Regular Functions on the Scaled Hypercomplex Numbers

In this paper, we study the regularity of \(\mathbb {R}\) $\mathbb{R}$ -differentiable functions on open connected subsets of the scaled hypercomplex numbers \(\left\{ \mathbb {H}_{t}\right\} _{t\in \mathbb {R}}\) ${\left\{{\mathbb{H}}_t\right\}}_{t\in \mathbb{R}}$ by studying the kernels of suitable differential operators \(\left\{ \nabla _{t}\right\} _{t\in \mathbb {R}}\) ${\left\{{\mathrm{\nabla }}_t\right\}}_{t\in \mathbb{R}}$ , up to scales in the real field \(\mathbb {R}\) $\mathbb{R}$ .