On the equation f(n^2 − Dnm + m^2) = f^2(n) − Df(n)f(m) + f^2(m)

Abstract

We give all solutions f : N -> C of the functional equation f(n(2) Dnm + m(2)) = f(2)(n) Df(n)f(m) + f(2)(m); where D is an element of {1, 2}.