On an Identity of Ramanujan

In this paper our aim is to determine all the solutions of the functional equation f(a + b + c) + f(b + c + d) + f(a - d) = f(a + b + d) + f(a + c + d) + f(b - c), where a, b, c, d Zsatisfy ad = bc. This equation is a generalization of one of the identities of Ramanujan. He found two solutions, f(x) = x2, and f(x) = x4. We prove that every solution of the equation can be written as a linear combination of 11 independent solutions.