Regular Functions on the Scaled Hypercomplex Numbers

Abstract

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