LAUSR

In this paper we consider a generalization of d’Alembert’s equation and Wilson’s equation on commutative semigroups using only the semigroup operation, ie. we consider the functional equation $$\begin{aligned}&h(x+2y)+h(x)=2f(y)h(x+y),\ x,y\in S, \end{aligned}$$ h ( x + 2 y ) + h ( x ) = 2 f ( y ) h ( x + y ) , x , y ∈ S , where $$f,h:S\rightarrow \mathbb {K}$$ f , h : S → K , $$(S,+)$$ ( S , + ) is a commutative semigroup, $$\mathbb {K}$$ K is a quadratically closed field, $$\text {char}\,\mathbb {K}\ne 2$$ char K ≠ 2 .