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Reconsidering a functional equation arising from number theory

In a recent paper Chavez and Sahoo considered the functional equation f(ux− vy, uy + v(x+ y)) = f(x, y)f(u, v), which arose in a number theoretical context. Unfortunately one of their results is incorrect. Here we reconsider the equation on various domains. We observe that it is in fact a multiplicative Cauchy equation in disguise. We also point out some remaining open problems.