Slice Fueter-regular functions, originally called slice Dirac-regular functions, are generalized holomorphic functions defined over the octonion algebra \({\mathbb {O}}\), recently introduced by M. Jin, G. Ren and I. Sabadini. A function \(f:\Omega _D\subset {\mathbb {O}}\rightarrow {\mathbb {O}}\) is called (quaternionic) slice Fueter-regular if, given any quaternionic subalgebra \({\mathbb {H}}_{\mathbb {I}}\) of \({\mathbb {O}}\) generated by a pair \({\mathbb {I}}=(I,J)\) of orthogonal imaginary units I and J (\({\mathbb {H}}_{\mathbb {I}}\) is a ‘quaternionic slice’ of \({\mathbb {O}}\)), the restriction of f to \({\mathbb {H}}_{\mathbb {I}}\) belongs to the kernel of the corresponding Cauchy–Riemann–Fueter operator \(\frac{\partial }{\partial x_0}+I\frac{\partial }{\partial x_1}+J\frac{\partial }{\partial x_2}+(IJ)\frac{\partial }{\partial x_3}\). The goal of this paper is to show that slice Fueter-regular functions are standard (complex) slice functions, whose stem functions satisfy a Vekua system having exactly the same form of the one characterizing axially monogenic functions of degree zero. The mentioned standard sliceness of slice Fueter-regular functions is able to reveal their ‘holomorphic nature’: slice Fueter-regular functions have Cauchy integral formulas, Taylor and Laurent series expansions, and a version of Maximum Modulus Principle, and each of these properties is global in the sense that it is true on genuine 8-dimesional domains of \({\mathbb {O}}\). Slice Fueter-regular functions are real analytic. Furthermore, we introduce the global concepts of spherical Dirac operator \(\Gamma \) and of slice Fueter operator \({\overline{\vartheta }}_F\) over octonions, which allow to characterize the slice Fueter-regular functions as the \({\mathscr {C}}^2\)-functions in the kernel of \({\overline{\vartheta }}_F\) satisfying a second order differential system associated with \(\Gamma \).

Slice Fueter-regular 函数，最初称为 slice Dirac-regular 函数，是定义在八元代数\({\mathbb {O}}\)上的广义全纯函数，最近由 M. Jin、G. Ren 和 I. Sabadini 引入。一个函数\(f:\Omega _D\subset {\mathbb {O}}\rightarrow {\mathbb {O}}\)被称为（四元数）切片 Fueter-regular if，给定任何四元子代数\({\mathbb { H}}_{\mathbb {I}}\)的\({\mathbb {O}}\)由一对\({\mathbb {I}}=(I,J)\)正交虚数单位生成I和J ( \({\mathbb {H}}_{\mathbb {I}}\)是\({\mathbb {O}}\)的“四元数切片”)，f对\({\mathbb {H}}_{\mathbb {I}}\) 的限制属于对应的 Cauchy-Riemann-Fueter 算子\(\frac{\partial }{\partial x_0}+I\frac{\partial }{\partial x_1}+J\frac{\partial }{\partial x_2}+(IJ)\frac{\partial }{\partial x_3}\). 本文的目的是证明切片 Fueter-regular 函数是标准（复杂）切片函数，其词干函数满足 Vekua 系统，该系统具有与表征零阶轴向单基因函数的形式完全相同的形式。上面提到的切片 Fueter-regular 函数的标准切片能够揭示它们的“全纯性质”：切片 Fueter-regular 函数具有柯西积分公式、泰勒和洛朗级数展开以及最大模量原理的一个版本，这些属性中的每一个都是全局，因为它在\({\mathbb {O}}\) 的真正 8 维域上是正确的。Slice Fueter-regular 函数是实解析的。此外，我们引入了球形狄拉克算子\(\Gamma \)的全局概念和切片 Fueter 算子\({\overline{\vartheta }}_F\)在八元上，这允许将切片 Fueter 正则函数表征为\({\mathscr {C}}^2\) -函数在\({\overline{\vartheta }}_F\) 的内核满足与\(\Gamma \)相关的二阶微分系统。