Abstract. In this paper we introduce the third order Jacobsthal quaternions and the third order Jacobsthal-Lucas quaternions and give some of their properties. We derive the relations between third order Jacobsthal numbers and third order Jacobsthal quaternions and we give the matrix representation of these quaternions.
Abstract. Let Vn denote the third order linear recursive sequence defined by the initial values V 0 , V 1 and V 2 and the recursion Vn = rV n−1 +sV n−2 +tV n−3 if n ≥ 3, where r, s, and t are real constants. The {Vn} n≥0 are generalized Tribonacci numbers and reduce to the usual Tribonacci numbers when r = s = t = 1 and to the 3-bonacci numbers when r = s = 1 and t = 0. In this study, we introduced a quaternion sequence which has not been introduced before. We show that the new quaternion sequence that we introduced includes the previously introduced Tribonacci, Padovan, Narayana and Third order Jacobsthal quaternion sequences. We obtained the Binet formula, summation formula and the norm value for this new quaternion sequence.2010 Mathematics Subject Classification. 11R52, 11B37, 11B39, 11B83.Keywords and phrases. Quaternion, Generalized Tribonacci sequence, Narayana sequence, Third order Jacobsthal sequence. Generalized Tribonacci sequenceWe consider the generalized Tribonacci Sequence, {V n (V 0 , V 1 , V 2 ; r, s, t)} n≥0 , or briefly {V n } n≥0 , defined as follows:(1) V n = rV n−1 + sV n−2 + tV n−3 , n ≥ 3, . If we set r = s = t = 1 and V 0 = 0 = V 1 , V 2 = 1, then {V n } is the well-known Tribonacci sequence which has been considered extensively (see, for example, [3]).As the elements of this Tribonacci-type number sequence provide third order iterative relation, its characteristic equation is x 3 − rx 2 − sx − t = 0, whose roots are α = α(r, s, t) = r 3 + A + B, ω 1 = r 3 + ǫA + ǫ 2 B and ω 2 = r 3 + ǫ 2 A + ǫB, where, B = r 3 27 + rs 6 + t 2 − √ ∆ 1 3, with ∆ = ∆(r, s, t) = r 3 t 27 − r 2 s 2 108 + rst 6 − s 3 27 + t 2 4 and ǫ = − 1 2 + i √ 3 2 . In this paper, ∆(r, s, t) > 0, then the equation (1) has one real and two nonreal solutions, the latter being conjugate complex. Thus, the Binet formula for the generalized Tribonacci numbers can be expressed as:In fact, the generalized Tribonacci sequence is the generalization of the wellknown sequences like Tribonacci, Padovan, Narayana and third order Jacobsthal. For example, {V n (0, 0, 1; 1, 1, 1)} n≥0 , {V n (0, 1, 0; 0, 1, 1)} n≥0 , are Tribonacci and Padovan sequences, respectively. The Binet formula for the generalized Tribonacci sequence is expressed as follows:
In this paper, we first give new generalizations for third-order Jacobsthal {J (3) n } n∈N and third-order Jacobsthal-Lucas {j (3) n } n∈N sequences for Jacobsthal and Jacobsthal-Lucas numbers. Considering these sequences, we define the matrix sequences which have elements of {J (3) n } n∈N and {j (3) n } n∈N . Then we investigate their properties.
In this paper, the third-order Jacobsthal generalized quaternions are introduced. We use the well-known identities related to the thirdorder Jacobsthal and third-order Jacobsthal-Lucas numbers to obtain the relations regarding these quaternions. Furthermore, the third-order Jacobsthal generalized quaternions are classified by considering the special cases of quaternionic units. We derive the relations between third-order Jacobsthal and third-order Jacobsthal-Lucas generalized quaternions.Mathematical subject classification: Primary: 11B37; Secondary: 11R52, 11Y55.
In [19], M. Özdemir defined a new non-commutative number system called hybrid numbers. In this paper, we define the hybrid Fibonacci and Lucas numbers. This number system can be accepted as a generalization of the complex (i 2 = −1), hyperbolic (h 2 = 1) and dual Fibonacci number (ε 2 = 0) systems. Furthermore, a hybrid Fibonacci number is a number created with any combination of the complex, hyperbolic and dual numbers satisfying the relation ih = −hi = ε + i. Then we used the Binet's formula to show some properties of the hybrid Fibonacci numbers. We get some generalized identities of the hybrid Fibonacci and hybrid Lucas numbers.
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