Indian J. Pure Appl. Math., 50(4): 979-998, December 2019 °c Indian National Science Academy DOI: 10.1007/s13226-019-0368-x PROPERTIES OF k-FIBONACCI AND k-LUCAS OCTONIONS A. D. Godase Department of Mathematics, V. P. College Vaijapur, Aurangabad 423 701, (MH), India e-mail: [email protected] (Received 9 April 2018; after final revision 18 July 2018; accepted 18 October 2018) We investigate some binomial and congruence properties for the k-Fibonacci and k-Lucas hyper- bolic octonions. In addition, we present several well-known identities such as Catalan’s, Cassini’s and d’Ocagne’s identities for k-Fibonacci and k-Lucas hyperbolic octonions. Key words : Fibonacci sequence; k-Fibonacci sequence; k-Lucas sequence. 1. INTRODUCTION The Fibonacci and Lucas sequences are generalised by changing the initial conditions or changing the recurrence relation. The k-Fibonacci sequence is the generalization of the Fibonacci sequence, which is first introduced by Falcon and Plaza [2]. The k-Fibonacci sequence is defined by the numbers which satisfy the second order recurrence relation Fk;n = kFk;n¡1+Fk;n¡2 with the initial conditions Fk;0 = 0 and Fk;1 = 1. Falcon [3] defined the k-Lucas sequence that is companion sequence of k- Fibonacci sequence defined with the k-Lucas numbers which are defined with the recurrence relation Lk;n = kLk;n¡1 + Lk;n¡2 with the initial conditions Lk;0 = 2 and Lk;1 = k. Binet’s formulas for the k-Fibonacci and k-Lucas numbers are n n r1 ¡ r2 Fk;n = r1 ¡ r2 and n n Lk;n = r1 + r2 p p k+ k2+4 k¡ k2+4 respectively, where r1 = 2 and r2 = 2 are the roots of the characteristic equation 2 x ¡ kx ¡ 1 = 0. The characteristic roots r1 and r2 satisfy the properties p p 2 r1 ¡ r2 = k + 4 = ±, r1 + r2 = k, r1r2 = ¡1: 980 A. D. GODASE The reader can refer to [1, 4-9] for properties and applications of k- Fibonacci and k- Lucas numbers. The quaternions are generalized numbers. The quaternions first introduced by Irish mathemati- cian William Rowan Hamilton in 1843. Hamilton [10] introduced the set of quaternions form a 4-dimensional real vector space with a multiplicative operation. The quaternions are used in applied sciences such as physics, computer science and Clifford algebras in mathematics. In particular, they are important in mechanics [11], chemistry [12], kinematics [13], quantum mechanics [14], differ- ential geometry and pure algebra. A quaternion a, with real components a0, a1, a2, a3 and basis 1; i; j; k, is an element of the form ­ ® a = a0 + a1i + a2j + a3k = a0; a1; a2; a3 ; where i2 = j2 = k2 = ijk = ¡1; ij = k = ¡ji; jk = i = ¡kj; ki = j = ik: Horadam [15] defined the nth Fibonacci and nth Lucas quaternions as ­ ® F¯n = Fn + Fn+1i + Fn+2j + Fn+3k = Fn;Fn+1;Fn+2;Fn+3 and ­ ® L¯n = Ln + Ln+1i + Ln+2j + Ln+3k = Ln;Ln+1;Ln+2;Ln+3 respectively. Ramirez [16] has defined and studied the k-Fibonacci and k-Lucas quaternions as ­ ® Fk;n¯ = Fk;n + Fk;n+1i + Fk;n+2j + Fk;n+3k = Fk;n;Fk;n+1;Fk;n+2;Fk;n+3 and ­ ® L¯k;n = Lk;n + Lk;n+1i + Lk;n+2j + Lk;n+3k = Lk;n;Lk;n+1;Lk;n+2;Lk;n+3 th th respectively, where Fk;n is the n k-Fibonacci sequence and Lk;n is the n k-Lucas sequence. Different quaternions of sequences have been studied by different researchers. For example, Iyer [17, 18] obtained various relations containing the Fibonacci and Lucas quaternions. Halici [19] studied some combinatorial properties of Fibonacci quaternions. Akyigit et al. [20, 21] established PROPERTIES OF k-FIBONACCI AND k-LUCAS OCTONIONS 981 and investigated the Fibonacci generalized quaternions and split Fibonacci quaternions. Catarino [22] obtained different properties of the h(x)-Fibonacci quaternion polynomials. Polatli and Kesim [23] have introduced quaternions with generalized Fibonacci and Lucas number components. Hyperbolic k-Fibonacci and k-Lucas Quaternions ­ ® A hyperbolic quaternion h is an expression of the form h = h1i1+h2i2+h3i3+h4i4 = h1; h2; h3; h4 ; with real components h1, h2, h3, h4 and i1; i2; i3; i4 are hyperbolic quaternionic units that satisfy the non-commutative multiplication rules 2 2 2 i2 = i3 = i4 = i2i3i4 = +1; i1 = 1; i2i3 = i4 = ¡i3i2; i3i4 = i2 = ¡i4i3; i4i2 = i3 = ¡i2i4: ¡! The scalar and the vector part of a hyperbolic quaternion h are denoted by Sh = h1 and V h = ¡! h2i2 + h3i3 + h4i4, respectively. Thus, a hyperbolic quaternion h is given by h = Sh + V h. For any (1) (1) (1) (1) (1) (2) (2) (2) two hyperbolic quaternions h = h1 i1 + h2 i2 + h3 i3 + h4 i4 and h = h1 i1 + h2 i2 + (2) (2) h3 i3 + h4 i4. Addition and subtraction of the hyperbolic quaternions is defined by (1) (2) ¡ (1) (1) (1) (1) ¢ h § h = h1 i1 + h2 i2 + h3 i3 + h4 i4 ¡ (2) (2) (2) (2) ¢ § h1 i1 + h2 i2 + h3 i3 + h4 i4 ¡ (1) (2)¢ ¡ (1) (2)¢ ¡ (1) (2)¢ ¡ (1) (2)¢ ¢ = h1 § h1 i1 + h2 § h2 i2 + h3 § h3 i3 + h4 § h4 i4 Multiplication of the hyperbolic quaternions is defined by (1) (2) ¡ (1) (1) (1) (1) ¢ h ¢ h = h1 i1 + h2 i2 + h3 i3 + h4 i4 ¡ (2) (2) (2) (2) ¢ ¢ h1 i1 + h2 i2 + h3 i3 + h4 i4 ¡ (1) (2) (1) (2) (1) (2) (1) (2)¢ = h1 h1 + h2 h2 + h3 h3 + h4 h4 ¡ (1) (2) (1) (2) (1) (2) (1) (2)¢ + h1 h2 + h2 h1 + h3 h4 ¡ h4 h3 i2 ¡ (1) (2) (1) (2) (1) (2) (1) (2)¢ + h1 h3 ¡ h2 h4 + h3 h1 + h4 h2 i3 ¡ (1) (2) (1) (2) (1) (2) (1) (2)¢ + h1 h4 + h2 h3 ¡ h3 h2 + h4 h1 i4: The conjugate of hyperbolic quaternion h is denoted by h¯ and it is ­ ® h¯ = h1i1 ¡ h2i2 ¡ h3i3 ¡ h4i4 = h1; ¡h2; ¡h3; ¡h4 : The norm of h is defined as ¯ 2 2 2 2 Nh = h ¢ h = h1 ¡ h2 ¡ h3 ¡ h4 : 982 A. D. GODASE F L In [24], 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 ; th th respectively, where Fk;n is n k-Fibonacci sequence and Lk;n is n k- Lucas sequence. Here, i1, i2, i3, i4 are hyperbolic quaternionic 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: The Binet formulas for the hyperbolic k-Fibonacci and k-Lucas quaternions are n n F r¯1r1 ¡ r¯2r2 Q k;n = ; r1 ¡ r2 and L n n Q k;n =r ¯1r1 +r ¯2r2 where, 2 3 ­ 2 3® r¯1 = i1 + r1i2 + r1 i3 + r1 i4 = 1; r1; r1 ; r1 ; 2 3 ­ 2 3® r¯2 = i1 + r2i2 + r2 i3 + r2 i4 = 1; r2; r2 ; r2 ; and i1, i2, i3, i4 are hyperbolic quaternionic units. Different properties of the hyperbolic k-Fibonacci and k-Lucas quaternions are investigated in [24], some of these are F F F (1) Q k;n+2 = kQ k;n+1 + Q k;n; L L L (2) Q k;n+2 = kQ k;n+1 + Q k;n; L F L (3) Q k;n = Q k;n+1 + Q k;n¡1; ¡ ¢ X1 F L F F n Q k;0 + Q k;0Fk;t ¡Q k;t x (4) Q k;tnx = 2 t ; 1 ¡ xLk;t + x (¡1) n=0 ¡ ¢ X1 QL ¡ QL L ¡QL x (5) QL xn = k;0 k;0 k;t k;t ; k;tn 1 ¡ xL + x2(¡1)t n=0 k;t PROPERTIES OF k-FIBONACCI AND k-LUCAS OCTONIONS 983 X1 QF + (¡1)txQF (6) QF xn = k;s s;s¡t ; k;tn+s 1 ¡ xL + x2(¡1)t n=0 k;t X1 QL + (¡1)txQL (7) QL xn = k;s s¡t ; k;tn+s 1 ¡ xL + x2(¡1)t n=0 k;t 1 t t X QF r¯ er1 x ¡ r¯ er2 x (8) k;tn xn = 1 2 ; n! r ¡ r n=0 1 2 X1 L Q k;tn t t (9) xn =r ¯ er1 x +r ¯ er2 x; n! 1 2 n=0 µ ¶ Xn n (10) kiQF = QF ; i k;i k;2n i=0 µ ¶ Xn n (11) kiQL = QL ; i k;i k;2n i=0 F F F 2 n¡t ¡ (12) Q k;n¡tQ k;n+t ¡Q k;n = (¡1) Fk;t 0; ¡2Fk;t+1; ¡2Fk;t+2; ¢ ¡ 2Fk;t+3 + Fk;t¡3 + Fk;t+1 + Fk;t¡1 ; L L L 2 n¡t+1 ¡ (13) Q k;n¡tQ k;n+t ¡Q k;n = ±(¡1) Fk;t 0; ¡2Fk;t+1; ¡2Fk;t+2; ¢ ¡ 2Fk;t+3 + Fk;t¡3 + Fk;t+1 + Fk;t¡1 ; F F F 2 n¡ ¢ (14) Q k;n¡1Q k;n+1 ¡Q k;n = (¡1) 0; ¡2Fk;2; 2Fk;3;Fk;4 ; L L L 2 n¡1¡ ¢ (15) Q k;n¡1Q k;n+1 ¡Q k;n = ±(¡1) 0; ¡2Fk;2; 2Fk;3;Fk;4 ; F F F F n¡ (16) Q k;tQ k;n+1 ¡Q k;t+1Q k;n = (¡1) 0; ¡2Fk;t¡n¡1; 2Fk;t¡n¡2; ¢ Fk;t¡n+3 + Fk;t¡n¡3 + Fk;t¡n+1 + Fk;t¡n¡1 ; L L L L n+1 ¡ (17) Q k;tQ k;n+1 ¡Q k;t+1Q k;n = (¡1) ± 0; ¡2Fk;t¡n¡1; 2Fk;t¡n¡2; ¢ Fk;t¡n+3 + Fk;t¡n¡3 + Fk;t¡n+1 + Fk;t¡n¡1 ; 2 2 2 2(k + 5) (18) QF + QL = QL + ±(k2 + 5)L k;t k;t k k;2t k;2t+3 (k2 + 3)¡ ¢ + 2(¡1)t QL ¡ 2 ; ± k;0 2 2 2 2(k + 3) (19) QF ¡QL = QL + (k2 + 3)(k2 + 2)L k;t k;t ± k;2t k;2t+3 (k2 + 5)¡ ¢ + 2(¡1)t+1 QL ¡ 2 ; ± k;0 F L F L r+t¡ L ¢ (20) Q k;r+sQ k;r+t ¡Q k;r+tQ k;r+s = 2(¡1) Q k;0 ¡ 2 Fk;s¡t; F t F F (21) Q k;s+t + (¡1) Q k;s¡t = Q k;sLk;t; L t L L (22) Q k;s+t + (¡1) Q k;s¡t = Q k;sLk;t; F F F F s ¡ ¢ (23) Q k;sQ k;t ¡Q k;tQ k;s = 2(¡1) Fk;t¡s 0; ¡1; ¡k; 1 ; L L L L s+1 ¡ ¢ (24) Q k;sQ k;t ¡Q k;tQ k;s = 2(¡1) Fk;t¡s± 0; ¡1; ¡k; 1 ; 984 A.