Towards a function theory of complexified octonions

In this article we study function theory in the 16-dimensional space of complexified octonions OC=C⊗ O O C = C ⊗ O . We define the complexified octonionic Cauchy–Riemann operator D=∑ ⱼ₌₀⁷ eⱼ∂ zⱼ D = ∑ j = 0 7 e j ∂ z j where ∂ zⱼ:=∂ xⱼ+i∂ yⱼ ∂ z j : = ∂ x j + i ∂ y j for $$j=0,1,...,7$$ j = 0 , 1 , . . . , 7 . This operator together with its octonionic conjugate factorize the ultrahyperbolic operator L=DD̄=D̄D=∑ ⱼ₌₀⁷ ∂ zⱼ². L = D D ¯ = D ¯ D = ∑ j = 0 7 ∂ z j 2 . In real coordinates we obtain a Lichnerowicz-Weitzenböck type formula of the form L = Δ ₓ - Δ y + 2i ∂ ₓ, ∂ y L = Δ x - Δ y + 2 i ⟨ ∂ x , ∂ y ⟩ One main goal of this paper consists in investigating the fundamental solution and polynomial solutions of the operator L L . The set of polynomial solutions is completely described by proposing explicit basis constructions and dimension formulae. Another main goal that is to carefully fig