Contribution to the study of two functional equations (Q1271585)

The first functional equation studied in this paper is \(f_{1}(u+v)f_{2}(u-v)=\sum^{k}_{i=1}g_{i}(u)h_{i}(v)\) where \(f_{1}, f_{2}, g_{i}, h_{i}:G\rightarrow F\), with an Abelian group \(G\) satisfying \(2G=G\) and a commutative field \(F\). A result of \textit{M. Bonk} [Math. Ann. 298, No. 4, 591-610 (1994; Zbl 0791.39009)] dealing with the case \(G= {\mathbb R}^{n}\) and \(F={\mathbb C}\) is extended to this more general setting. The second functional considered is \(f(x+x', y+y', t+t'+(xy'-yx')/2)+f(x+y', y+x', t-t'+(xx'-yy')/2)=2f(x,y,t)f(x',y',t'), x,x',y,y',t,t'\in R\) where \(f:R^{3}\rightarrow G\), with \(R\) a 2-divisible ring and \(F\) a quadratically closed commutative field of characteristic different from 2. It is shown that the general solution is of the form \(f(x,y,t)=\chi_{1}(x+y){(\chi(x+y)+\chi(y-x))/2}\) with multiplicative functions \(\chi_{1},\chi:R\rightarrow F\).