On a Generalization for Tribonacci Quaternions

Abstract

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:

Tetranacci and Tetranacci-Lucas Quaternions

“…Cerda-Morales [17] defined and studied the generalized Tribonacci quaternion sequence that includes the previously introduced Tribonacci, Padovan, Narayana and third order Jacobsthal quaternion sequences. In [17], the author defined generalized Tribonacci quaternion as…”

On the Third-Order Jacobsthal and Third-Order Jacobsthal–Lucas Sequences and Their Matrix Representations

“…In [6] the Jacobsthal recurrence relation (1.1) is extended to higher order recurrence relations and the basic list of identities provided by A. F. Horadam [7] is expanded and extended to several identities for some of the higher order cases. In particular, third order Jacobsthal numbers, {J (3) n } n≥0 , and third order Jacobsthal-Lucas numbers, {j (3) n } n≥0 , are defined by…”

On the Bicomplex Generalized Tribonacci Quaternions

“…where F n and L n denote the n th Fibonacci and Lucas numbers, respectively [5]. There are many generalizations of the Fibonacci, and Fibonacci-like quaternions have been studied in the literature (see for example [5][6][7][8][9][10][11][12][13][14][15][16][17][18]). The generalized tribonacci sequence, {T n = T n (r, s, t; a, b, c)} n≥0 , is defined by the following recurrence relation:…”

“…The tribonacci quaternion sequences and some generalizations have been studied by many researchers in their papers (see for example [9]).…”