Left Regular Functions from the Quaternions to the Split-Quaternions

In this paper, we study the regularity of $\mathbb{R}$-differentiable functions on open connected subsets of the scaled hypercomplex numbers $\left\{ \mathbb{H}_{t}\right\} _{t\in[-1.1]}$, in terms of the well-defined differential operators $\left\{ \nabla_{t}\right\} _{t\in\left[-1,1\right]}$, where $\left[-1,1\right]$ is the closed interval in $\mathbb{R}$. To do that, we concentrate on studying the kernels $\left\{ \mathrm{ker}\nabla_{t}\right\} _{t\in\left[-1,1\right]}$ of $\left\{ \nabla_{t}\right\} _{t\in[-1,1]}$. And then, we define and study scale-depending differential operator $\nabla_{\left[-1,1\right]}$ acting on the direct product $\mathbb{R}$-algebra $\mathscr{H}\left[-1,1\right]=\underset{t\in\left[-1,1\right]}{\oplus^{a}}\mathbb{H}_{t}$, and the corresponding kernel $\mathrm{ker}\nabla_{\left[-1,1\right]}$, where $\oplus^{a}$ means the pure-algebraic direct product of $\mathbb{R}$-algebras.