On a linear functional equation (Q1262481)

The equation reads: \(f(x+ny)+\sum^{n-1}_{k=0}c_ k(y)f(x+ky)=0,\) where n is a fixed positive integer and \(c_ 0,...,c_{n-1}\) are given complex-valued functions defined on a group G. If G is either a finitely generated discrete Abelian group, or G is a locally compact compactly generated Abelian group in which the set of all compact elements is connected, then each continuous solution \(f:G\to {\mathbb{C}}\) is an exponential polynomial. This means that f is a linear combination of exponential functions (i.e. continuous homomorphisms of G into the multiplicative group of nonzero complex numbers) with (continuous) polynomial coefficients.