Ramanujan's identities and functional equations (Q1807878)

Motivated by Ramanujan's identity \(2(ab+ac+bc)^2 = a^4 + b^4 + c^4 \) whenever \( a+b+c= 0 \), the author considers an interesting problem in functional equations. Let \( (R, +)\) be a commutative ring, let \((G, +)\) be an Abelian group and let \(S(R, G)\) be the set of functions \( g: D_R : = \{ x^2 + xy + y^2 \;|\;x, y \in R \} \to G\) such that for all \( x, y \) in \(R\): \[ 2g ( x^2 + xy + y^2) = g ( x^2) + g (y^2) + g (( x+y)^2) . \tag{*} \] It is shown that when \(R\) is a field of characteristic zero and \(G\) is a linear space then the general solution of (*) is \( g(z)= a_2 (z^2) + a_1 (z)\) for all \( z\) in \( D_R\), where \(a_i : R \to G\), \( i =1,2, \) are additive functions. Some challenging open problems related to (*) are presented.