Hyperbolic K-Fibonacci and K-Lucas Octonions

Notes on Number Theory and Discrete Mathematics Print ISSN 1310–5132, Online ISSN 2367–8275 Vol. 26, 2020, No. 3, 176–188 DOI: 10.7546/nntdm.2020.26.3.176-188

Hyperbolic k-Fibonacci and k-Lucas octonions

A. D. Godase

Department of Mathematics, V. P. College Vaijapur Aurangabad(MH), India e-mail: [email protected]

Received: 16 October 2019 Revised: 31 July 2020 Accepted: 1 August 2020

Abstract: In this paper, we introduce the hyperbolic k-Fibonacci and k-Lucas octonions. We present Binet’s formulas, Catalan’s identity, Cassini’s identity, d’Ocagne’s identity and generating functions for the k-Fibonacci and k-Lucas hyperbolic octonions. Keywords: Fibonacci sequence, k-Fibonacci sequence, k-Lucas sequence. 2010 Mathematics Subject Classiﬁcation: 11B39, 11B37.

1 Introduction

Fibonacci sequence is a well-known sequence which satisfy the second order recurrence relation. The Fibonacci sequence is generalized in different ways by changing initial conditions or recurrence relation. The k-Fibonacci sequence is one generalization of Fibonacci sequence which is first introduced by Falcon and Plaza [10]. For properties and applications of k-Fibonacci and k-Lucas numbers one can refer the articles [2, 6–9, 11, 12, 27]. The quaternions were first introduced by the Irish mathematician William Rowan Hamilton in 1843. Hamilton [18] introduced the set of quaternions which form a 4-dimensional real vector space with a multiplicative operation. The quaternions have many applications in applied sciences such as physics, computer science and Clifford algebras in mathematics. They are important in mechanics [16], chemistry [13], kinematics [1], quantum mechanics [24], differential geometry and pure algebra. In [19], Horadam defined the n-th Fibonacci and n-th Lucas quaternions. In [23], Ramirez defined and studied the k-Fibonacci and k-Lucas quaternions. Quaternions of sequences have been studied by many researchers. Such as Iyer [20, 21] obtained various relations containing the Fibonacci and Lucas quaternions. Halici [17] studied combinatorial properties of Fibonacci quaternions. Akyigit et al. [25, 26] established and investigated the

176 Fibonacci generalized quaternions and split Fibonacci quaternions. Catarino [5] obtained properties of the h(x)-Fibonacci quaternion polynomials. Polatli and Kesim [22] have introduced quaternions with generalized Fibonacci and Lucas number components. F L In [15], the hyperbolic k-Fibonacci and k-Lucas quaternions Q k,n and Q k,n are deﬁned as F Q k,n = Fk,ni1 + Fk,n+1i2 + Fk,n+2i3 + Fk,n+3i4

= Fk,n,Fk,n+1,Fk,n+2,Fk,n+3 and

L Q k,n = Lk,ni1 + Lk,n+1i2 + Lk,n+2i3 + Lk,n+3i4

= Lk,n,Lk,n+1,Lk,n+2,Lk,n+3 ,

respectively, where Fk,n is the n-th k-Fibonacci sequence and Lk,n is n-th k-Lucas sequence.

Here, i1, i2, i3, i4 are hyperbolic quaternion units which satisfy the multiplication rule 2 2 2 i2 = i3 = i4 = i2i3i4 = +1, i1 = 1,

i2i3 = i4 = −i3i2, i3i4 = i2 = −i4i3, i4i2 = i3 = −i2i4. In [3, 4], A. Cariow, G. Cariow and J. Knapiski deﬁned hyperbolic octonions. A hyperbolic octonion O is an expression of the form

O = h0 + h1i1 + h2i2 + h3i3 + h4e4 + h5e5 + h6e6 + h7e7

= h0, h1, h2, h3, h4, h5, h6, h7 ,

with real components h0, h1, h2, h3, h4, h5, h6, h7 and i1, i2, i3 are quaternion imaginary units, 2 e4(e4 = 1) is a counter imaginary unit, and the bases of hyperbolic octonions are deﬁned as follows:

2 2 2 2 i1e4 = e5, i2e4 = e6, i3e4 = e7, e4 = e5 = e6 = e7 = 1. The bases of hyperbolic octonion O appeared in [3, 4] have multiplication rules as in Table 1.

· i1 i2 i3 e4 e5 e6 e7

i1 −1 i3 −i2 e5 e4 −e7 e6

i2 −e3 −1 i1 e6 e7 e4 −e5

i3 i2 −i1 −1 e7 −e6 e5 e4

e4 −e5 −e6 −e7 1 i1 i2 i3

e5 −e4 −e7 e6 −i1 1 i3 −i2

e6 e7 −e4 −e5 −i2 −i3 1 i1

e7 −e6 e5 −e4 −i3 i2 −i1 1

Table 1. Rules for multiplication of hyperbolic octonion bases

In [14], we derived some properties of the hyperbolic k-Fibonacci and k-Lucas octonions. In the current paper, our main aim is to prove the well-known identities like Binet’s formulas, Catalan’s identity, Cassini’s identity, d’Ocagne’s identity and generating functions for the k-Fibonacci and k-Lucas hyperbolic octonions.

177 2 Hyperbolic k-Fibonacci and k-Lucas octonions

In this section, some elementary properties of the hyperbolic k-Fibonacci and k-Lucas octonions are obtained.

F L Deﬁnition 2.1. For n ≥ 0, the hyperbolic k-Fibonacci and k-Lucas octonions O k,n and O k,n are deﬁned by

F O k,n = Fk,n + Fk,n+1i1 + Fk,n+2i2 + Fk,n+3i3 + Fk,n+4e4 + Fk,n+5e5 + Fk,n+6e6 + Fk,n+7e7

= Fk,n,Fk,n+1,Fk,n+2,Fk,n+3,Fk,n+4,Fk,n+5,Fk,n+6,Fk,n+7 , and

L O k,n = Lk,n + Lk,n+1i1 + Lk,n+2i2 + Lk,n+3i3 + Lk,n+4e4 + Lk,n+5e5 + Lk,n+6e6 + Lk,n+7e7

= Lk,n,Lk,n+1,Lk,n+2,Lk,n+3,Lk,n+4,Lk,n+5,Lk,n+6,Lk,n+7 ,

respectively, where Fk,n is the n-th k-Fibonacci sequence and Lk,n is the n-th k-Lucas sequence. 2 Here, i1, i2, i3 are quaternion imaginary units, e4(e4 = 1) is a counter imaginary unit, and the F L bases of hyperbolic octonions O k,n and O k,n are deﬁned as i1e4 = e5, i2e4 = e6, 2 2 2 2 F L i3e4 = e7, e4 = e5 = e6 = e7 = 1. The bases of hyperbolic octonions O k,n and O k,n have multiplication rules as in Table 1.

¯F Deﬁnition 2.2. For n ≥ 0, the conjugate of hyperbolic k-Fibonacci and k-Lucas octonions O k,n ¯L and O k,n are deﬁned by

¯F O k,n = Fk,n − Fk,n+1i1 − Fk,n+2i2 − Fk,n+3i3 − Fk,n+4e4 − Fk,n+5e5 − Fk,n+6e6 − Fk,n+7e7

= Fk,n, −Fk,n+1, −Fk,n+2, −Fk,n+3, −Fk,n+4, −Fk,n+5, −Fk,n+6, −Fk,n+7 , and

¯L O k,n = Lk,n − Lk,n+1i1 − Lk,n+2i2 − Lk,n+3i3 − Lk,n+4e4 − Lk,n+5e5 − Lk,n+6e6 − Lk,n+7e7

= Lk,n, −Lk,n+1, −Lk,n+2, −Lk,n+3, −Lk,n+4, −Lk,n+5, −Lk,n+6, −Lk,n+7 ,

F F F (i) O k,n+2 = kO k,n+1 + O k,n, L L L (ii) O k,n+2 = kO k,n+1 + O k,n L F F (iii) O k,n = O k,n+1 + O k,n−1 ¯F ¯F ¯F (iv) O k,n+2 = kO k,n+1 + O k,n, ¯L ¯L ¯L (v) O k,n+2 = kO k,n+1 + O k,n ¯L ¯F ¯F (vi) O k,n = O k,n+1 + O k,n−1.

178 Proof. (i). Using Deﬁnition 2.1, we have

F F kO k,n+1 + O k,n = k Fk,n+1 + Fk,n+2i1 + Fk,n+3i2 + Fk,n+4i3 + Fk,n+5e4 + Fk,n+6e5 + Fk,n+7e6 + Fk,n+8e7 + Fk,n + Fk,n+1i1 + Fk,n+2i2 + Fk,n+3i3 + Fk,n+4e4 + Fk,n+5e5 + Fk,n+6e6 + Fk,n+7e7 = kFk,n+1 + Fk,n + kFk,n+2 + Fk,n+1 i1 + kFk,n+3 + Fk,n+2 i2 + kFk,n+4 + Fk,n+3 i3 + kFk,n+5 + Fk,n+4 e4 + kFk,n+6 + Fk,n+5 e5 + kFk,n+7 + Fk,n+6 e6 + kFk,n+8 + Fk,n+7 e7

= Fk,n+2 + Fk,n+3i1 + Fk,n+4i2 + Fk,n+5i3 + Fk,n+6e4

+ Fk,n+7e5 + Fk,n+8e6 + Fk,n+9e7 F = O k,n+2.

The proofs of (ii), (iii), (iv), (v) and (vi) are similar to (i), using Deﬁnition 2.1.

Theorem 2.4 (Binet Formulas). For all n ≥ 0, we have

n n F r¯1r1 − r¯2r2 (i) O k,n = r1 − r2 L n n (ii) O k,n =r ¯1r1 +r ¯2r2 , r¯ r n − r¯ r n ¯F 3 1 4 2 (iii) O k,n = r1 − r2 ¯L n n (iv) O k,n =r ¯3r1 +r ¯4r2 ,

where 2 3 4 5 6 7 2 3 4 5 6 7 r¯1 = 1 + r1i1 + r1 i2 + r1 i3 + r1 e4 + r1 e5 + r1 e6 + r1 e7 = 1, r1, r1 , r1 , r1 , r1 , r1 , r1 , 2 3 4 5 6 7 2 3 4 5 6 7 r¯2 = 1 + r2i1 + r2 i2 + r2 i3 + r2 e4 + r2 e5 + r2 e6 + r2 e7 = 1, r2, r2 , r2 , r2 , r2 , r2 , r2 , 2 3 4 5 6 7 r¯3 = 1 − r1i1 − r1 i2 − r1 i3 − r1 e4 − r1 e5 − r1 e6 − r1 e7 2 3 4 5 6 7 = 1, −r1, −r1 , −r1 , −r1 , −r1 , −r1 , −r1 , 2 3 4 5 6 7 r¯4 = 1 + r2i1 − r2 i2 − r2 i3 − r2 e4 − r2 e5 − r2 e6 − r2 e7 2 3 4 5 6 7 = 1, −r2, −r2 , −r2 , −r2 , −r2 , −r2 , −r2 .

2 Here, i1, i2, i3 are quaternion imaginary units, e4(e4 = 1) is a counter imaginary unit, and the F L bases of hyperbolic octonions O k,n and O k,n are deﬁned as i1e4 = e5, i2e4 = e6, 2 2 2 2 F L i3e4 = e7, e4 = e5 = e6 = e7 = 1. The bases of hyperbolic octonions O k,n and O k,n have multiplication rules as in Table 1.

Proof. (i). Using Deﬁnition 2.1 and the Binet formulas of k-Fibonacci and k-Lucas sequences, we have

F O k,n = Fk,n + Fk,n+1i1 + Fk,n+2i2 + Fk,n+3i3 + Fk,n+4e4 + Fk,n+5e5 + Fk,n+6e6 + Fk,n+7e7

179 n n n+1 n+1 n+2 n+2 n+3 n+3 r1 − r2 r1 − r2 r1 − r2 r1 − r2 = + i1 + i2 + i3 r1 − r2 r1 − r2 r1 − r2 r1 − r2 n+4 n+4 n+5 n+5 n+6 n+6 n+7 n+7 r1 − r2 r1 − r2 r1 − r2 r1 − r2 + e4 + e5 + e6 + e7 r1 − r2 r1 − r2 r1 − r2 r1 − r2 n r1 2 3 4 5 6 7 = 1 + r1i1 + r1 i2 + r1 i3 + r1 e4 + r1 e5 + r1 e6 + r1 e7 r1 − r2 n r2 2 3 4 5 6 7 − 1 + r2i1 + r2 i2 + r2 i3 + r2 e4 + r2 e5 + r2 e6 + r2 e7 r1 − r2 r¯ r n − r¯ r n = 1 1 2 2 r1 − r2 (ii). We have,

L O k,n = Lk,n + Lk,n+1i1 + Lk,n+2i2 + Lk,n+3i3 + Lk,n+4e4 + Lk,n+5e5 + Lk,n+6e6 + Lk,n+7e7 n n n+1 n+1 n+2 n+2 n+3 n+3 = r1 + r2 + r1 + r2 i1 + r1 + r2 i2 + r1 r2 i3 n+4 n+4 n+5 n+5 n+6 n+6 n+7 n+7 + r1 r2 e4 + r1 r2 e5 + r1 r2 e6 + r1 r2 e7 n 2 3 4 5 6 7 = r1 1 + r1i1 + r1 i2 + r1 i3 + r1 e4 + r1 e5 + r1 e6 + r1 e7 n 2 3 4 5 6 7 + r2 1 + r2i1 + r2 i2 + r2 i3 + r2 e4 + r2 e5 + r2 e6 + r2 e7 n n =r ¯1r1 +r ¯2r2 .

The proofs of (iii) and (iv) are similar to these of (i) and (ii) using Deﬁnition 2.1.

Some interesting properties of r¯1, r¯2, r¯3 and r¯4 are listed in [14]. Some of these are listed in Lemma 2.5.

2 3 4 5 6 7 2 3 4 Lemma 2.5. For r¯1 = 1, r1, r1 , r1 , r1 , r1 , r1 , r1 , r¯2 = 1, r2, r2 , r2 , r2 , 5 6 7 2 3 4 5 6 7 r2 , r2 , r2 , r¯3 = 1, −r1, −r1 , −r1 , −r1 , −r1 , −r1 , −r1 and r¯4 = 1, −r2, 2 3 4 5 6 7 − r2 , −r2 , −r2 , −r2 , −r2 , −r2 , we have √ F (1)r ¯1 − r¯2 = δO k,0, L (2)r ¯1 +r ¯2 = O k,0, 2 2 3 4 (3)r ¯1r¯2 = 2, −2(r1 − 2r2), −2(r1 − 2r2 ), 2(r1 − r2 + r2 ), 2r1 , 3 3 5 2 2 6 7 7 4 4 2(r1 − r2 ) + 2r1 , −2(r1 − r2 + r1 ), r1 + r2 − (r1 − r2)(r1 + r2 − 1) =u ¯1, 2 2 3 4 (4)r ¯2r¯1 = 2, −2(r2 − 2r1), −2(r2 − 2r1 ), 2(r2 − r1 + r1 ), 2r2 , 3 3 5 2 2 6 7 7 4 4 2(r2 − r1 ) + 2r2 , −2(r2 − r1 + r2 ), r2 + r1 − (r2 − r1)(r1 + r2 − 1) =u ¯2, 2 2 4 6 8 10 12 14 (5)r ¯1 = − 1 − r1 − r1 − r1 + r1 + r1 + r1 + r1 + 2r ¯1 =u ¯3, 2 2 4 6 8 10 12 14 (6)r ¯2 = − 1 − r2 − r2 − r2 + r2 + r2 + r2 + r2 + 2r ¯2 =u ¯4, L (7)r ¯1r¯2 +r ¯2r¯1 = 2O k,0, √ 2 2 (8)r ¯1r¯2 − r¯2r¯1 = 2 δ 0, −3, −3k, (1 − k ), k(k + 2), 4 2 5 3 4 k + 5k + 3, k + 4k + k, −(k + 4k + 1) =u ¯5, √ 2 2 13 11 9 7 5 3 F (9)r ¯1 − r¯2 = δ k + 13k + 66k + 165k + 208k + 116k + 16k + 2O k,0 =u ¯6, 2 2 14 12 10 8 6 4 2 L (10)r ¯1 +r ¯2 = k + 15k + 90k + 275k + 448k + 364k + 112k + 2O k,0 =u ¯7,

180 (11)r ¯3 = 2 − r¯1 and r¯4 = 2 − r¯2, 2 4 6 8 10 12 14 (12)r ¯1r¯3 =r ¯3r¯1 = 1 + r1 + r1 + r1 − r1 − r1 − r1 − r1 = u8, 2 4 6 8 10 12 14 (13)r ¯2r¯4 =r ¯4r¯2 = 1 + r2 + r2 + r2 − r2 − r2 − r2 − r2 = u9, 2 2 3 3 (14)r ¯1r¯4 = 0, 4(r1 − r2), 4(r1 − r2 ), −2(r1 − r2 − r1 − r2 ), 0, 3 3 2 2 6 7 7 4 4 − 2(r1 − r2 ), 2(r1 − r2 + 2r1 ), r1 − r2 + (r1 − r2)(r1 + r2 − 1) =u ¯10, 2 2 4 4 3 3 (15)r ¯4r¯1 = 0, −2(r1 − r2), −2(r1 − r2 ), 2(r1 − r2), −2(r1 − r2 ), 2(r1 − r2 5 5 2 2 6 6 7 7 4 4 − r2 + r1 ), −2(r1 − r2 − r1 − r2 ), r1 − r2 − (r1 − r2)(r1 + r2 − 1) =u ¯11, 2 2 3 3 4 4 (16)r ¯2r¯3 = 0, −2(r1 − r2), −2(r1 − r2 ), 2(r1 − r2 − r1 + r2 ), −2(r1 − r2 ), 3 3 2 2 6 7 7 4 4 2(r1 − r2 ), −2(r1 − r2 − 2r2 ), −r1 + r2 − (r1 − r2)(r1 + r2 − 1) =u ¯12, 2 2 4 4 3 3 (17)r ¯3r¯2 = 0, 2(r1 − r2), 2(r1 − r2 ), −2(r1 − r2), −2(r1 − r2 ), −2(r1 − r2 5 5 2 2 6 6 7 7 4 4 + r1 − r2 ), 2(r1 − r2 + r1 + r2 ), −r1 + r2 + (r1 − r2)(r1 + r2 − 1) =u ¯13, L (18)r ¯3r¯4 = 4 − 2O k,0 +u ¯1, L (19)r ¯4r¯3 = 4 − 2O k,0 +u ¯2,

(20)r ¯3r¯4 − r¯4r¯3 =u ¯1 − u¯2, 2 2 3 3 4 4 (21)r ¯3r¯2 = 0, −2(r1 − r2), −2(r1 − r2 ), −2(r1 − r2 ), −4(r1 − r2 ), 5 5 6 6 7 7 − 2(r1 − r2 ), 2(r1 + 3r2 ), −2(r1 − r2 ) =u ¯14, 2 2 3 3 (22)r ¯3r¯2 − r¯2r¯3 = 0, 6(r1 − r2), 6(r1 − r2 ), −2(2r1 − 2r2 − r1 + r2 ), 0, 3 3 5 5 2 2 6 6 4 4 − 2(2r1 − 2r2 + r1 − r2 ), 2(2r1 − 2r2 + r1 − r2 ), 2(r1 − r2)(r1 + r2 − 1)

=u ¯15, 2 2 3 3 4 4 (23)r ¯1r¯4 − r¯4r¯1 = 0, 6(r1 − r2), 6(r1 − r2 ), −2(2r1 − 2r2 − r1 − r2 ), 2(r2 − r1 ), 3 3 5 5 2 2 6 6 4 4 − 2(2r1 − 2r2 + r1 − r2 ), 2(2r1 − 2r2 + r1 − r2 ), 2(r1 − r2)(r1 + r2 − 1)

=u ¯16, 2 (24)r ¯3 = 4r ¯3 +u ¯3 − 4 =u ¯17, 2 (25)r ¯4 = 4r ¯4 +u ¯4 − 4 =u ¯18.

Theorem 2.6. For all s, t ∈ Z+,s ≥ t and n ∈ N, the generating functions for the hyperbolic F L k-Fibonacci and k-Lucas quaternions O k,tn and O k,tn are

∞ F L F X O k,0 + O k,0Fk,t − O k,t x (i) OF xn = , k,tn 1 − xL + x2(−1)t n=0 k,t ∞ L L L X O k,0 − O k,0Lk,t − O k,t x (ii) OL xn = , k,tn 1 − xL + x2(−1)t n=0 k,t ∞ F t F X O k,s + (−1) xO s,s−t (iii) OF xn = , k,tn+s 1 − xL + x2(−1)t n=0 k,t ∞ L t L X O k,s + (−1) xO s−t (iv) OL xn = , k,tn+s 1 − xL + x2(−1)t n=0 k,t and the exponential generating functions for the hyperbolic k-Fibonacci and k-Lucas quaternions

181 F L O k,tn and O k,tn are

∞ t t F r1 x r2 x X O k,tn r¯1e − r¯2e (v) xn = , n! r − r n=0 1 2 ∞ L X O k,tn t t (vi) xn =r ¯ er1 x +r ¯ er2 x. n! 1 2 n=0 Proof. (i). Using Theorem 2.4, we obtain

∞ ∞ tn tn X X r¯1r1 − r¯2r2 OF xn = xn k,tn r − r n=0 n=0 1 2

∞ ∞ r¯1 X n r¯2 X n = r t xn − r t xn r − r 1 r − r 2 1 2 n=0 1 2 n=0

r¯1 1 r¯2 1 = t − t r1 − r2 1 − r1 x r1 − r2 1 − r2 x

1 r¯ − r¯ + r¯ r t − r¯ r tx = 1 2 2 1 1 2 t t 2 t r1 − r2 1 − r1 + r2 x + x (r1r2)

1 r¯ − r¯ + r¯ r t − r¯ r t +r ¯ r t − r¯ r t +r ¯ r t − r¯ r tx = 1 2 2 1 2 2 2 2 1 1 1 1 1 2 t t 2 t r1 − r2 1 − r1 + r2 x + x (r1r2)

r¯ − r¯ r t − r t r¯ r t − r¯ r t 1 2 + r¯ +r ¯ 1 2 − 1 1 2 2 x r − r 1 2 r − r r − r = 1 2 1 2 1 2 . t t 2 t 1 − r1 + r2 x + x (r1r2)

Using Theorem 2.4 and Lemma 2.5, we obtain

F L F O k,0 + O k,0Fk,t − O k,t x = 2 t . 1 − xLk,t + x (−1) The proofs of (ii), (iii), (iv), (v) and (vi) are similar to (i), using Theorem 2.4.

Theorem 2.7. For all n ∈ N, we have

n n X i F F (i) i k O k,i = O k,2n, i=0 n n X i L L (ii) i k O k,i = O k,2n. i=0 Proof. (i). Using Theorem 2.4, we obtain

n n i i X n X n r¯1r1 − r¯2r2 i kiOF = i ki k,i r − r i=0 i=0 1 2

182 n n r¯1 X n r¯2 X n = i (kr )i − i (kr )i r − r 1 r − r 2 1 2 i=0 1 2 i=0

r¯1 n r¯2 n = 1 + kr1 − 1 + kr2 r1 − r2 r1 − r2 r¯ r 2n − r¯ r 2n = 1 1 2 2 r1 − r2

F = O k,2n.

The proof of (ii) is similar to (i), using Theorem 2.4.

Lemma 2.8. For all t ≥ 0, we have t t u¯1r2 − u¯2r1 ¯ (i) = Uk,t, r1 − r2 where ¯ Uk,t = − 2Fk,t, −2Fk,t−1 − 4Fk,t+1, 2Fk,t−2 − 4Fk,t+2, 2Fk,t−2 + 2Fk,t+1 − 2Fk,t+3,

−2Fk,t−4, 2Fk,t−3 + 2Fk,t+3 + 2Fk,t−5, 2Fk,t−2 − 2Fk,t+2 + 2Fk,t−6,

Fk,t−7 − Fk,t+7 − Lk,t+4 − Lk,t−4 + Lk,t ,

m−n m−n r1 u¯1 − r2 u¯2 ¯ (ii) = Vk,m−n, r1 − r2 where ¯ Vk,m−n = 2Fk,m−n, −2Fk,m−n+1 − 4Fk,m−n−1, −2Fk,m−n+2 + 4Fk,m−n−2,

2Fk,m−n+1 + 2Fk,m−n−1 − 2Fk,m−n−3, 2Fk,m−n+4,

2Fk,m−n+3 + 2Fk,m−n−3 + Fk,m−n+5, −2Fk,m−n+2 + 2Fk,m−n−2 − 2Fk,m−n+6,

Fk,m−n+7 − Fk,m−n−7 − Lk,m−n+4 − Lk,m−n−4 + Lk,m−n ,

t t ¯ (iii) u¯3r1 +u ¯4r2 = Wk,t, where ¯ L Wk,t = − Lk,t − Lk,t+2 − Lk,t+4 − Lk,t+6 + Lk,t+8 + Lk,t+10 + Lk,t+12 + Lk,t+14 + 2O k,t,

t t u¯3r1 − u¯4r2 ¯ (iv) = Xk,t, r1 − r2 where ¯ F Xk,t = − Fk,t − Fk,t+2 − Fk,t+4 − Fk,t+6 + Fk,t+8 + Fk,t+10 + Fk,t+12 + Fk,t+14 + 2O k,t,

t t u¯8r1 − u¯9r2 (v) = Lk,t+1 + Lk,t+3 − Lk,t+9 − Lk,t+13, r1 − r2

t t u¯17r1 − u¯18r2 ¯ (vi) = Yk,t, r1 − r2 where ¯ F Yk,t = −Fk,t −Fk,t+2 −Fk,t+4 −Fk,t+6 +Fk,t+8 +Fk,t+10 +Fk,t+12 +Fk,t+14 +4−2O k,t.

183 Theorem 2.9 (Catalan’s Identity). For any integer t and s, we have

F F F 2 n−t ¯ (i) O k,n−tO k,n+t − O k,n = (−1) Fk,tUk,t, L L L 2 n−t+1 ¯ (ii) O k,n−tO k,n+t − O k,n = δ(−1) Fk,tVk,m−n.

Proof. Using Theorem 2.4, we have

n−t n−t n+t n+t n n F F F 2 r¯1r1 − r¯2r2 r¯1r1 − r¯2r2 r¯1r1 − r¯2r2 2 O k,n−tO k,n+t − O k,n = − r1 − r2 r1 − r2 r1 − r2 1 2 2n n−t n+t n−t n+t 2 2n = 2 r¯1 r1 − r¯1r¯2r1 r2 − r¯2r¯1r2 r1 +r ¯2 r2 (r1 − r2) 2 2n n n 2 2n − r¯1 r1 +r ¯1r¯2(r1r2) +r ¯2r¯1(r1r2) − r¯2 r2 n (r1r2) t −t t −t t −t t −t = 2 r¯1r¯2r1 r1 − r¯2r¯1r1 r2 − r¯1r¯2r2 r2 +r ¯2r¯1r2 r2 (r1 − r2) n (r1r2) t −t −t t −t −t = 2 r1 [(r ¯1r¯2)r1 − (r ¯2r¯1)r2 ] − r2 [(r ¯1r¯2)r1 − (r ¯2r¯1)r2 ] (r1 − r2) t t −t −t nr1 − r2 (r ¯1r¯2)r1 − (r ¯2r¯1)r2 = (r1r2) r1 − r2 r1 − r2 t t t t n−tr1 − r2 (r ¯1r¯2)r1 − (r ¯2r¯1)r2 = (r1r2) . r1 − r2 r1 − r2

Using Lemma 2.8 and r1r2 = −1, we obtain n−t ¯ = (−1) Fk,tUk,t. The proof of (ii) is similar to (i), using Theorem 2.4 and Lemma 2.8.

Theorem 2.10 (Cassini’s Identity). For all n ≥ 1, we have

F F F 2 n−1 ¯ (i) O k,n−1O k,n+1 − O k,n = (−1) Fk,tUk,1, L L L 2 n ¯ (ii) O k,n−1O k,n+1 − O k,n = δ(−1) Fk,tVk,1.

Theorem 2.11 (d’Ocagne’s Identity). Let n be any non-negative integer and t a natural number. If t ≥ n + 1, then we have

F F F F n ¯ (i) O k,tO k,n+1 − O k,t+1O k,n = (−1) Vk,t−n, L L L L n+1 ¯ (ii) O k,tO k,n+1 − O k,t+1O k,n = (−1) δVk,t−n.

Proof. Using Theorem 2.4, we get

t t n+1 n+1 F F F F r¯1r1 − r¯2r2 r¯1r1 − r¯2r2 O k,tO k,n+1 − O k,t+1O k,n = r1 − r2 r1 − r2 r¯ r t+1 − r¯ r t+1 r¯ r n − r¯ r n − 1 1 2 2 1 1 2 2 r1 − r2 r1 − r2 1 2 n+t+1 t n+1 t n+1 2 n+t+1 = 2 r¯1 r1 − r¯1r¯2r1 r2 − r¯2r¯1r2 r1 +r ¯2 r2 (r1 − r2) 2 n+t+1 t+1 n n t+1 2 n+t+1 − r¯1 r1 +r ¯1r¯2(r1 r2 ) +r ¯2r¯1(r1 r2 ) − r¯2 r2

184 n (r1r2) t−n t−n = 2 r¯1r¯2r1 (r1 − r2) − r¯2r¯1r2 (r1 − r2) (r1 − r2) t−n t−n nr¯1r¯2r1 − r¯2r¯1r2 = (r1r2) . r1 − r2

Using Lemma 2.8 and r1r2 = −1, we obtain

n ¯ = (−1) Vk,t−n.

The proof of (ii) is similar to (i), using Theorem 2.4 and Lemma 2.8.

Theorem 2.12. For any integer t, we have

2 2 1 (i) OF + OL = (1 + δ)W¯ + (δ − 1)(−1)tOL , k,t k,t δ k,2t k,0 2 2 1 (ii) OF − OL = (1 − δ)W¯ − (1 + δ)(−1)tOL . k,t k,t δ k,2t k,0 Proof. Using Theorem 2.4, we get

t t F 2 L 2 r¯1r1 − r¯2r2 2 t t2 O k,t + O k,t+1 = + r¯1r1 +r ¯2r2 r1 − r2 1 2 2t 2 2t t t = 2 (r ¯1) r1 + (r ¯2) r2 − r¯1r¯2r1r2 − r¯2r¯1r1r2 (r1 − r2) 2 2t 2 2t t t + (r ¯1) r1 + (r ¯2) r2 +r ¯1r¯2r1r2 +r ¯2r¯1r1r2 (1 + δ) (δ − 1)(−1)t = (r ¯ )2r 2t +r ¯ )2r 2t + r¯ r¯ +r ¯ r¯ . δ 1 1 2 2 δ 1 2 2 1 Using Lemma 2.5, we obtain 1 = (1 + δ)W¯ + (δ − 1)(−1)tOL . δ k,2t k,0 The proof of (ii) is similar to (i), using Theorem 2.4 and Lemma 2.5.

Theorem 2.13. For any integer r, s ≥ t, we have

F L F L r+t L O k,r+sO k,r+t − O k,r+tO k,r+s = 2(−1) O k,0 − 2 Fk,s−t.

Proof. Using Theorem 2.4 and r1r2 = −1, we get

F L F L 1 r+s r+s O k,r+sO k,r+t − O k,r+tO k,r+s = r¯1r1 − r¯2r2 · r1 − r2 r+t r+t r+t r+t r+s r+s r¯1r1 +r ¯2r2 − r¯1r1 − r¯2r2 r¯1r1 +r ¯2r2 r (r1r2) s t t s = (r ¯1r¯2 +r ¯2r¯1)r1 r2 − (r ¯1r¯2 +r ¯2r¯1)r1 r2 r1 − r2 r (r1r2) s t t s = r¯1r¯2 +r ¯2r¯1 r1 r2 − r1 r2 . r1 − r2 Using Lemma 2.5, we obtain r+t L = 2(−1) O k,0 − 2 Fk,s−t.

185 Theorem 2.14. For any integer s, and t, we have

F t F F (i) O k,s+t + (−1) O k,s−t = O k,sLk,t, L t L L (ii) O k,s+t + (−1) O k,s−t = O k,sLk,t. Proof. Using Theorem 2.4, we get

F t F 1 s+t s+t t s−t s−t O k,s+t + (−1) O k,s−t = r¯1r1 − r¯2r2 + (−1) r¯1r1 +r ¯2r2 r1 − r2 1 s+t s+t s t t s = r¯1r1 − r¯2r2 +r ¯1r1 r2 − r¯2r1 r2 r1 − r2 1 s t t s t t = r¯1r1 (r1 + r2 ) − r¯2r2 (r1 + r2 ) r1 − r2 s s r¯1r1 − r¯2r2 t t = ( r1 + r2 . r1 − r2 Using Theorem 2.4, we obtain

F = O k,sLk,t. The proof of (ii) is similar to (i), using Theorem 2.4. Theorem 2.15. For any integer s ≤ t, we have

F F F F s − 1 (i) O k,sO k,t − O k,tO k,s = (−1) δ 2 u¯5Fk,t−s, L L L L t 1 (ii) O k,sO k,t − O k,tO k,s = (−1) δ 2 u¯5Fk,s−t, F L F L s L (iii) O k,tO k,s − O k,sO k,t = 2(−1) Fk,t−sO k,0, F L L F s ¯ (iv) O k,tO k,s − O k,tO k,s = 2(−1) Vk,t−s. Proof. (i). Using Theorem 2.4, we have s s t t t t s s L L L L r¯1r1 − r¯2r2 r¯1r1 − r¯2r2 r¯1r1 − r¯2r2 r¯1r1 − r¯2r2 O k,sO k,t − O k,tO k,s = − r1 − r2 r1 − r2 r1 − r2 r1 − r2 1 t s s t = 2 r1 r2 − r1 r2 r¯1r¯2 − r¯2r¯1 (r1 − r2) t−s t−s s −1 r1 − r2 = (r1r2) (r1 − r2) r¯1r¯2 − r¯2r¯1 r1 − r2 Using Lemma 2.5, we obtain s − 1 = (−1) δ 2 u¯5Fk,t−s. The proof of (ii), (iii) and (iv) is similar to (i), using Theorem 2.4 and Lemma 2.5. Theorem 2.16. For any integer n ≥ 0, we have

1 F ¯L ¯F F n − (i) O k,nO k,n − O k,nO k,n = (−1) δ 2 (2u ¯10 − u¯14), 1 F ¯L ¯F F − (ii) O k,nO k,n + O k,nO k,n = 2δ 2 (Lk,t+1 + Lk,t+3 − Lk,t+9 − Lk,t+13) n − 1 + (−1) δ 2 (2u ¯15 − u¯16), 1 F L ¯F ¯F ¯ − n ¯ (iii) O k,nO k,n − O k,nO k,n = Xk,2n + δ 2 (−1) u¯5 − u¯1 +u ¯2 − Yk,2n.

186 Proof. (i). Using Theorem 2.4, we have r¯ r n − r¯ r n r¯ r n − r¯ r n F ¯L ¯F F 1 1 2 2 n n 3 1 4 2 n n O k,nO k,n − O k,nO k,n = r¯3r1 +r ¯4r2 − r¯1r1 +r ¯2r2 r1 − r2 r1 − r2 n (r1r2) = 2r ¯1r¯4 − r¯2r¯3 − r¯3r¯2 (r1 − r2) Using Lemma 2.5, we obtain n − 1 = (−1) δ 2 (2u ¯10 − u¯14).

The proof of (ii) and (iii) is similar to (i), using Theorem 2.4 and Lemma 2.5.

3 Conclusion

This study introduce hyperbolic k-Fibonacci and k-Lucas octonions. Moreover, we obtain Binet‘s formulas, Catalan‘s identity, Cassini‘s identity and d‘Ocagne‘s identity for hyperbolic k-Fibonacci and k-Lucas octonions.

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