Ramanujan's identities and functional equations

Abstract

Let $ R(+,\cdot) $ be a commutative ring, and $ G(+) $ be an Abelian group. Let $ S(R,G) $ denote the set of all functions $ g\,:\,D_R\,:=\{x^2 + xy + y^2 \mid x,y \,\epsilon \,R\} \rightarrow G $ for which the functional equation $ 2g(x^2 + xy + y^2) = g(x^2) + g(y^2) + g((x + y)^2) $ holds for all $ x,y\in R $ . We give the full characterization of $ S(R,G) $ if R is a field of characteristic zero, and G is a linear space over the field of the rationals. Partial results are given for certain structures R and G.