Complex Jacobsthal Numbers in Two Dimension

In this paper, we present a new approach to the generalization of Jacobsthal sequences to the complex plane. It is shown that the Jacobsthal numbers are generalized to two dimensions. For special entries of this new sequence, some relations to the classical Jacobsthal sequences are constructed. Binet formula, the generating function, the explicit closed formula, the sum formula for the new two dimensional Gaussian Jacobsthal sequence are investigated. The relation with classical Jacobsthal Lucas numbers and two dimensional Gaussian Jacobsthal numbers are obtained by using Binet formula. From matrix algebra, the matrix representation of two dimensional Gaussian Jacobsthal sequences is obtained.