Left Regular Functions from the Quaternions to the Split-Quaternions

In this paper, we study the regularity of R-differentiable functions on open connected subsets of the scaled hypercomplex numbers \ Hₜ\ t∈[-1.1], in terms of the well-defined differential operators \ ∇ₜ\ t∈[-1,1], where [-1,1] is the closed interval in R. To do that, we concentrate on studying the kernels \ ker∇ₜ\ t∈[-1,1] of \ ∇ₜ\ t∈[-1,1]. And then, we define and study scale-depending differential operator ∇[-1,1] acting on the direct product R-algebra H[-1,1]=t∈[-1,1]⊕ᵃHₜ, and the corresponding kernel ker∇[-1,1], where ⊕ᵃ means the pure-algebraic direct product of R-algebras.