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 √ √ 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 √ 2 r1 − r2 = k + 4 = δ, 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, ¡ ¢ X∞ 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 ¡ ¢ X∞ 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

X∞ QF + (−1)txQF (6) QF xn = k,s s,s−t , k,tn+s 1 − xL + x2(−1)t n=0 k,t X∞ QL + (−1)txQL (7) QL xn = k,s s−t , k,tn+s 1 − xL + x2(−1)t n=0 k,t ∞ t t X QF r¯ er1 x − r¯ er2 x (8) k,tn xn = 1 2 , n! r − r n=0 1 2 X∞ 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 = δ(−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 = δ(−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) δ 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 + δ(k2 + 5)L k,t k,t k k,2t k,2t+3 (k2 + 3)¡ ¢ + 2(−1)t QL − 2 , δ k,0 2 2 2 2(k + 3) (19) QF − QL = QL + (k2 + 3)(k2 + 2)L k,t k,t δ k,2t k,2t+3 (k2 + 5)¡ ¢ + 2(−1)t+1 QL − 2 , δ 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−sδ 0, −1, −k, 1 , 984 A. D. GODASE

F L F L s ¡ L ¢ (25) Q k,tQ k,s − Q k,sQ k,t = 2(−1) Fk,t−s Q k,0 − 2 , F L L F s £ F t−s¡ 2 ¢¤ (26) Q k,tQ k,s − Q k,tQ k,s = 2(−1) r¯2 Q k,t−s − r2 0, 1, k, k + 1 .

Hyperbolic k-Fibonacci and k-LucasOctonions 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 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

Cariow and Cariow [26, 27] state low multiplicative complexity algorithm for multiplying two hyperbolic octonions. −→ The scalar and the vector part of a hyperbolic octonion O are denoted by SO = h0 and V O = h1i1 + h2i2 + h3i3 + h4e4 + h5e5 + h6e6 + h7e7, respectively. Thus, a hyperbolic octonion O is −→ (h) given by O = SO + V O. For any two hyperbolic octonions O = h0 + h1i1 + h2i2 + h3i3 + (H) h4e4 + h5e5 + h6e6 + h7e7 and O = H0 + H1i1 + H2i2 + H3i3 + H4e4 + H5e5 + H6e6 + H7e7 addition and subtraction of the hyperbolic octonions is defined by

(h) (H) ¡ ¢ O ± O = h0 + h1i1 + h2i2 + h3i3 + h4e4 + h5e5 + h6e6 + h7e7 ¡ ¢ ± H0 + H1i1 + H2i2 + H3i3 + H4e4 + H5e5 + H6e6 + H7e7 ¡ ¢ ¡ ¢ ¡ ¢ ¢ ¡ ¢ = h0 ± H0 + h1 ± H1 i1 + h2 ± H2 i2 + h3 ± H3 i3 PROPERTIES OF k-FIBONACCI AND k-LUCAS OCTONIONS 985

¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ + h4 ± H4 e4 + h5 ± H5 e5 + h6 ± H6 e6 + h7 ± H7 e7.

Multiplication of the hyperbolic octonions is defined by

(h) (H) ¡ ¢ O ·O = h0 + h1i1 + h2i2 + h3i3 + h4e4 + h5e5 + h6e6 + h7e7 ¡ ¢ · H0 + H1i1 + H2i2 + H3i3 + H4e4 + H5e5 + H6e6 + H7e7

= O0 + O1i1 + O2i2 + O3i3 + O4e4 + O5e5 + O6e6 + O7e7, where,

O0 = h0H0 − h1H1 − h2H2 − h3H3 + h4H4 + h5H5 + h6H6 + h7H7,

O1 = h0H1 + h1H0 + h2H3 − h3H2 + h4H5 − h5H4 + h6H7 − h7H6,

O2 = h0H2 − h1H3 + h2H0 + h3H1 + h4H6 − h5H7 − h6H4 + h7H5,

O3 = h0H3 + h1H2 − h2H1 + h3H0 + h4H7 + h5H6 − h6H5 − h7H4,

O4 = h0H4 + h1H5 + h2H6 + h3H7 + h4H0 − h5H1 − h6H2 − h7H3,

O5 = h0H5 + h1H4 − h2H7 + h3H6 − h4H1 + h5H0 − h6H3 + h7H2,

O6 = h0H6 + h1H7 + h2H4 − h3H5 − h4H2 + h5H3 + h6H0 − h7H1,

O7 = h0H7 − h1H6 + h2H5 + h3H4 − h4H3 − h5H2 + h6H1 + h7H0.

The conjugate of hyperbolic octonion O is denoted by O¯ and it is

O¯ = h0 − h1i1 − h2i2 − h3i3 − h4e4 − h5e5 − h6e6 − h7e7.

The norm of O is defined as

2 2 2 2 2 2 2 2 NO = O· O¯ = h0 − h1 − h2 − h3 + h4 + h5 + h6 + h7 .

F L In [25], the hyperbolic k-Fibonacci and k-Lucas octonions O k,n and O k,n are defined as

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 , 986 A. D. GODASE

¯F ¯L respectively. The conjugate of hyperbolic k-Fibonacci and k-Lucas octonions O k,n and O k,n are defined as

¯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 ,

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

F L The Binet Formulas for the hyperbolic k-Fibonacci and k-Lucas octonions O k,n and O k,n are n n F r¯1r1 − r¯2r2 (i) O k,n = r1 − r2 L n n (ii) O k,n =r ¯1r1 +r ¯2r2 , n n ¯F r¯3r1 − r¯4r2 (iii) O k,n = r1 − r2 ¯L n n (iv) O k,n =r ¯3r1 +r ¯4r2 , where, 2 3 4 5 6 7 r¯1 = 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¯2 = 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 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 . PROPERTIES OF k-FIBONACCI AND k-LUCAS OCTONIONS 987

The properties presented here are the extension of Cassini, Catalan and d’Ocagne’s identity to hyperbolic k-Fibonacci and k-Lucas octonions.

Theorem 1.1 (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,tU¯k,t, L L L 2 n−t+1 (ii) O k,n−tO k,n+t − O k,n = δ(−1) Fk,tV¯k,m−n.

In Catalan‘s identity, taking t = 1, it reduces to Cassini’s Identity.

Theorem 1.2 (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,tU¯k,1, L L L 2 n (ii) O k,n−1O k,n+1 − O k,n = δ(−1) Fk,tV¯k,1.

Theorem 1.3 (d’Ocagene’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) V¯k,t−n, L L L L n+1 (ii) O k,tO k,n+1 − O k,t+1O k,n = (−1) δV¯k,t−n.

Different properties of the hyperbolic k-Fibonacci and k-Lucas octonions are investigated in [25]. These properties are the generalisations of Cassini, Catalan and d’Ocagne’s identity to hyperbolic k- Fibonacci and k-Lucas octonions.

F F F (1) O k,n+2 = kO k,n+1 + O k,n, L L L (2) O k,n+2 = kO k,n+1 + O k,n, L F F (3) O k,n = O k,n+1 + O k,n−1, ¯F ¯F ¯F (4) O k,n+2 = kO k,n+1 + O k,n, ¯L ¯L ¯L (5) O k,n+2 = kO k,n+1 + O k,n, ¯L ¯F ¯F (6) O k,n = O k,n+1 + O k,n−1, ¡ ¢ X∞ F L F F n O k,0 + O k,0Fk,t − O k,t x (7) O k,tnx = 2 t , 1 − xLk,t + x (−1) n=0 ¡ ¢ X∞ OL − OL L − OL x (8) OL xn = k,0 k,0 k,t k,t , k,tn 1 − xL + x2(−1)t n=0 k,t 988 A. D. GODASE

X∞ OF + (−1)txOF (9) OF xn = k,s s,s−t , k,tn+s 1 − xL + x2(−1)t n=0 k,t X∞ OL + (−1)txOL (10) OL xn = k,s s−t , k,tn+s 1 − xL + x2(−1)t n=0 k,t ∞ t t X OF r¯ er1 x − r¯ er2 x (11) k,tn xn = 1 2 , n! r − r n=0 1 2 X∞ L O k,tn t t (12) xn =r ¯ er1 x +r ¯ er2 x, n! 1 2 n=0 µ ¶ Xn n (13) kiOF = OF , i k,i k,2n i=0 µ ¶ Xn n (14) kiOL = OL , i k,i k,2n i=0 2 2 1£ ¤ (15) OF + OL = (1 + δ)W¯ + (δ − 1)(−1)tOL , k,t k,t δ k,2t k,0 2 2 1£ ¤ (16) OF − OL = (1 − δ)W¯ − (1 + δ)(−1)tOL , k,t k,t δ k,2t k,0 F L F L r+t¡ L ¢ (17) O k,r+sO k,r+t − O k,r+tO k,r+s = 2(−1) O k,0 − 2 Fk,s−t, F t F F (18) O k,s+t + (−1) O k,s−t = O k,sLk,t, L t L L (19) O k,s+t + (−1) O k,s−t = O k,sLk,t, F F F F s − 1 (20) O k,sO k,t − O k,tO k,s = (−1) δ 2 u¯5Fk,t−s, L L L L t 1 (21) O k,sO k,t − O k,tO k,s = (−1) δ 2 u¯5Fk,s−t, F L F L s L (22) O k,tO k,s − O k,sO k,t = 2(−1) Fk,t−sO k,0, F L L F s (23) O k,tO k,s − O k,tO k,s = 2(−1) V¯k,t−s, 1 F ¯L ¯F F n − (24) O k,nO k,n − O k,nO k,n = (−1) δ 2 (2u ¯10 − u¯14), 1 F ¯L ¯F F − (25) 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 (26) O k,nO k,n − O k,nO k,n = X¯k,2n + δ 2 (−1) u¯5 − u¯1 +u ¯2 − Y¯k,2n,

where, (u¯1- u¯18) are defined in Lemma 2.5 and U¯k,t, V¯k,t, W¯ k,t, X¯k,t and Y¯k,t are defined in Lemma 2.8 of [25]. In current paper, our main goal is to investigate some binomial and congruence properties for the k-Fibonacci and k-Lucas hyperbolic octonions. PROPERTIES OF k-FIBONACCI AND k-LUCAS OCTONIONS 989

2. PROPERTIES OF HYPERBOLIC k-FIBONACCIAND k-LUCAS OCTONIONS

In this section, we explore some binomial and congruence properties of the hyperbolic k-Fibonacci and k-Lucas octonions.

Lemma 2.1 — For n ≥ 0, we have

n (i) r1 = r1Fk,n + Fk,n−1, n (ii) r2 = r2Fk,n + Fk,n−1, 2n n n (iii) r1 = r1 Lk,n − (−1) , 2n n n (iv) r2 = r2 Lk,n − (−1) ,

tn n Fk,tn n Fk,(t−1)n (v) r1 = r1 − (−1) − , Fk,n Fk,n

tn n Fk,tn n Fk,(t−1)n (vi) r2 = r2 − (−1) − , Fk,n Fk,n sn rn sn (vii) r1 Fk,rn − r1 Fk,sn = (−1) Fk,(r−s)n, sn rn sn (viii) r2 Fk,rn − r2 Fk,sn = (−1) Fk,(r−s)n, 2(2n+1+1) 2(2n+1) (ix) 1 + kr1 + r1 = Lk,2n+1 r1 , 2(2n+1+1) 2(2n+1) (x) 1 + kr2 + r2 = Lk,2n+1 r1 , Xn (xi) For every n, t ≥ 1 and ln = Lk,2i , we have i=1  l n−1  n−1 2  r1 ; ln−2  Pt 2n ln−1 2n−t 1 (a) 1 + r = r1 − ln−1 , for t = 2, 3, 4, . . . , n − 2 ; , 1 l l  n−t−1 i=2 n−i  nP−1 1 l r 2 − l n−1 1 n−1 l  i=2 n−i l n−1  n−1 2  r2 ; ln−2  Pt 2n ln−1 2n−t 1 (b) 1 + r2 = r2 − ln−1 , for t = 2, 3, 4, . . . , n − 2 ; , ln−t−1 i=2 ln−i  n−1  2 P 1 ln−1r2 − ln−1 i=2 ln−i

F √ L (xii) r2t = k,2t r δ − k,2t−1 , 1 k 1 k F √ L (xiii) r2t = − k,2t r δ − k,2t−1 . 2 k 2 k 990 A. D. GODASE

L F √ (xiv) r2t+1 = k,2t+1 r − k,2t δ, 1 k 1 k L F √ (xv) r2t+1 = k,2t+1 r + k,2t δ. 2 k 2 k

PROOF : (i). We use induction principle on n, for n = 1, we have

1 r1 = 1 · r1 + 0 = r1Fk,1 + Fk,0.

2 For n = 2, since r1 is root of r − kr − 1 = 0 therefore we have

2 r1 = kr1 + 1 = r1Fk,2 + Fk,1. Now, consider n+1 n 2 r1 = r1 · r1 = (r1Fk,n + Fk,n−1)r1 = r1 Fk,n + r1Fk,n−1. 2 Using r1 = kr1 + 1, we have

= (kr1 + 1)Fk,n + r1Fk,n−1 = r1(kFk,n + Fk,n−1) + Fk,n = r1Fk,n+1 + Fk,n.

This complete the proof of (i). (iii). Using (i), we have

2n n+1 n r1 = Fk,nr1 + r1 Fk,n−1 n = Fk,n(r1Fk,n+1 + Fk,n) + r1 Fk,n−1 n 2 = r1Fk,nFk,n+1 + Fk,n−1r1 + Fk,n n n 2 = (r1 − Fk,n−1)Fk,n+1 + Fk,n−1r1 + Fk,n n 2 = r1 (Fk,n+1 + Fk,n−1) + Fk,n − Fk,nFk,n−1.

2 n Using Fk,n−1Fk,n+1 − Fk,n = (−1) and Fk,n+1 + Fk,n−1 = Lk,n, we obtain

2n n n r1 = Lk,nr1 − (−1) .

This complete the proof of (iii).

The proofs of (ii), (iv), (v), (vii), (viii), (ix), (x), (xi),(xii), (xiii), (xiv) and (xv) are similar to (i) and (iii). 2 PROPERTIES OF k-FIBONACCI AND k-LUCAS OCTONIONS 991

Theorem 2.2 — For all n, r, s, t ≥ 1, we have

F F F (i) O k,n+t = Fk,nO k,t+1 + Fk,n−1O k,t, F F n F (ii) O k,2n+t = Lk,nO k,n+t − (−1) O k,t,

F Fk,sn F n Fk,(s−1)n F (iii) O k,sn+t = O k,n+t − (−1) O k,t, Fk,n Fk,n F F sn F (iv) O k,sn+tFk,rn − O k,rn+tFk,sn = (−1) O k,tFk,(r−s)n, F F F F O k,t + kO k,t+1 + O k,t+2n+2+2 (v) O k,t+2n+1+2 = , Lk,2n+1 L L L L O k,t + kO k,t+1 + O k,t+2n+2+2 (vi) O k,t+2n+1+2 = , Lk,2n+1 Xn (vii) For every n, t ≥ 1 and ln = Lk,2i , we have i=1

 ln−1 F F  O k,t+2n−1 − O k,t; ln−2  Ps  ln−1 F 1 F  O k,t+2n−s − ln−1 (1 + )O k,t, F ln−t−1 ln−i (a) O n = i=2 , k,t+2   If s = 2, 3, 4, . . . , n − 2 ;   n−1  F P 1 F ln−1O k,t+2 − ln−1 ( + 1)O k,t. i=2 ln−i  ln−1 L L  O k,t+2n−1 − O k,t; ln−2  Ps  ln−1 L 1 L  O k,t+2n−s − ln−1 (1 + )O k,t, L ln−t−1 ln−i (b) O n = i=2 , k,t+2   If s = 2, 3, 4, . . . , n − 2 ;   n−1  L P 1 L ln−1O k,t+2 − ln−1 ( + 1)O k,t. i=2 ln−i

 P ¡ ¢  n lr−1 i j F  i ( ) (−1) O k,2r−1i+t; i+j=n lr−2  P ¡ ¢ P  n lr−1 i j s lr−1 j F  ( ) (−1) ( h=2(1 + ) O k,2n−si+t, F i l l (c) O r = i+j=n r−s−1 r−h , k,2 n+t    If s = 2, 3, 4, . . . , n − 2 ;  P ¡ ¢ P  n i j s lr−1 j F  i (lr−1) (−1) ( h=2(1 + ) O k,2i+t. i+j=n lr−h 992 A. D. GODASE

 P ¡ ¢  n lr−1 i j L  i ( ) (−1) O k,2r−1i+t; i+j=n lr−2  P ¡ ¢ P  n lr−1 i j s lr−1 j L  ( ) (−1) ( h=2(1 + ) O k,2n−si+t, L i l l (d) O r = i+j=n r−s−1 r−h , k,2 n+t    If s = 2, 3, 4, . . . , n − 2 ;  P ¡ ¢ P  n i j s lr−1 j L  i (lr−1) (−1) ( h=2(1 + ) O k,2i+t. i+j=n lr−h F L (viii) OF = k,2t OL − k,2t−1 OF , k,s+2t k k,s+1 k k,s F L (ix) OL = k,2t δOF − k,2t−1 OL , k,s+2t k k,s+1 k k,s F F (x) OF − k,2t OF + k,2t−2 OF = 0, k,s+2t k k,s+2 k k,s F F (xi) OL − k,2t OL + k,2t−2 OL = 0, k,s+2t k k,s+2 k k,s L F (xii) OF = k,2t+1 OF − k,2t OL , k,s+2t+1 k k,s+1 k k,s L F (xiii) OL = k,2t+1 OL − δ k,2t OF , k,s+2t+1 k k,s+1 k k,s L F (xiv) OF − k,2t+1 OF + k,2t−2 OL = 0, k,s+2t+1 k(k2 + 3) k,s+3 k(k2 + 3) k,s L F (xv) OL − k,2t+1 OL + k,2t−2 δOF = 0. k,s+2t+1 k(k2 + 3) k,s+3 k(k2 + 3) k,s

PROOF : (i). From (i) and (ii) of Lemma 2.1, we have

n r1 = r1Fk,n + Fk,n−1, (2.1)

n r2 = r2Fk,n + Fk,n−1. (2.2)

r¯ r t r¯ r t Multiplying (2.1) by 1 1 and (2.2) by 2 2 , we get r1 − r2 r1 − r2

n+t t+1 t r¯1r1 r¯1r1 r¯1r1 = Fk,n + Fk,n−1, (2.3) r1 − r2 r1 − r2 r1 − r2

n+t t+1 t r¯2r2 r¯2r2 r¯2r2 = Fk,n + Fk,n−1. (2.4) r1 − r2 r1 − r2 r1 − r2 Subtracting (2.3) and (2.4), we obtain

n+t n+t t+1 t+1 t t r¯1r1 − r¯2r2 r¯1r1 − r¯2r2 r¯1r1 − r¯2r2 = Fk,n + Fk,n−1. r1 − r2 r1 − r2 r1 − r2 PROPERTIES OF k-FIBONACCI AND k-LUCAS OCTONIONS 993

F Using Binet formula of the hyperbolic k-Fibonacci octonion O k,n, we get

F F F O k,n+t = Fk,nO k,t+1 + Fk,n−1O k,t.

The proofs of (ii)-(xv) are similar to (i) using Lemma 2.1. 2

Theorem 2.3 — For all n, r, s, t ≥ 1, we have

µ ¶ Xn n (i) OF = F i F n−i OF , k,rn+t i k,r k,r−1 k,i+t i=0 µ ¶ Xn n (ii) OL = F i F n−i OL , k,rn+t i k,r k,r−1 k,i+t i=0 µ ¶ Xn n (iii) OF = (−1)(n−i)(r+1)Li OF , k,2rn+t i k,r k,ri+t i=0 µ ¶ Xn n (iv) OL = (−1)(n−i)(r+1)Li OL , k,2rn+t i k,r k,ri+t i=0 µ ¶ 1 Xn n (v) OF = (−1)(n−i)(r+1)F n−i F i OF , k,trn+l F n i k,(t−1)r k,tr k,ri+l k,r i=0 µ ¶ 1 Xn n (vi) OL = (−1)(n−i)(r+1)F n−i F i OL , k,trn+l F n i k,(t−1)r k,tr k,ri+l k,r i=0 µ ¶ Xn n (vii) (−1)iOF F i = OF F n , i k,r(n−i)+i+t k,r k,t k,r−1 i=0 µ ¶ Xn n (viii) (−1)iOL F i = OL F n , i k,r(n−i)+i+t k,r k,t k,r−1 i=0 µ ¶ Xn n (ix) (−1)(n−i)OF F (n−i) = OF F n , i k,ri+t k,r−1 k,n+t k,r i=0 µ ¶ Xn n (x) (−1)(n−i)OL F (n−i) = OL F n , i k,ri+t k,r−1 k,n+t k,r i=0 µ ¶ Xn n (xi) (−1)(n−i)F (n−i)F (i) OF = (−1)smnOF F n , i k,sm k,rm k,m[rn+i(s−r)]+t k,t k,(r−s)m i=0 µ ¶ Xn n (xii) (−1)(n−i)F (n−i)F (i) OL = (−1)smnOL F n , i k,sm k,rm k,m[rn+i(s−r)]+t k,t k,(r−s)m i=0 X µ ¶ F n −n j+s i F (xiii) O = k (−1) L r+1 O r+1 , k,n+t i, j k,2 k,2 (i+2j)+2(i+j)+t i+j+s=n 994 A. D. GODASE

X µ ¶ L n −n j+s i L (xiv) O = k (−1) L r+1 O r+1 , k,n+t i, j k,2 k,2 (i+2j)+2(i+j)+t i+j+s=n X µ ¶ F n j j+s i F (xv) O r+2 = k (−1) L r+1 O r+1 , k,(2 +2)n+t i, j k,2 k,(2 +2)i+j+t i+j+s=n X µ ¶ L n j j+s i L (xvi) O r+2 = k (−1) L r+1 O r+1 , k,(2 +2)n+t i, j k,2 k,(2 +2)i+j+t i+j+s=n X µ ¶ F n j −n F (xvii) O r+1 = k L r+1 O r+1 , k,(2 +2)n+t i, j k,2 k,(2 +2)i+j+t i+j+s=n X µ ¶ F n j −n F (xviii) O r+1 = )k L r+1 O r+1 , k,(2 +2)n+t i, j k,2 k,(2 +2)i+j+t i+j+s=n µ ¶ Xn n (xix) k(i−n)(L )(n−i)OF i k,2t−1 k,2ti+s i=0   −n n n F k (Fk,2t) δ 2 O k,n+s, if n is even; = ,  −n n n−1 L k (Fk,2t) δ 2 O k,n+s, if n is odd, µ ¶ Xn n (xx) k(i−n)(L )(n−i)OL i k,2t−1 k,2ti+s i=0   −n n n L k (Fk,2t) δ 2 O k,n+s, if n is even; = ,  −n n n+1 F k (Fk,2t) δ 2 O k,n+s, if n is odd. µ ¶ Xn n (xxi) (−1)(n−i)k−i(L )iOF i k,2t+1 k,2t(n−i)+n i=0   n F δ 2 O k,0, if n is even; = ,  n−1 L δ 2 O k,0, if n is odd, µ ¶ Xn n (xxii) (−1)(n−i)k−i(L )iOL i k,2t+1 k,2t(n−i)+n i=0   n L δ 2 O k,0, if n is even; =  n+1 F δ 2 O k,0, if n is odd.

PROOF : (i). From (i) and (ii) of Lemma 2.1, we have

r r1 = Fk,rr1 + Fk,r−1, r r2 = Fk,rr2 + Fk,r−1. PROPERTIES OF k-FIBONACCI AND k-LUCAS OCTONIONS 995

Now, by the binomial theorem, we have µ ¶ Xn n rrn = (F r + F )n = F i F n−i ri , (2.5) 1 k,r 1 k,r−1 i k,r k,r−1 1 i=0 µ ¶ Xn n rrn = (F r + F )n = F i F n−i ri . (2.6) 2 k,r 2 k,r−1 i k,r k,r−1 2 i=0 r¯ r¯ Multiplying (2.5) by 1 and (2.6) by 2 and subtracting, we obtain r1 − r2 r1 − r2 µ ¶ r¯ rrn+t − r¯ rrn+t Xn n r¯ ri+t − r¯ ri+t 1 1 2 2 = F i F n−i ( 1 1 2 2 ). r − r i k,r k,r−1 r − r 1 2 i=0 1 2 F F Using Binet formula of O k,rn+t and O k,i+t, we get µ ¶ Xn n OF = F i F n−i OF . k,rn+t i k,r k,r−1 k,i+t i=0

(ii). Multiplying (2.5) by r¯1 and (2.6) by r¯2 and adding, we obtain

µ ¶ Xn n r¯ rrn+t +r ¯ rrn+t = F i F n−i (r ¯ ri+t +r ¯ ri+t). 1 1 2 2 i k,r k,r−1 1 1 2 2 i=0 L L Using Binet formula of O k,rn+t and O k,i+t, we get µ ¶ Xn n OL = F i F n−i OL . k,rn+t i k,r k,r−1 k,i+t i=0 The proofs of (iii)-(xxii) are similar to (i) and (ii) using Lemma 2.1. 2

Next theorem deals with congruence properties of the hyperbolic k-Fibonacci and k-Lucas octo- nions. F L Theorem 2.4 — For n, t ≥ 1 and Gk,n = O k,n or O k,n, we have Xn µ ¶ F n −n n F ¡ ¢ (i) O − k (−1) O r+2 ≡ 0 mod L r+1 , k,n+t j k,(2 +2)j+t k,2 j=0 Xn µ ¶ L n −n n L ¡ ¢ (ii) O − k (−1) O r+2 ≡ 0 mod L r+1 , k,n+t j k,(2 +2)j+t k,2 j=0 Xn µ ¶ F n j n F (iii) O r+2 − k (−1) O ≡ 0 (mod L r+1 ), k,(2 +2)n+t j k,j+t k,2 j=0 Xn µ ¶ L n j n L (iv) O r+2 − k (−1) O ≡ 0 (mod L r+1 ). k,(2 +2)n+t j k,j+t k,2 j=0 996 A. D. GODASE

PROOF : (i). From (xiii) of Theorem 2.3, for all n, t ≥ 1, we have

X µ ¶ F n −n j+s i F O = k (−1) L r+1 O r+1 k,n+t i, j k,2 k,2 (i+2j)+2(i+j)+t i+j+s=n;i6=0 X µ ¶ n −n j+s i F + k (−1) L r+1 O r+1 , i, j k,2 k,2 (i+2j)+2(i+j)+t i+j+s=n;i=0 X µ ¶ n −n j+s i F = k (−1) L r+1 O r+1 i, j k,2 k,2 (i+2j)+2(i+j)+t i+j+s=n;i6=0 Xn µ ¶ n −n n F + k (−1) O r+2 . j k,(2 +2)j+t j=0 Xn µ ¶ F n −n n F O − k (−1) O r+2 k,n+t j k,(2 +2)j+t j=0 X µ ¶ n −n j+s i F = k (−1) L r+1 O r+1 , i, j k,2 k,2 (i+2j)+2(i+j)+t i+j+s=n;i6=0 Xn µ ¶ F n −n n F ∴ L divides (O − k (−1) O r+2 ), k,2 k,n+t j k,(2 +2)j+t j=0 Xn µ ¶ F n −n n F ∴ O − k (−1) O r+2 ≡ 0 (modL ). k,n+t j k,(2 +2)j+t k,2 j=0

The proofs of (ii), (iii) and (iv) are similar to (i), using Theorem 2.3. 2

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