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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] in- troduced the set of quaternions which form a 4-dimensional real vector space with a multiplica- tive 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 ; 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. Theorem 2.3. For all n ≥ 0, we have 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.
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