On generalized cocycle equations (Q1312260)

The cocycle equation \(F(x + y,z) + F(x,y) = F(x,y + z) + F(y,z)\) is generalized to \(F_ 1(x + y,z) + F_ 2 (y + z,x) + F_ 3 (z + x,y) + F_ 4 (x,y) + F_ 5 (y,z) + F_ 6 (z,x) = 0\), where all the \(F_ i\) are unknown functions mapping the Cartesian product of an abelian group into a vector space over the rationals. The general solution is found and proofs are essentially based on some results on Cauchy differences of higher orders obtained by the authors and \textit{K. J. Heuvers} [ibid. 42, No. 2/3, 137-153 (1991; Zbl 0744.39003)].