The aim of this work is to show that given \(u\in {\mathbb {H}}{\setminus }{\mathbb {R}}\), there exists a differential operator \(G^{-u}\) whose solutions expand in quaternionic power series expansion \( \sum _{n=0}^\infty (x-u)^n a_n\) in a neighborhood of \(u\in {\mathbb {H}}\). This paper also presents Stokes and Borel-Pompeiu formulas induced by \(G^{-u}\) and as consequence we give some versions of Cauchy’s Theorem and Cauchy’s Formula associated to these kind of regular functions.

中文翻译：

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关于一些四元数级数

这项工作的目的是证明给定\(u\in {\mathbb {H}}{\setminus }{\mathbb {R}}\)，存在微分算子\(G^{-u}\ ） ，其解在 \(u\in {\mathbb {H}}\)的邻域中以四元数幂级数展开\( \sum _{n=0}^\infty (xu)^n a_n\)展开。本文还提出了由\(G^{-u}\)导出的 Stokes 和 Borel-Pompeiu 公式，因此我们给出了与此类正则函数相关的柯西定理和柯西公式的一些版本。