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 R²ⁿ R 2 n . 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 WCᵈ W C d in [End(R²ⁿ)]ᵈ [ End ( R 2 n ) ] d and we extend the slice topology τ ₛ τ s to this cone. Slice regular functions can be defined on open sets in ( τ ₛ,WCᵈ) τ s , W C d 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.