The Representation, Generalized Binet Formula and Sums of The Generalized Jacobsthal p-Sequence

Abstract

The Representation, Generalized Binet Formula and Sums of The Generalized Jacobsthal p-Sequence Correspondence to matrix theory has played an important and effective role. Quite apart from pursuing the discovery of the additional formulas by the matrix technics, the different matrices for obtaining new results can be introduced. Chen and Louck have investigated an nxn companion matrix and shown the combinatorial representation of the sequence generated by the nth power of the matrix [6]. Considering the matrix theory, Koken and Bozkurt have presented the Jacobsthal F-matrix and some results [7]. In the literature, there exist many other references on the subject which are not given here.The object of this article is to give a new definition for the generalization of the usual Jacobsthal sequence. The generating matrix, the Binet formula, characteristic equations, generating functions, combinatorial representations and sums of the terms of the generalized Jacobsthal sequence are respectively studied. Generalized Jacobsthal p-sequence Generalization of Jacobsthal SequenceFirst of all, the generalization of the usual Jacobsthal sequence is denoted by J p (n) and defined as follows: for p Z + and n > p+1,

On split r-Jacobsthal quaternions

“…Hence we get q = S q + V q . The conjugate of the split quaternion denoted by q, is given by q = a 0 − a 1 i − a 2 j − a 3 k. The norm of q is defined as (4) N (q) = qq = a 2 0 + a 2 1 − a 2 2 − a 2 3 . The split quaternions are elements of a 4-dimensional associative algebra.…”

“…Many authors introduced and studied some generalizations of the recurrence of the Jacobsthal sequence, see [4,5]. The second order recurrence (6) has been generalized in two ways: first, by preserving the initial conditions and second, by preserving the recurrence relation.…”

On a two-parameter generalization of Jacobsthal numbers and its graph interpretation

“…Introduction. The Jacobsthal sequence {J n } is defined by the second order linear recurrence (1) J n = J n−1 + 2J n−2 for n ≥ 2…”

“…There are many generalizations of this sequence in the literature. The second order recurrence (1) has been generalized in two ways: first, by preserving the initial conditions and second, by preserving the recurrence relation. We recall some of such generalizations: 1) k-Jacobsthal sequence {j k,n } [5], j k,n+1 = kj k,n + 2j k,n−1 for k ≥ 1 and n ≥ 1 with j k,0 = 0, j k,1 = 1, 2) k-Jacobsthal sequence {J k,n } [3], J k,n+1 = J k,n + kJ k,n−1 for k ≥ 1 and n ≥ 1 with J k,0 = 0, J k,1 = 1, 3) generalized Jacobsthal p-sequence {J p } [1], for any p ∈ Z + and n > p + 1…”

“…In this paper we introduce a new generalization of the classical Jacobsthal numbers. Unlike other variations, this generalization depends on two integer parameters used in the recurrence relation (1). Let n, s, p ≥ 0 be integers.…”