Characterization of arithmetical functions with functional equation

Abstract. The functional equation of type $$ f(\alpha+n^3+m^3)=g(\alpha)+h(n^3)+h(m^3) $$ is investigated, where \(n,m\in{\Bbb N}, \alpha\in{\mathcal A}\) and \({\mathcal A}\subseteq {\Bbb N}\) satisfies some conditions. It follows from our results that if \({\mathcal A}={\mathcal P}\) (the set of all prime numbers), then there exist numbers \(A,D, Q\) such that \(h(n^3)=An^3+D\) and \(g(p)=Ap+Q\) for every \(p\in{\mathcal P}\), \(n\in{\Bbb N}\). Similarly, if \({\mathcal A}=\{n^2\vert n\in{\Bbb N}\}\), then \(h(n^3)=An^3+D\) and \(g(m^2)=Am^2+R\) for every \(n,m\in{\Bbb N}\), where \(A,D,R\) are suitable numbers.