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 + 2z +··· +nz. For the case n = 2, others solutions appear by considering the pe function of Weierstrass. © 2007 Elsevier Inc. All rights reserved