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}. This is a new setting which contains and encompasses in a nontrivial way other cases already studied in the literature and which requires new tools. To this end, we define a cone {mathcal {W}}_{mathcal {C}}^d in [{text {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}}^dright) 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.