Some generalization of Cauchy’s and Wilson’s functional equations on abelian groups

Abstract

We find the solutions f, g, h: G → X, α: G → K of the functional equation

λ∈K f(x + λy) = |K|g(x) + α(x)h(y), x,y∈ G, where (G, +) is an abelian group, K is a finite, abelian subgroup of the automorphism group of G,X is a linear space over the field K ∈ {R,C}.