An extensions of Kannappan's and Van Vleck's functional equations on semigroups

This paper treats two functional equations, the Kannppan-Van Vleck functional equation $$\mu(y)f(x\tau(y)z_0)\pm f(xyz_0) =2f(x)f(y), \;x,y\in S$$ and the following variant of it $$\mu(y)f(\tau(y)xz_0)\pm f(xyz_0) = 2f(x)f(y), \;x,y\in S,$$ in the setting of semigroups $S$ that need not be abelian or unital, $\tau$ is an involutive morphism of $S$, $\mu$ : $S\longrightarrow \mathbb{C}$ is a multiplicative function such that $\mu(x\tau(x))=1$ for all $x\in S$ and $z_0$ is a fixed element in the center of $S$. We find the complex-valued solutions of these equations in terms of multiplicative functions and solutions of d'Alembert's functional equation.