Miskolc Mathematical Notes HU e-ISSN 1787-2413 Vol. 20 (2019), No. 2, pp. 1315–1330 DOI: 10.18514/MMN.2019.2628

BICOMPLEX GENERALIZED k HORADAM

YASIN YAZLIK, SURE KOME,¨ AND CAHIT KOME¨ Received 12 May, 2018

Abstract. This study provides a broad overview of the generalization of the various quaternions, especially in the context of its enhancing importance in the disciplines of mathematics and phys- ics. By the help of bicomplex , in this paper, we define the bicomplex generalized k Horadam quaternions. Fundamental properties and mathematical preliminaries of these qua- ternions are outlined. Finally, we give some basic conjucation identities, generating function, the Binet formula, summation formula, matrix representation and a generalized identity, which is generalization of the well-known identities such as Catalan’s identity, Cassini’s identity and d’Ocagne’s identity, of the bicomplex generalized k Horadam quaternions in detail. 2010 Mathematics Subject Classification: 11B39; 05A15 Keywords: bicomplex generalized k Horadam quaternions, recurrence relations, quaternions

1. INTRODUCTION Bicomplex numbers emerge in various scientific areas such as quantum mechan- ics, digital signal processing, electromagnetic waves and curved structures, determ- ination of antenna patterns, fractal structures and many related fields. Recently, sev- eral remarkable studies have been conducted related with bicomplex numbers (see [6, 11, 13, 19, 22, 24, 25, 27, 29, 30, 35]). For instance, Rochon and Tremblay, in [29], studied the bicomplex Schrodinger¨ equations. They also mentioned that the bicom- plex quantum mechanics are the generalization of both the classical and hyperbolic quantum mechanics. Kabadayi and Yayli, in [19], represented a curve by means of bicomplex numbers in a hypersurface in E4 and then they defined the homothetic motion of this curve. Lavoie et al., in [21], determined the eigenkets and eigenvalues of the bicomplex quantum harmonic oscillator Hamiltonian. They asserted that these eigenvalues and eigenkets, first in the literature, were derived with a system larger than C. The bicomplex number q, which extends the complex numbers, can be defined as

C2 q q1 iq2 jq3 ijq4 q1;q2;q3;q4 R ; (1.1) D f D C C C j 2 g where i;j and ij satisfy the multiplication rules i2 j2 1; ij ji: (1.2) D D D c 2019 Miskolc University Press

1316 YASIN YAZLIK, SURE KOME,¨ AND CAHIT KOME¨

The conjugations of the bicomplex numbers are defined in [28] as: ? ? q q1 iq2 jq3 ijq4; q q1 iq2 jq3 ijq4 i D C j D C and ? q q1 iq2 jq3 ijq4 ij D C and the basic properties of the conjugations are as follows: ?? ? ? ? ? ? ? ? ? q q; .q1q2/ q q ; .q1 q2/ q q ; .q1/ q D D 2 1 C D 1 C 2 D 1 and ? ? ? .q1 q2/ q q ; ˙ D 1 C 2 where q1;q2 C2 and ; R. Furthermore, three different norms for the bicom- plex numbers2 are given by 2 q 2 2 2 2 Nq q qi q q q q 2j.q1q3 q2q4/ ; i D jj  jj D j 1 C 2 3 4 C C j q 2 2 2 2 Nq q qj q q q q 2i.q1q2 q3q4/ ; j D jj  jj D j 1 2 C 3 4 C C j q 2 2 2 2 Nq q qij q q q q 2ij.q1q4 q2q3/ : ij D jj  jj D j 1 C 2 C 3 C 4 C j Quaternions, which are a number system that extends the complex numbers, arise in quantum mechanics, physics, mathematics, computer science and related areas (see [1,2,5,9, 10, 12, 14, 15, 26, 31–33]). They were first introduced by in 1843 [14]. In general, a q, which is member of a noncommutative , is defined by

H q q0 iq1 jq2 kq3 q0;q1;q2;q3 R ; (1.3) D f D C C C j 2 g where i;j and k satisfy the multiplication rules i2 j2 k2 1; ij ji k; jk kj i; ki ik j: (1.4) D D D D D D D D D Note that, although quaternions are noncommutative, the bicomplex numbers and bicomplex quaternions are commutative. The conjugate of a quaternion q is defined by N q q0 iq1 jq2 kq3; (1.5) D where i;j and k satisfy the rules (1.4). The quaternions have been studied by several authors in the recent years (see [1,2,4,5,9, 10, 12, 15, 26, 31–33]). For example, Horadam, in [15], defined the Fibonacci quaternions. Motivated by Horadam’s study, Halici, in [9], examine some basic properties of Fibonacci and Lucas quaternions. She also gave the generat- ing functions, the Binet formulas and derived some sums formulas for these qua- ternions. Liana and Wloch [31] introduced the Jacobsthal and Jacobsthal-Lucas quaternions and they gave some properties and matrix representations of these qua- ternions. Tan proposed the biperiodic Fibonacci quaternions whose coefficients are the biperiodic Fibonacci numbers in [33]. Later, Tan et al. described the biperiodic BICOMPLEX GENERALIZED k HORADAM QUATERNIONS 1317 Lucas quaternions and gave the generating functions, the Binet formulas and Cassini and Catalan like identities [32]. Later, by using the bicomplex numbers, Aydın, in [1], defined the bicomplex Fibonacci and Lucas quaternions as:

Qn Fn iFn 1 jFn 2 ijFn 3; (1.6) D C C C C C C where Fn is the nth Fibonacci number. She also studied addition, subtraction, multi- plication of the bicomplex Fibonacci quaternions and then gave several properties of this quaternion. Although, she mentioned the bicomplex Lucas quaternions in The- orem 2.5 in [1], she didn’t give any definition of the bicomplex Lucas quaternions. For n N0, the Fibonacci and Lucas numbers are defined by the recurrence rela- tions 2

Fn 2 Fn 1 Fn;F0 0; F1 1 (1.7) C D C C D D and

Ln 2 Ln 1 Ln;L0 2; L1 1; (1.8) C D C C D D respectively. Recently, many researchers have studied several applications and gen- eralizations for the number sequences(see [7,8, 16–18, 20, 36]). For further inform- ation, we specially refer to book in [20]. For example, Yazlik and Taskara, in [36], defined the generalized k Horadam sequence, which is generalization of many num- 2 ber sequences in the literature. For n N0 and f .k/ 4g.k/ > 0, the generalized k Horadam sequence defined by 2 C Hk;n 2 f .k/Hk;n 1 g.k/Hk;n;Hk;0 a; Hk;1 b: (1.9) C D C C D D Note that, the Binet formula of the generalized k Horadam sequence is given by, for n N0, 2 n n Xr1 Y r2 Hk;n ; (1.10) D r1 r2 where X b ar2 and Y b ar1. In thisD paper, by analogyD to generalizations of Fibonacci and Lucas quaternions explained for example in [12, 26, 32, 33], we generalize families of the Fibonacci and Lucas quaternions. Hence, the next section describes the bicomplex generalized k Horadam quaternions which are both generalization of the results in [1] and they include several bicomplex quaternions which are not defined before.

2. BICOMPLEXGENERALIZED k HORADAM QUATERNIONS Definition 1. For n N0, the bicomplex generalized k Horadam quaternion is defined by 2 Q Hk;n Hk;n iHk;n 1 jHk;n 2 ijHk;n 3; (2.1) D C C C C C C where H is the generalized k Horadam numbers which is defined in (1.9). k;n 1318 YASIN YAZLIK, SURE KOME,¨ AND CAHIT KOME¨

TABLE 1. The bicomplex generalized k Horadam quaternions Bicomplex generalized k Horadam quaternions Q Hk;n Hk;n iHk;n 1 jHk;n 2 ijHk;n 3 f .k/ g.k/ a b D C C C C C C Hk;n f .k/Hk;n 1 g.k/Hk;n 2, D C Hk;0 a and Hk;1 b Bicomplex FibonacciD quaternionsD [1] 1 1 0 1 Qn Fn iFn 1 jFn 2 ijFn 3 D C C C C C C Fn Fn 1 Fn 2, F0 0 and F1 1 DBicomplex C Lucas quaternionsD D Q 1 1 2 1 Ln Ln iLn 1 jLn 2 ijLn 3 D C C C C C C Ln Ln 1 Ln 2, L0 2 and L1 1 BicomplexD C k Pell quaternionsD [3]D P 2 k 0 1 BCk;n Pk;n iPk;n 1 jPk;n 2 ijPk;n 3 D C C C C C C Pk;n 2Pk;n 1 kPk;n 2, Pk;0 0 and Pk;1 1 D Bicomplex C Pell quaternionsD [3] D Q 2 1 0 1 Pn Pn iPn 1 jPn 2 ijPn 3 D C C C C C C Pn 2Pn 1 Pn 2, P0 0 and P1 1 DBicomplex C Pell-Lucas D quaternionsD Q 2 1 2 2 PLn Qn iQn 1 jQn 2 ijQn 3 D C C C C C C Qn 2Qn 1 Qn 2, Q0 2 and Q1 2 DBicomplex C Jacobsthal quaternionsD D Q 1 2 0 1 Jn Jn iJn 1 jJn 2 ijJn 3 D C C C C C C Jn Jn 1 2Jn 2, J0 0 and J1 1 BicomplexD Jacobsthal-LucasC D quaternionsD Q 1 2 2 1 JLn jn ijn 1 jjn 2 ijjn 3 D C C C C C C jn jn 1 2jn 2, j0 2 and j1 1 D C D D It is not difficult to see from the following table that the bicomplex generalized k Horadam quaternions can be reduced into several quaternions for the special cases off .k/;g.k/;a and b. Q Q For n;m N0, H and H be two bicomplex generalized k Horadam qua- 2 k;n k;m ternions. Thus, the addition and subtraction of these two quaternions can be given as: Q Q  Hk;n Hk;m Hk;n iHk;n 1 jHk;n 2 ijHk;n 3 ˙ D C C C C C C  Hk;m iHk;m 1 jHk;m 2 ijHk;m 3 ˙ C  C C C C  C Hk;n Hk;m i Hk;n 1 Hk;m 1 D ˙ C  C ˙ C  j Hk;n 2 Hk;m 2 ij Hk;n 3 Hk;m 3 : C C ˙ C C C ˙ C BICOMPLEX GENERALIZED k HORADAM QUATERNIONS 1319 TABLE 2. The bicomplex generalized k Horadam quaternions (continued from previous page)

Bicomplex k Fibonacci quaternions [34] QFk;n Fk;n iFk;n 1 jFk;n 2 ijFk;n 3 k 1 0 1 D C C C C C C Fk;n kFk;n 1 Fk;n 2, D C Fk;0 0 and Fk;1 1 BicomplexDk Lucas quaternionsD Q Lk;n Lk;n iLk;n 1 jLk;n 2 ijLk;n 3 k 1 2 k D C C C C C C Lk;n kLk;n 1 Lk;n 2, D C Lk;0 2 and Lk;1 k BicomplexD Horadam quaternionsD Q p q a b Hn Hn iHn 1 jHn 2 ijHn 3 D C C C C C C Hn pHn 1 qHn 2, H0 a and H1 b D C D D On the other hand, the multiplication of these two quaternions can be computed as: H Q H Q k;n  k;m  Hk;n iHk;n 1 jHk;n 2 ijHk;n 3 D C C C C C C  Hk;m iHk;m 1 jHk;m 2 ijHk;m 3  C C C C C C  Hk;nHk;m Hk;n 1Hk;m 1 Hk;n 2Hk;m 2 Hk;n 3Hk;m 3 D C C C C C C C  i Hk;nHk;m 1 Hk;n 1Hk;m Hk;n 2Hk;m 3 Hk;n 3Hk;m 2 C C C C C C C C  j Hk;nHk;m 2 Hk;n 2Hk;m Hk;n 1Hk;m 3 Hk;n 3Hk;m 1 C C C C C C C C  ij Hk;nHk;m 3 Hk;n 3Hk;m Hk;n 1Hk;m 2 Hk;n 2Hk;m 1 C C C C C C C C C C H Q H Q : D k;m  k;n In addition, the conjugations of the bicomplex generalized k Horadam quaternions are defined as: ?  Q Á Hk;n Hk;n iHk;n 1 jHk;n 2 ijHk;n 3 (2.2) i D C C C C ?  Q Á Hk;n Hk;n iHk;n 1 jHk;n 2 ijHk;n 3 (2.3) j D C C C C ?  Q Á Hk;n Hk;n iHk;n 1 jHk;n 2 ijHk;n 3: (2.4) ij D C C C C Theorem 1. For any given two bicomplex generalized k Horadam quaternions Q Q H and H , where n;m N0, the following conjugation identities hold. k;n k;m 2  Á?  Á?  Á?  Á?  Á? H Q H Q H Q H Q H Q H Q ; (2.5) k;n k;m i D k;m i k;n i D k;n i k;m i 1320 YASIN YAZLIK, SURE KOME,¨ AND CAHIT KOME¨  Á?  Á?  Á?  Á?  Á? H Q H Q H Q H Q H Q H Q ; (2.6) k;n k;m j D k;m j k;n j D k;n j k;m j  Á?  Á?  Á?  Á?  Á? H Q H Q H Q H Q H Q H Q : (2.7) k;n k;m ij D k;m ij k;n ij D k;n ij k;m ij Proof. By considering the Eqs. (2.2), (2.3) and (2.4), the theorem can be proved easily.   Á?  Á?  Á? Theorem 2. Let H Q , H Q and H Q be three conjugations of the k;n i k;n j k;n ij generalized k Horadam quaternion. Then we obtain the following relations ? Q  Q Á 2 2 2 2 Hk;n Hk;n Hk;n Hk;n 1 Hk;n 2 Hk;n 3 i D C C C C  2j Hk;nHk;n 2 Hk;n 1Hk;n 3 (2.8) C C C C C ? Q  Q Á 2 2 2 2 Hk;n Hk;n Hk;n Hk;n 1 Hk;n 2 Hk;n 3 j D C C C C  2i Hk;nHk;n 1 Hk;n 2Hk;n 3 (2.9) C C C C C ? Q  Q Á 2 2 2 2 Hk;n Hk;n Hk;n Hk;n 1 Hk;n 2 Hk;n 3 ij D C C C C C C  2ij Hk;nHk;n 3 Hk;n 1Hk;n 2 : (2.10) C C C C Proof. The theorem can be proved easily by using the Eqs. (2.1), (2.2), (2.3) and (2.4). Hence we omit the proof. 

Theorem 3. For n N0, the generalized k Horadam quaternion satisfies the recurrence relation 2 Q Q Q Hk;n 2 f .k/Hk;n 1 g.k/Hk;n: (2.11) C D C C Proof. By considering the right hand side of the Eq. (2.11), we get Q Q f .k/Hk;n 1 g.k/Hk;n C C  f .k/ Hk;n 1 iHk;n 2 jHk;n 3 ijHk;n 4 D C C C C C C C g.k/ Hk;n iHk;n 1 jHk;n 2 ijHk;n 3 C C C C C C C  f .k/Hk;n 1 g.k/Hk;n i f .k/Hk;n 2 g.k/Hk;n 1 D C C C  C C C  j f .k/Hk;n 3 g.k/Hk;n 2 ij f .k/Hk;n 4 g.k/Hk;n 3 C C C C C C C C Hk;n 2 iHk;n 3 jHk;n 4 ijHk;n 5 D C C C C C C C Q Hk;n 2; D C which is the desired result.  BICOMPLEX GENERALIZED k HORADAM QUATERNIONS 1321 The following theorem explains the generating function of the bicomplex general- ized k Horadam quaternions. Theorem 4. For n N0, the generating function for the generalized k Horadam quaternion is 2 Q  Q Q Á Hk;0 Hk;1 f .k/Hk;0 t H.t/ C : (2.12) D 1 f .k/t g.k/t 2 Q Proof. We use the formal power series to find the generating function of Hk;n. Now, we define

X1 Q Q Q X1 Q H.t/ H t n H H t H t n: (2.13) D k;n D k;0 C k;1 C k;n n 0 n 2 D D Multiplying the Eq. (2.13) both f .k/t and g.k/t 2, we get

X1 Q Q X1 Q f .k/tH.t/ f .k/H t n 1 f .k/H t f .k/H t n (2.14) D k;n C D k;0 C k;n 1 n 0 n 2 D D and X1 Q g.k/t 2H.t/ g.k/H t n: (2.15) D k;n 2 n 2 D By considering the above equations and doing some basic operations, we get 1 f .k/t g.k/t 2H.t/ H Q H Q t f .k/H Q t D k;0 C k;1 k;0 0 1 X1 Q Q Q BH f .k/H g.k/H Ct n: C @ k;n k;n 1 k;n 2A n 2 „ ƒ‚ … D 0 Therefore, we obtain Q  Q Q Á Hk;0 Hk;1 f .k/Hk;0 t H.t/ C ; (2.16) D 1 f .k/t g.k/t 2 which is the desired result.  Now, we give the Binet formula for the bicomplex generalized k Horadam qua- ternions by means of the Binet formula of the generalized k Horadam numbers which is defined in the Eq. (1.10). Theorem 5. The Binet formula for the generalized k Horadam quaternion is ? n ? n Q ˛ Xr1 ˇ Y r2 Hk;n ; (2.17) D r1 r2 1322 YASIN YAZLIK, SURE KOME,¨ AND CAHIT KOME¨

? 2 3 ? 2 3 where ˛ 1 ir1 jr ijr , ˇ 1 ir2 jr ijr , X b ar2, Y b ar1, D C C 1 C 1 D C C 2 C 2 D D f .k/ pf .k/2 4g.k/ f .k/ pf .k/2 4g.k/ r1 C C and r2 C . D 2 D 2 Proof. By considering the Binet formula of the generalized k Horadam numbers, we get Q Hk;n Hk;n iHk;n 1 jHk;n 2 ijHk;n 3 D C C C C C C Xrn Y rn Xrn 1 Y rn 1 Xrn 2 Y rn 2 1 2 i 1 C 2 C j 1 C 2 C D r1 r2 C r1 r2 C r1 r2 Xrn 3 Y rn 3 ij 1 C 2 C C r1 r2 n ! n ! Xr1 2 3 Y r2 2 3 1 ir1 jr1 ijr1 1 ir2 jr2 ijr2 D r1 r2 C C C r1 r2 C C C ! 1 ? n ? n ˛ Xr1 ˇ Y r2 ; D r1 r2 ? 2 3 ? 2 3 where ˛ 1 ir1 jr ijr , ˇ 1 ir2 jr ijr , X b ar2, Y b D C C 1 C 1 D C C 2 C 2 D D f .k/ pf .k/2 4g.k/ f .k/ pf .k/2 4g.k/ ar1, r1 C C and r2 C . Therefore the proof is D 2 D 2 completed.  The matrix representation of the bicomplex generalized k Horadam quaternions can be given in the following theorem. Theorem 6. Let n 1 be . Then  Q Q ! Q Q ! 2 Ãn 1 Hk;2n l Hk;2.n 1/ l Hk;2 l Hk;l f .k/ 2g.k/ 1 C C C ; H Q H Q D H Q C H Q g.k/2 0 k;2.n 1/ l k;2n l k;4 l k;2 l C C C C C (2.18) where l 0;1 . 2 f g Proof. We prove the theorem by induction on n. If n 1 then the result is clear. Now we assume that, for any integer m such as 1 m nD, Ä Ä Q Q ! Q Q ! 2 Ãm 1 H H H H f .k/ 2g.k/ 1 k;2m l k;2.m 1/ l k;2 l k;l : Q C Q C Q C Q g.k/C 2 0 Hk;2.m 1/ l Hk;2m l D Hk;4 l Hk;2 l C C C C C Then, for n m 1, we get D C Q Q ! 2 Ãm Hk;2 l Hk;l f .k/ 2g.k/ 1 Q C Q g.k/C 2 0 Hk;4 l Hk;2 l C C BICOMPLEX GENERALIZED k HORADAM QUATERNIONS 1323 Q Q ! 2 Ãm 1  2 à Hk;2 l Hk;l f .k/ 2g.k/ 1 f .k/ 2g.k/ 1 Q C Q g.k/C 2 0 g.k/C 2 0 D Hk;4 l Hk;2 l C C Q Q ! 2 à Hk;2m l Hk;2.m 1/ l f .k/ 2g.k/ 1 Q C Q C g.k/C 2 0 D Hk;2.m 1/ l Hk;2m l C C C Q Q ! Hk;2.m 1/ l Hk;2m l Q C C Q C ; D Hk;2.m 2/ l Hk;2.m 1/ l C C C C where l 0;1 . Therefore, the proof is completed. 2 f g  Theorem 7. For n N0, the summation formula for the generalized k Horadam quaternion2 is Q Q Q Q n 8 H g.k/H H g.k/H k;n 1C k;n k;1 k;0 X Q < C f .k/ g.k/ 1 ; if f .k/ g.k/ 1 0 Hk;s Q Q C C ¤ D g.k/Hk;n Hk;1 .n 1/Œg.k/a b.1 i+j+ij/ s 1 : C C C C ; if f .k/ g.k/ 1 0 D 1 g.k/ C D C (2.19) Proof. We prove the theorem with two cases. First we assume that f .k/ g .k/ 1 0. By using the definition of the generalized k Horadam quaternion, weC have ¤ n n n n n X Q X X X X Hk;s Hk;s i Hk;s 1 j Hk;s 2 ij Hk;s 3: (2.20) D C C C C C C s 1 s 1 s 1 s 1 s 1 D D D D D Now, we compute each term on the right hand side of the Eq. (2.20). By using the Eq. (12) in [23], we get n X Hk;n 1 g.k/Hk;n Hk;1 g.k/Hk;0 H C C : (2.21) k;s D f .k/ g.k/ 1 s 1 D C Moreover, n X Hk;s 1 C s 1 D n n X X f .k/ Hk;s g.k/ Hk;s 1 D C s 1 s 1 D D n ! X f .k/ Hk;s 1 Hk;n 1 Hk;1 D C C C s 1 D n ! X g.k/ Hk;s 1 Hk;n 1 Hk;n Hk;0 Hk;1 C C C C C s 1 D 1324 YASIN YAZLIK, SURE KOME,¨ AND CAHIT KOME¨

g.k/Hk;n 1 g.k/Hk;n g.k/Hk;0 g.k/Hk;1 f .k/Hk;n 1 f .k/Hk;1 C C C C D .f .k/ g.k/ 1/ C Hk;n 2 g.k/Hk;n 1 Hk;2 g.k/Hk;1 C C C : (2.22) D f .k/ g.k/ 1 C By doing the similar operations as in the Eq. (2.22), we obtain n X Hk;n 3 g.k/Hk;n 2 Hk;3 g.k/Hk;2 Hk;s 2 C C C (2.23) C D f .k/ g.k/ 1 s 1 D C and n X Hk;n 4 g.k/Hk;n 3 Hk;4 g.k/Hk;3 Hk;s 3 C C C : (2.24) C D f .k/ g.k/ 1 s 1 D C By substituting the Equations (2.21), (2.22), (2.23) and (2.24) in the Eq. (2.20), one can obtain that n Q Q Q Q X Q Hk;n 1 g.k/Hk;n Hk;1 g.k/Hk;0 H C C : (2.25) k;s D f .k/ g.k/ 1 s 1 D C Next we assume that f .k/ g .k/ 1 0. By using the Eq. (13) in [23], we get C D n X g.k/Hk;n .n 1/Œg.k/a b Hk;1 H C C C : (2.26) k;s D 1 g.k/ s 1 D C Furthermore, we have n n X X Hk;s 2 g.k/ Hk;s 1 nŒg.k/a b C C C D C s 1 s 1 Dn D n X X Hk;s 1 Hk;n 2 Hk;2 g.k/ Hk;s 1 nŒg.k/a b C C C C C D C s 1 s 1 D n D X .1 g.k// Hk;s 1 .n 1/Œg.k/a b .g.k/a b/ Hk;n 2 Hk;2 C C D C C C C C s 1 n D X g.k/Hk;n 1 .n 1/Œg.k/a b Hk;2 Hk;s 1 C C C C : (2.27) C D 1 g.k/ s 1 D C By doing the similar operations as in the Eq. (2.27), we obtain n X g.k/Hk;n 2 .n 1/Œg.k/a b Hk;3 Hk;s 2 C C C C (2.28) C D 1 g.k/ s 1 D C BICOMPLEX GENERALIZED k HORADAM QUATERNIONS 1325 and n X g.k/Hk;n 3 .n 1/Œg.k/a b Hk;4 Hk;s 3 C C C C : (2.29) C D 1 g.k/ s 1 C Therefore, byD substituting the Equations (2.26), (2.27), (2.28) and (2.29) in the Eq. (2.20), one can obtain that n Q Q X Q g.k/Hk;n Hk;1 .n 1/Œg.k/a b.1 i+j+ij/ H C C C C : (2.30) k;s D 1 g.k/ s 1 D C  Now, we give a theorem which generalizes the well-known identities such as Catalan’s identity, Cassini’s identity and d’Ocagne’s identity. Theorem 8. The generalized k Horadam quaternions satisfy the identity that is Q Q Q Q Hk;nHk;n r s Hk;n sHk;n r C C n r   . g.k// bHk;r aHk;r 1 bHk;s aHk;s 1 C C D b2 a2g.k/ abf .k/ " 1 g.k/ g.k/2 g.k/3 i1 g.k/2f .k/  C C # j.1 g.k//f .k/2 2g.k/ ijf .k/f .k/2 2g.k/ : (2.31) C C C C C Proof. By using the Theorem 7 in [36] and definition of the generalized k Horadam numbers, we get Q Q Q Q Hk;nHk;n r s Hk;n sHk;n r C C  Á Hk;n iHk;n 1 jHk;n 2 ijHk;n 3 D C C C C C C  Á Hk;n r s iHk;n r s 1 jHk;n r s 2 ijHk;n r s 3  C C C C C C C C C C  Á Hk;n s iHk;n s 1 jHk;n s 2 ijHk;n s 3 C C C C C C C C C C  Á Hk;n r iHk;n r 1 jHk;n r 2 ijHk;n r 3  C C C C C C Hk;nHk;n r s Hk;n 1Hk;n r s 1 Hk;n 2Hk;n r s 2 D C C C C C C C Hk;n 3Hk;n r s 3 Hk;n sHk;n r Hk;n s 1Hk;n r 1 C C C C C C C C C Hk;n s 2Hk;n r 2 Hk;n s 3Hk;n r 3 C  C C C C C C i Hk;n 1Hk;n r s Hk;nHk;n r s 1 C C C C C C 1326 YASIN YAZLIK, SURE KOME,¨ AND CAHIT KOME¨

Hk;n 2Hk;n r s 3 Hk;n 3Hk;n r s 2 Hk;n sHk;n r 1 C C C C C C C C Á Hk;n s 1Hk;n r Hk;n s 2Hk;n r 3 Hk;n s 3Hk;n r 2 C C C C C C C C C C  j Hk;n 2Hk;n r s Hk;nHk;n r s 2 Hk;n 1Hk;n r s 3 C C C C C C C C C Hk;n 3Hk;n r s 1 Hk;n s 2Hk;n r Hk;n sHk;n r 2 C C C C C Á C C Hk;n s 1Hk;n r 3 Hk;n s 3Hk;n r 1 C C C C C C C C  ij Hk;nHk;n r s 3 Hk;n 3Hk;n r s Hk;n 1Hk;n r s 2 C C C C C C C C C C Hk;n 2Hk;n r s 1 Hk;n sHk;n r 3 Hk;n s 3Hk;n r C C C C C C Á C C Hk;n s 1Hk;n r 2 Hk;n s 2Hk;n r 1 C C C C C C n r   . g.k// bHk;r aHk;r 1 bHk;s aHk;s 1 C C D b2 a2g.k/ abf .k/ " # 1 g.k/ g.k/2 g.k/3  C n r   . g.k// bHk;r aHk;r 1 bHk;s aHk;s 1 i C C C b2 a2g.k/ abf .k/ " # 1 g.k/2f .k/  n r   . g.k// bHk;r aHk;r 1 bHk;s aHk;s 1 j C C C b2 a2g.k/ abf .k/ " # .1 g.k//f .k/2 2g.k/  C C n r   . g.k// bHk;r aHk;r 1 bHk;s aHk;s 1 ij C C C b2 a2g.k/ abf .k/ " # f .k/f .k/2 2g.k/  C n r   . g.k// bHk;r aHk;r 1 bHk;s aHk;s 1 C C D b2 a2g.k/ abf .k/ " 1 g.k/ g.k/2 g.k/3 i1 g.k/2f .k/  C C BICOMPLEX GENERALIZED k HORADAM QUATERNIONS 1327 # j.1 g.k//f .k/2 2g.k/ ijf .k/f .k/2 2g.k/ : C C C C C  Corollary 1. By taking s m n r in the Theorem8, we obtain the following identity: D C Q Q Q Q Hk;nHk;m Hk;m r Hk;n r C n r   . g.k// bHk;r aHk;r 1 bHk;m n r aHk;m n r 1 C C C C D b2 a2g.k/ abf .k/ " 1 g.k/ g.k/2 g.k/3 i1 g.k/2f .k/  C C # j.1 g.k//f .k/2 2g.k/ ijf .k/f .k/2 2g.k/ : C C C C C Corollary 2. (Catalan’s identity) By taking m n in Corollary1, we obtain Catalan’s identity for the generalized k Horadam quaternionsD as: 2  Q Á Q Q Hk;n Hk;n r Hk;n r C n r 2 . g.k// bHk;r aHk;r 1 C D b2 a2g.k/ abf .k/ " 1 g.k/ g.k/2 g.k/3 i1 g.k/2f .k/  C C # j.1 g.k//f .k/2 2g.k/ ijf .k/f .k/2 2g.k/ : C C C C C Corollary 3. (Cassini’s identity) By taking m n and r 1 in Corollary1, we obtain Cassini’s identity for the generalized k HoradamD quaternionsD as: 2  Q Á Q Q Hk;n Hk;n 1Hk;n 1 C n 1 2 2  . g.k// b a g.k/ abf .k/ D " 1 g.k/ g.k/2 g.k/3 i1 g.k/2f .k/  C C # j.1 g.k//f .k/2 2g.k/ ijf .k/f .k/2 2g.k/ : C C C C C 1328 YASIN YAZLIK, SURE KOME,¨ AND CAHIT KOME¨

Corollary 4. (d’Ocagne’s identity) By taking n n 1 and r 1 in Corollary1, we obtain d’Ocagne’s identity for the generalized kD HoradamC quaternionsD as: Q Q Q Q Hk;n 1Hk;m Hk;m 1Hk;n C C n  . g.k// bHk;m n aHk;m n 1 D " C 1 g.k/ g.k/2 g.k/3 i1 g.k/2f .k/  C C # j.1 g.k//f .k/2 2g.k/ ijf .k/f .k/2 2g.k/ : C C C C C

3. CONCLUSION This study presents the bicomplex generalized k Horadam quaternions which are generalization of the results in [1,3, 34]. Moreover, for the special cases of f .k/;g.k/;a and b, we obtain several new quaternions(see Table1) which are not defined before in the literature. We derive the Binet formula, generating function, matrix representation and the summation formula for this quaternion. We also give Theorem8 which is the generalization of the Catalan’s identity, Cassini’s identity and d’Ocagne’s identity. Since this study includes some new generalized results for the bicomplex quaternions, it contributes to the literature by providing essential inform- ation on the generalization of the bicomplex quaternions.

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Authors’ addresses

Yasin Yazlik Nevs¸ehir Hacı Bektas¸Veli University, Department of Mathematics, 50300 Nevs¸ehir, Turkey E-mail address: [email protected]

Sure Kome¨ Nevs¸ehir Hacı Bektas¸Veli University, Department of Mathematics, 50300, Turkey E-mail address: [email protected]

Cahit Kome¨ Nevs¸ehir Hacı Bektas¸Veli University, Department of Mathematics, 50300 Nevs¸ehir, Turkey E-mail address: [email protected]