The slice Dirac operator over octonions is a slice counterpart of the Dirac operator over quaternions. It involves a new theory of stem functions, which is the extension from the commutative $ O(1) $ case to the non-commutative $ O(3) $ case. For functions in the kernel of the slice Dirac operator over octonions, we establish the representation formula, the Cauchy integral formula (and, more in general, the Cauchy-Pompeiu formula), and the Taylor as well as the Laurent series expansion formulas.

中文翻译：

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在八元数上对 Dirac 算子进行切片

八元数上的切片狄拉克算子是四元数上狄拉克算子的切片对应物。它涉及到一个新的词干函数理论，它是从可交换$O(1)$情况到非交换$O(3)$情况的扩展。对于八元数上的切片狄拉克算子核中的函数，我们建立了表示公式、柯西积分公式（以及更一般的柯西-庞贝公式）、泰勒级数和洛朗级数展开公式。