1, j e 2, k e } satisfying multiplication rules q r 1 = q r, e 1 e 2 = e 2 e 1 = e, e 2 e = e e 2 = e 1 and e e 1 = e 1 e = e 2. Furthermore, the real octonion algebra denoted by O is an 8-dimensional real linear space with basis {e 0 = 1, e 1 = i, e 2 = j, e = k, e 4 = e, e 5 = ie, e 6 = je, e 7 = ke}, 1.1) where e 0 e s = e s s = 1,..., 7) and q r e 0 = q r q r R). The space O becomes an algebra via multiplication rules listed in the table 1, see [27]. Table 1: The multiplication table for the basis of O. e 1 e 2 e e 4 e 5 e 6 e 7 e 1 1 e e 2 e 5 e 4 e 7 e 6 e 2 e 1 e 1 e 6 e 7 e 4 e 5 e e 2 e 1 1 e 7 e 6 e 5 e 4 e 4 e 5 e 6 e 7 1 e 1 e 2 e e 5 e 4 e 7 e 6 e 1 1 e e 2 e 6 e 7 e 4 e 5 e 2 e 1 e 1 e 7 e 6 e 5 e 4 e e 2 e 1 1 A variety of new results on Fibonacci-like quaternion and octonion numbers can be found in several papers [8, 12, 1, 16, 17, 18, 19, 21, 2, 26]. The origin of the topic of number sequences in division algebra can be traced back to the works by Horadam in [18] and by Iyer in [21]. A. F. Horadam [18] defined the quaternions with the classic Fibonacci and Lucas number components as and QF n = F n + F n+1 i + F n+2 j + F n+ k QL n = L n + L n+1 i + L n+2 j + L n+ k, respectively, where F n and L n are the n-th classic Fibonacci and Lucas numbers, respectively, and the author studied the properties of these quaternions. Several interesting and useful extensions of many of the familiar quaternion numbers such as the Fibonacci and Lucas quaternions [, 16, 18], Pell quaternion [7, 12], Jacobsthal quaternions [26] and third order Jacobsthal quaternion [8]) have been considered by several authors. There has been an increasing interest on quaternions and octonions that play an important role in various areas such as computer sciences, physics, differential geometry, quantum physics, signal, color image processing and geostatics for more, see [1, 6, 14, 15, 24, 25]). In this paper, we define two families of the octonions, where the coefficients in the terms of the octonions are determined by the third order Jacobsthal and

3 PROPERTIES 59 third order Jacobsthal-Lucas numbers. These two families of the octonions are called as the third order Jacobsthal and third order Jacobsthal-Lucas octonions, respectively. We mention some of their properties, and apply them to the study of some identities and formulas of the third order Jacobsthal and third order Jacobsthal-Lucas octonions. Here, our approach for obtaining some fundamental properties and characteristics of third order Jacobsthal and third order Jacobsthal-Lucas octonions is to apply the properties of the third order Jacobsthal and third order Jacobsthal-Lucas numbers introduced by Cook and Bacon [10]. This approach was originally proposed by Horadam and Iyer in the articles [18, 21] for Fibonacci quaternions. The methods used by Horadam and Iyer in that papers have been recently applied to the other familiar octonion numbers by several authors [2, 7, 9, 1, 2]. This paper has three main sections. In Section 2, we provide the basic definitions of the octonions and the third order Jacobsthal and third order Jacobsthal-Lucas numbers. Section is devoted to introducing third order Jacobsthal and third order Jacobsthal-Lucas octonions, and then to obtaining some fundamental properties and characteristics of these numbers. 2 Preliminaries The Jacobsthal numbers have many interesting properties and applications in many fields of science see, e.g., [5, 20]). The Jacobsthal numbers J n are defined by the recurrence relation J 0 = 0, J 1 = 1, J n+1 = J n + 2J n 1, n ) Another important sequence is the Jacobsthal-Lucas sequence. This sequence is defined by the recurrence relation j 0 = 2, j 1 = 1, j n+1 = j n + 2j n 1, n ) see, [20]). In [10] the Jacobsthal recurrence relation is extended to higher order recurrence relations and the basic list of identities provided by A. F. Horadam [20] is expanded and extended to several identities for some of the higher order cases. For example, the third order Jacobsthal numbers, {J n ) } n 0, and third order Jacobsthal-Lucas numbers, {j n ) } n 0, are defined by and n+ = n+2 + ) n+1 + 2J n, 0 = 0, 1 = 2 = 1, n 0, 2.) n+ = j) n+2 + j) n+1 + 2j) n, 0 = 2, 1 = 1, 2 = 5, n 0, 2.4)

4 PROPERTIES 60 respectively. The following properties given for third order Jacobsthal numbers and third order Jacobsthal-Lucas numbers play important roles in this paper see [8, 10]). J n ) + j n ) = 2 n+1, 2.5) j n ) J n ) = 2 n, 2.6) { ) 2 if n 1 mod ) n+2 4J n =, 2.7) 1 if n 1 mod ) n 4 n = n n+2 = n k=0 n k=0 k = k = and n+1 + j) ) 2 n + J ) { n { 2 if n 0 mod ) if n 1 mod ) 1 if n 2 mod ), 2.8) n = n+2, 2.9) 1 if n 0 mod ) 1 if n 1 mod ) 0 if n 2 mod ) n, 2.10) n = 4 n, 2.11) n+1 if n 0 mod ) n+1 1 if n 0 mod ), 2.12) n+1 2 if n 0 mod ) n if n 0 mod ) 2.1) ) 2 9 n ) 2 = 2 n+2 n. 2.14) Using standard techniques for solving recurrence relations, the auxiliary equation, and its roots are given by x x 2 x 2 = 0; x = 2, and x = 1 ± i. 2 Note that the latter two are the complex conjugate cube roots of unity. Call them ω 1 and ω 2, respectively. Thus the Binet formulas can be written as n = 2 7 2n + 2i ω1 n 2i ω2 n 2.15) 21 21

5 PROPERTIES 61 and j n ) = 8 7 2n + + 2i ω1 n + 2i ω2 n, 2.16) 7 7 respectively. In the following we will study the important properties of the octonions. We refer to [4] for a detailed analysis of the properties of the next octonions p = 7 a se s and q = 7 b se s where the coefficients a s and b s, s = 0, 1,..., 7, are real. We recall here only the following facts The sum and subtract of p and q is defined as p ± q = a s ± b s )e s, 2.17) where p O can be written as p = R p + I p, and R p = a 0 and 7 a se s are called the real and imaginary parts, respectively. The conjugate of p is defined by p = R p I p = a 0 a s e s 2.18) and this operation satisfies p = p, p + q = p + q and p q = q p, for all p, q O. The norm of an octonion, which agrees with the standard Euclidean norm on R 8 is defined as Nr 2 p) = p p = p p = a 2 s, Nrp) = 7 a 2 s R ) The inverse of p 0 is given by p 1 = definitions it is deduced that p Nr 2 p). From the above two Nr 2 p q) = Nr 2 p)nr 2 q) and p q) 1 = q 1 p ) O is non-commutative and non-associative but it is alternative, in other words p p q) = p 2 q, p q) q = p q 2, p q) p = p q p) = p q p, where denotes the product in the octonion algebra O. 2.21)

6 PROPERTIES 62 The third order Jacobsthal Octonions and third order Jacobsthal-Lucas Octonions In this section, we define new kinds of sequences of octonion number called as third order Jacobsthal octonions and third order Jacobsthal-Lucas octonions. We study some properties of these octonions. We obtain various results for these classes of octonion numbers included recurrence relations, summation formulas, Binet s formulas and generating functions. In [8], the author introduced the so-called third order Jacobsthal quaternions, which are a new class of quaternion sequences. They are defined by JQ ) n = n+se s = n + n+se s, n 1 = n ).1) where J n ) is the n-th third order Jacobsthal number, e 2 1 = e 2 2 = e 2 = 1 and e 1 e 2 e = 1. We now consider the usual third order Jacobsthal and third order Jacobsthal- Lucas numbers, and based on the definition.1) we give definitions of new kinds of octonion numbers, which we call the third order Jacobsthal octonions and third order Jacobsthal-Lucas octonions. In this paper, we define the n-th third order Jacobsthal octonion and third order Jacobsthal-Lucas octonion numbers, respectively, by the following recurrence relations and n = n + n+se s, n 0 = n + n+1 e 1 + n+2 e 2 + n+ e + n+4 e 4 + n+5 e 5 + n+6 e 6 + n+7 e 7 n = n + n+se s, n 0 = n + n+1 e 1 + n+2 e 2 + n+ e + n+4 e 4 + n+5 e 5 + n+6 e 6 + n+7 e 7,.2).) where J n ) and j n ) are the n-th third order Jacobsthal number and third order Jacobsthal-Lucas number, respectively. Here {e s : s = 0, 1,..., 7} satisfies the multiplication rule given in the Table 1. The equalities in 2.17) gives n ± n = n+s ± j n+s)e ) s..4)

7 PROPERTIES 6 From 2.18), 2.19),.2) and.) an easy computation gives n = n Nr 2 n ) = J n+se ) s, jo n ) = j n ) n+s) 2 and Nr 2 jo n ) ) = n+se s,.5) n+s) 2..6) By some elementary calculations we find the following recurrence relations for the third order Jacobsthal and third order Jacobsthal-Lucas octonions from.2),.), 2.), 2.4) and.4): n+1 + JO) n + 2 n 1 = n+s+1 + n+s + 2 = n+2 + n+s+2 e s = n+2 n+s 1 )e s.7) and similarly n+2 = jo) n+1 + jo) n + 2 n 1, for n 1. Now, we will state Binet s formulas for the third order Jacobsthal and third order Jacobsthal-Lucas octonions. Repeated use of 2.) in.2) enables one to write for α = 7 2s e s, ω 1 = 7 ωs 1e s and ω 2 = 7 ωs 2e s n = n + = n+se s 2 7 2n+s + 2i 21 = 2 7 α2n + 2i 21 ω n+s 1 2i 21 ω 1 ω1 n 2i ω 2 ω2 n 21 and similarly making use of 2.4) in.) yields ω n+s 2 ) e s.8) n = n + = n+se s 8 7 2n+s + + 2i 7 = 8 7 α2n + + 2i 7 ω n+s 1 + 2i 7 ω 1 ω1 n + 2i ω 2 ω2 n. 7 ω n+s 2 ) e s.9)

8 PROPERTIES 64 The formulas in.8) and.9) are called as Binet s formulas for the third order Jacobsthal and third order Jacobsthal-Lucas octonions, respectively. The recurrence relations for the third order Jacobsthal octonions and the norm of the n-th third order Jacobsthal octonion are expressed in the following theorem. Theorem.1. For n 0, we have the following identities: n+2 4JO) n = n+2 + JO) n+1 + JO) n = 2 n+1 α,.10) ) 1 + e2 + e + e 5 + e 6 if n 0 mod ) 2e 1 + e 4 + e 7 ) ) Nr 2 JO n ) ) = 1 49 where α = 7 2s e s. e1 + e 2 + e 4 + e 5 + e e + e 6 ) ) 1 + e1 + e + e 4 + e 6 +e 7 2e 2 + e 5 ) Proof. Consider 2.17) and.2) we can write n+2 + JO) n+1 + JO) n = if n 1 mod ) if n 2 mod ) n n + 41 if n 0 mod ) n n + 8 if n 1 mod ) n n + if n 2 mod ),.11),.12) n+s+2 + n+s+1 + J n+s)e ) s..1) Using the identity n+2 + n+1 + J n ) = 2 n+1 in.1), the above sum can be calculated as n+2 + JO) n+1 + JO) n = 2 n+s+1 e s, which can be simplified as n+2 + JO) n+1 + JO) n = 2 n+1 α, where α = 7 2s e s. Now, using 2.7) and.2) we have n+2 4JO) n = 7 ) J 4J n+s)e ) s, then n+2 4JO) n = n+s+2 4J ) n+s)e s n+s+2 = n+2 4J ) n + n+ 4J ) n+1 )e n+9 4J ) n+7 )e 7 = e 1 + e 2 + e 4 + e 5 + e e + e 6 )

9 PROPERTIES 65 if n 1mod ). Similarly in the other cases, this proves.11). Finally, observing that Nr 2 JO n ) ) = 7 ) J n+s) 2 from the Binet formula 2.15) we have Nr 2 n ) = n ) 2 + n+1 )2 + + n+7 )2 = n+1 aω n 1 + bω n 2 )) n+8 aω n bω n+7 2 )) 2), where a = 1 + 2i and b = 1 2i. It is easy to see that, 2 if n 0 mod ) aω1 n + bω2 n = if n 1 mod ),.14) 1 if n 2 mod ) because ω 1 and ω 2 are the complex conjugate cube roots of unity i.e. ω1 = ω2 = 1). Then, if we consider first n 0mod ), we obtain 2 n+1 Nr 2 JO n ) )) = 1 2 ) n+2 + ) n+ 1 ) n+4 2 ) n+5 + ) n+6 1 ) n+7 2 ) n+8 + ) 2 = n n ). The other identities are clear from equation.14). The recurrence relations for the third order Jacobsthal-Lucas octonions and the norm of the n-th third order Jacobsthal-Lucas octonion are given in the following theorem. Theorem.2. Let n 0 be integer. Then, n+2 4jO) n = Nr 2 jo n ) ) = 1 49 where α = 7 2s e s. n+2 + jo) n+1 + jo) n = 2 n+ α,.15) ) 1 + e2 + e + e 5 + e 6 ) if n 0 mod ) +6e 1 + e 4 + e 7 ) ) e1 + e 2 + e 4 + e 5 + e 7 ) if n 1 mod ), e + e 6 ) ) 1 + e1 + e + e 4 + e 6 ) if n 2 mod ) e 7 + 6e 2 + e 5 ).16) n n + 69 if n 0 mod ) n n + 42 if n 1 mod ) n n if n 2 mod ),.17)

10 PROPERTIES 66 The proofs of the identities.15).17) of this theorem are similar to the proofs of the identities.10).12) of Theorem.1, respectively, and are omitted here. The following theorem deals with two relations between the third order Jacobsthal and third order Jacobsthal-Lucas octonions. Theorem.. Let n 0 be integer. Then, jo n ) n+2 = n 4 n = n+ JO) n+ = 2jO) n,.18) n + n+1 = JO) n+2,.19) 1 e 1 + e e 4 + e 6 e 7 if n 0 mod ) 1 + e 2 e + e 5 e 6 if n 1 mod ) e 1 e 2 + e 4 e 5 + e 7 if n 2 mod ) ) 2 e1 + e 2 + 2e e 4 + e 5 + 2e 6 e 7 ) + e1 + 2e 2 e +e 4 + 2e 5 e 6 + e 7 ) 1 + 2e1 e 2 + e +2e 4 e 5 + e 6 + 2e 7 Proof. The following recurrence relation n+ JO) n+ = can be readily written considering that n = n + n+s+ n+se s and n = n + if n 0 mod ) if n 1 mod ) if n 2 mod ),.20)..21) J ) n+s+ )e s.22) n+se s. Notice that ) n+ J n+ = 2j) n from 2.6) see [10]), whence it follows that.22) can be rewritten as n+ JO) n+ = 2jO) n from which the desired result.18) of Theorem.. In a similar way we can show the second equality. By using the identity j n ) + ) n+1 = J n+2 we have jo n ) + ) n+1 = J n+2 e 0 + n+ e n+9 e 7), which is the assertion.19) of theorem.

11 PROPERTIES 67 By using the identity j n ) n+2 = 1 from 2.10) see [10]) we have jo n ) n+2 = j) n n+2 )e 0 + n+1 JO) n+ )e n+7 n+9 )e 7 = 1 e 1 + e e 4 + e 6 e 7 if n 0mod ), the other identities are clear from equation 2.10). Finally, the proof of Eq..21) is similar to.20) by using 2.8). Theorem.4. Let n 0 be integer such that n 0mod ). Then, 49 n n ) = n n )e n n + 6)e n n 12)e n n 24)e.2) + 2 2n n + 6)e n n 12)e n n 24)e n n + 6)e 7 and 49 n n ) = n n )e n n + 6)e n n 12)e n n 24)e.24) + 2 2n n + 6)e n n 12)e n n 24)e n n + 6)e 7. Proof. In view of the multiplication table 1 and the definitions.2) and.), we obtain jo n ) JO n ) = j n ) J n ) n+1 n+1 j) n+2 n+2 j) n+ n+ j) n+4 n+4 n+5 n+5 j) n+6 n+6 j) n+7 n+7 )e 0 + j n ) n+1 + j) n+1 J n ) + n+2 n+ j) n+ n+2 + j) n+4 n+5 n+5 n+4 j) n+6 n+7 + j) n+7 n+6 )e 1 + j n ) n+2 j) n+1 n+ + j) n+2 J n ) + n+ n+1 + j) n+4 n+6 + n+5 n+7 j) n+6 n+4 j) n+7 n+5 )e 2. + j n ) n+7 j) n+1 n+6 + j) n+2 n+5 + j) n+ n+4 j) n+4 n+ n+5 n+2 + j) n+6 n+1 + j) n+7 J n ) )e 7.

12 PROPERTIES 68 Let a = 1 + 2i and b = 1 2i, using Eq..14) and the following formula 49 n+r n+s) = 2 2n+4+r+s 2 n+r+ aω n+s 1 + bω n+s 2 ) + 2 n+s+1 aω n+r 1 + bω n+r 2 ) a 2 ω 2n+r+s 1 + b 2 ω 2n+r+s 2 ) 7ω r 1ω s 2 + ω s 1ω r 2), we get the required result.2) if n 0mod ). In the same way, from the multiplication JO n ) jo n ) and the formula 49 n+r n+s) = 2 2n+4+r+s + 2 n+r+1 aω n+s 1 + bω n+s 2 ) we reach.24). 2 n+s+ aω n+r 1 + bω n+r 2 ) a 2 ω 2n+r+s 1 + b 2 ω 2n+r+s 2 ) 7ω r 1ω s 2 + ω s 1ω r 2), Based on the Binet s formulas given in.8) and.9) for the third order Jacobsthal and third order Jacobsthal-Lucas octonions, now we give some quadratic identities for these octonions. Theorem.5. For every nonnegative integer number n we get ) 2 jo n ) ) + JO n+ jo) n+ = 4n+ α 2 + 2n+1 25α ɛ n 1 ɛ n α),.25) 49 where α = 7 2s e s and 21 + e + e 6 ) e 1 + e 4 + e 7 ) + e 2 + e 5 ) if n 0 mod ) ɛ n = 1 + e + e 6 ) + e 1 + e 4 + e 7 ) + 2e 2 + e 5 ) if n 1 mod ) 1 + e + e 6 ) + 2e 1 + e 4 + e 7 ) e 2 + e 5 ) if n 2 mod ) Proof. Let α = 7 2s e s O. Using the relation in.8) and.9) for the third order Jacobsthal and third order Jacobsthal-Lucas octonions, the left side of equality.25) can be written as = n ) 2 ) + JO n+ jo) n+ ) n+ α + ɛ n ) 1 7 2n+4 α ɛ n+ ) = n+6 α n+1 α ɛ n + ɛ n α) + 9 ɛ n ) 2 ) n+10 α n+4 α ɛ n+ 4ɛ n+ α) ɛ n ) 2 ) ) ) 1 7 2n+6 α + ɛ n+ ) = 2 2n+6 α n+1 α ɛ n + ɛ n α) + 2 n+4 α ɛ n+ 4ɛ n+ α) ), 49.

13 PROPERTIES 69 where ) ) ɛ n = 1 + 2i ω 1 ω1 n + 1 2i ω 2 ω2 n ) ) = 1 + 2i ω1 n+s + 1 2i = ω n+s e + e 6 ) e 1 + e 4 + e 7 ) + e 2 + e 5 ) if n 0 mod ) 1 + e + e 6 ) + e 1 + e 4 + e 7 ) + 2e 2 + e 5 ) if n 1 mod ) 1 + e + e 6 ) + 2e 1 + e 4 + e 7 ) e 2 + e 5 ) if n 2 mod ) Note that ɛ n = ɛ n+ for all n 0, which can be simplified as n ) 2 + JO ) n+ jo) n+ ) e s.26). = 2 2n+6 α 2 + 2n+1 49 = 4 n+ α 2 + 2n+1 49 Thus, we get the required result in.25). α ɛ n + ɛ n α + 24α ɛ n 2 ɛ n α) 25α ɛ n 1 ɛ n α). Theorem.6. For every nonnegative integer number n we get n ) 2 9 n where α = 7 2s e s and ɛ n as in.26). ) 2 = 1 7 4n+1 α n+1 α ɛ n + ɛ n α)),.27) The proofs of quadratic identities for the third order Jacobsthal and third order Jacobsthal-Lucas octonions in this theorem are similar to the proof of the identity.25) of Theorem.5, and are omitted here. References [1] S.L. Adler, Quaternionic quantum mechanics and quantum fields, New York: Oxford University Press, [2] I. Akkus and O. Keçilioglu, Split Fibonacci and Lucas octonions, Adv. Appl. Clifford Algebras, 25), 2015), [] M. Akyigit, H.H. Kösal and M. Tosun, Split Fibonacci quaternions, Adv. Appl. Clifford Algebras, 2, 201),

14 PROPERTIES 70 [4] J.C. Baez, The octonions, Bull. Am. Math. Soc., 9, 2002), [5] P. Barry, Triangle geometry and Jacobsthal numbers, Irish Math. Soc. Bull., 51, 200), [6] K. Carmody, Circular and Hyperbolic Quaternions, Octonions and Sedenions, Appl. Math. Comput., 28, 1988), [7] P. Catarino, The modified Pell and the modified k-pell quaternions and octonions, Adv. Appl. Clifford Algebras, 26, 2016), [8] G. Cerda-Morales, Identities for Third Order Jacobsthal Quaternions, Advances in Applied Clifford Algebras, 272), 2017), [9] G. Cerda-Morales, On a Generalization of Tribonacci Quaternions, Mediterranean Journal of Mathematics, 14: ), [10] Ch. K. Cook and M. R. Bacon, Some identities for Jacobsthal and Jacobsthal-Lucas numbers satisfying higher order recurrence relations, Annales Mathematicae et Informaticae, 41, 201), [11] W. Y. C. Chen and J. D. Louck, The combinatorial power of the companion matrix, Linear Algebra Appl., 22, 1996), [12] C.B. Çimen and A. İpek, On Pell quaternions and Pell-Lucas quaternions, Adv. Appl. Clifford Algebras, 261), 2016), [1] C.B. Çimen and A. İpek, On Jacobsthal and Jacobsthal-Lucas Octonions, Mediterranean Journal of Mathematics, 14:7, 2017), 1 1. [14] M. Gogberashvili, Octonionic Geometry, Adv. Appl. Clifford Algebras, 15, 2005), [15] M. Gogberashvili, Octonionic electrodynamics, J. Phys. A: Math. Gen., 9, 2006), [16] S. Halici, On Fibonacci quaternions, Adv. Appl. Clifford Algebras, 22, 2012), [17] S. Halici, On complex Fibonacci quaternions, Adv. Appl. Clifford Algebras, 2, 201), [18] A. F. Horadam, Complex Fibonacci numbers and Fibonacci quaternions, Am. Math. Month., 70, 196), [19] A. F. Horadam, Quaternion recurrence relations, Ulam Quarterly, 2, 199), 2.

15 PROPERTIES 71 [20] A. F. Horadam, Jacobsthal representation numbers, Fibonacci Quarterly, 4, 1996), [21] M.R. Iyer, A note on Fibonacci quaternions, Fibonacci Quaterly, 7), 1969), [22] D. Kalman, Generalized Fibonacci numbers by matrix methods, Fibonacci Quaterly, 201), 1982), [2] O. Keçilioğlu and I. Akkus, The Fibonacci Octonions, Adv. Appl. Clifford Algebras, 251), 2015), [24] J. Köplinger, Signature of gravity in conic sedenions, Appl. Math. Computation, 188, 2007), [25] J. Köplinger, Hypernumbers and relativity, Appl. Math. Computation, 188, 2007), [26] A. Szynal-Liana and I. W loch, A Note on Jacobsthal Quaternions, Adv. Appl. Clifford Algebras, 26, 2016), [27] Y. Tian, Matrix representations of octonions and their applications, Adv. Appl. Clifford Algebras, 101), 2000), Gamaliel CERDA-MORALES, Institute of Mathematics, Pontificia Universidad Católica de Valparaíso, Blanco Viel 596, Cerro Barón, Valparaíso, Chile. gamaliel.cerda.m@mail.pucv.cl

16 PROPERTIES 72