The kernel of the second order Cauchy difference on semigroups

Let S be a semigroup, H a 2-torsion free, abelian group and (C^2f) the second order Cauchy difference of a function (f:S rightarrow H). Assuming that H is uniquely 2-divisible or S is generated by its squares we prove that the solutions f of (C^2f = 0) are the functions of the form (f(x) = j(x) + B(x,x)), where j is a solution of the symmetrized additive Cauchy equation and B is bi-additive. Under certain conditions we prove that the terms j and B are continuous, if f is. We relate the solutions f of (C^2f = 0) to Fréchet’s functional equation and to polynomials of degree less than or equal to 2.