On \(J(r,n)\)-Jacobsthal hybrid numbers (Q2057303)

The authors start from the generalized Jacobsthal numbers \(J(r, n)\), with \(r\in\mathbb{N}\), defined by the recurrence \[ J(r, n) = 2^rJ(r, n-1) + (2^r + 4^r)J(r, n-2) \qquad \text{for } n \geq 2 \] with the initial values \[ J(r, 0) = 1 \qquad\text{and}\qquad J(r, 1) = 1 + 2^{r+1}. \] Then, they extend these numbers to the non-commutative ring of hybrid numbers. Specifically, they define the Jacobsthal hybrid numbers \(JH^r_n\) setting \[ JH^r_n = J(r, n) + \mathbf{i} J(r, n + 1) + \varepsilon J(r, n + 2) + \mathbf{h}J(r, n+ 3) \] where the symbols \(\mathbf{i}\), \(\varepsilon\) and \(\mathbf{h}\) satisfy the conditions \[ \mathbf{i}^2 = -1, \quad \varepsilon^2 = 0, \quad \mathbf{h}^2 = 1 \quad\text{and}\quad \mathbf{i}\mathbf{h} = -\mathbf{h}\mathbf{i} = \varepsilon + \mathbf{i}. \] The rest of the paper is devoted to obtaining the first properties of these numbers, such as their character, a Binet's formula, a summation formula and their generating function.