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

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 \(\left( \tau _s,{\mathcal {W}}_{\mathcal {C}}^d\right) \) 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.