WEIGHTED REPRESENTATION FUNCTIONS ON Zm

Let m, k(1), and k(2) be three integers with m >= 2. For A subset of Z(m) and n is an element of Z(m), let r(k1,k2) (A, n) denote the number of solutions of the equation n = k(1)a(1) + k(2)a(2) with a(1), a(2) is an element of A. In this paper, we characterize all m, k(1), k(2), and A for which r(k1,k2) (Z(m) \ A, n) = r(k1,k2) (A, n) for all n is an element of Z(m). As a corollary, we prove that there exists A subset of Z(m) such that r(k1,k2)(Zm \ A, n) = r(k1,k2) (A, n) for all n is an element of Z(m) if and only if 2d vertical bar m, where d = (k(1), m)(k(2), rn) / (k(1), k(2), m)(2). We also pose several problems for further research.