A variant of Wilson's functional equation on semigroups
We determine the complex-valued solutions of the following functional equation \[f(xy)+μ(y)f(σ(y)x) = 2f(x)g(y),\quad x,y\in S,\] where $S$ is a semigroup and $σ$ an automorphism, $μ:S\rightarrow \mathbb{C}$ is a multiplicative function such that $μ(xσ(x))=1$ for all $x\in S$.
math.FA↗