TY  - JOUR
T1  - Left Regular Functions from the Quaternions to the Split-Quaternions
AU  - Cho, Ilwoo
PY  - 2025
DA  - September
Y2  - 2025
DO  - 10.33401/fujma.1588231
JF  - Fundamental Journal of Mathematics and Applications
JO  - Fundam. J. Math. Appl.
PB  - Fuat USTA
WT  - DergiPark
SN  - 2645-8845
SP  - 115
EP  - 147
VL  - 8
IS  - 3
LA  - en
AB  - 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.
KW  - Scaled Hypercomplexes
KW  - Differential Operators.
KW  - Regular functions
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UR  - https://doi.org/10.33401/fujma.1588231
L1  - https://dergipark.org.tr/en/download/article-file/4379651
ER  -