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 $$\mu (y)f(x\tau (y)z_0)\pm f(xy{z}_0)=2f(x)f(y),x,y\in S$$ and the following variant of it $$\mu (y)f(\tau (y)x{z}_0)\pm f(xy{z}_0)=2f(x)f(y),x,y\in S,$$ in the setting of semigroups $S$ that need not be abelian or unital, $\tau$ is an involutive morphism of $S$, $\mu$ : $S⟶\mathbb{C}$ 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.