A representation formula for slice regular functions over slice-cones in several variables

Abstract

The aim of this paper is to extend the so called slice analysis to a general case in which the codomain is a real vector space of even dimension, i.e. is of the form $\mathbb{R}^{2n}$. We define a cone $\mathcal{W}_\mathcal{C}^d$ in $[End(\mathbb{R}^{2n})]^d$ and we extend the slice-topology $\tau_s$ to this cone. Slice regular functions can be defined on open sets in $\left(\tau_s,\mathcal{W}_\mathcal{C}^d\right)$ and a number of results can be proved in this framework, among which a representation formula. This theory can be applied to some real algebras, called left slice complex structure algebras. These algebras include quaternions, octonions, Clifford algebras and real alternative $*$-algebras but also left-alternative algebras and sedenions, thus providing brand new settings in slice analysis.