On the equation \(f(n^2)=g^2(n)\) for \(q\)-additive
functions \(f\) and \(g\)

Abstract. We prove that if the \(q\)-additive functions \(f\) and \(g\) satisfy the equation \(f(n^2)=g^2(n)\) for every \(n\in{\mathbb{N}}\) and \(\{a\in \{0,1,\dots,q-1\}, k\in{\mathbb{N}}~\vert~g(aq^k)\neq 0\}\) is an infinite set, then there is a non-zero complex number \(c\) such that \(g(n)=cn\) and \(f(n^2)=c^2n^2\) for every \(n\in{\mathbb{N}}\).