On split r-Jacobsthal quaternions

Dorota Bród

doi:10.17951/a.2020.74.1.1-14

Abstract

In this paper we introduce a one-parameter generalization of the split Jacobsthal quaternions, namely the split r-Jacobsthal quaternions. We give a generating function, Binet formula for these numbers. Moreover, we obtain some identities, among others Catalan, Cassini identities and convolution identity for the split r-Jacobsthal quaternions.

Highlights

A quaternion p is a hyper-complex number represented by an equation p = a + bi + cj + dk, where a, b, c, d ∈ R and {1, i, j, k} is an orthonormal basis in R4, which satisfies the quaternion multiplication rules: i2 = j2 = k2 = ijk = −1, ij = k = −ji, jk = i = −kj, ki = j = −ik
We introduce and study split r-Jacobsthal quaternions
We present the Binet formula for the split r-Jacobsthal quaternions

Summary

Introduction

We introduce and study split r-Jacobsthal quaternions. Another generalization of the split Jacobsthal quaternions was studied in [8]. For n ≥ 0, the split r-Jacobsthal quaternion JSQrn we define by JSQrn = J(r, n) + iJ(r, n + 1) + jJ(r, n + 2) + kJ(r, n + 3), where J(r, n) is the nth r-Jacobsthal number, defined by (7) and i, j, k are split quaternions units which satisfy the multiplication rules (2) and (3). Using the formula J (0, n) = Jn+2, we obtain J SQ0n = J SQn+2, where JSQn is the nth split Jacobsthal quaternion introduced in [11].

Results

Conclusion