The d'Alembert functional equation on metabelian groups

Consider the d'Alembert functional equation f(xy) + f(xy-1) = 2f(x)f(y) for f : G → K where G is a group and K is a field with characteristic ≠ 2. Pl. Kannappan has proved that for K = ℂ, the field of complex numbers, any non-zero solution of d'Alembert's equation which satisfies the condition f(xyz) = f(xzy), ∀x, y, z ∈ G has the form f(x) = g(x) + [g(x)]-1/2 where g is a homomorphism of G into the multiplicative group of ℂ. Investigations of d'Alembert's equation on non-abelian groups led to solutions of the equation not having the form (*). In the present note we obtain the general solution of d'Alembert's equation when G is a metabelian group, and we show that there exist solutions which do not have the form (*). © Birkhäuser Verlag, Basel, 1999.