Can we solve the follwing functional equation
$$f(xy)=g(x)h(y)+g(y)h(x)$$
on semigroups for unknown complex valued functions $f,g,h$ ?



A pexiderization of the sine addition law on semigroups
I corrected a typo on the right-hand-side of the equation in the OP, I'm unsure whether the left-hand-side has a typo, but the generalized sine addition law on semigroups is known in the form
$$g(xy)=g(x)h(y)+h(x)g(y),$$
so in terms of two unknown functions $g$ and $h$, generalizing
$$\sin(x+y)=\sin x\cos y+\cos x\sin y,$$
see [Ebanks - The sine addition and subtraction formulas on semigroups](https://doi.org/10.1007/s10474-021-01149-3) (2021).


