We show that solutions of analytic elliptic partial differential equations of the form

formula here

in a simply connected domain Ω ⊂ ℝn can be extended holomorphically into [Nscr ] (Ω) ⊂ [Copf ]n, the so-called kernel of the harmonicity hull of Ω. This extends results of Avanissian, Aronszajn etc., on polyharmonic functions and also results of Vekua, Khavinson and Shapiro in ℝ2. We also find the domain of influence for solutions of a certain subclass of these operators, in terms of their Cauchy data on analytic hypersurfaces in [Copf ]n (a complex Huygens' principle). As an application, we investigate reflection properties of these solutions and, in particular, solutions of the Helmholtz equation in ℝ3.