A note on the functional equation F(z)+F(2z)+⋯+F(nz)=0

In this note we prove the existence of a family of basic solutions of the complex functional equation F(z)+F(2z)+⋯+F(nz)=0, which form an infinite dimensional vector space generated by the complex power functions zβ, where β ranges on the set of zeros of the entire function Gn(z)≡1+z2+⋯+nz. For the case n=2, others solutions appear by considering the pe function of Weierstrass.