An extension of Van Vleck's functional equation for the sine

In \cite{St3} H. Stetk\ae r obtained the solutions of Van Vleck’s functional equation for the sine $$f(x\tau (y)z_0)-f(xy{z}_0)=2f(x)f(y),x,y\in G,$$ where $G$ is a semigroup, $\tau$ is an involution of $G$ and $z_0$ is a fixed element in the center of $G$. The purpose of this paper is to determine the complex-valued solutions of the following extension of Van Vleck’s functional equation for the sine $$\mu (y)f(x\tau (y)z_0)-f(xy{z}_0)=2f(x)f(y),x,y\in G,$$ where $\mu$ : $G⟶\mathbb{C}$ is a multiplicative function such that $\mu (x\tau (x))=1$ for all $x\in G$. Furthermore, we obtain the solutions of a variant of Van Vleck’s functional equation for the sine $$\mu (y)f(\sigma (y)x{z}_0)-f(xy{z}_0)=2f(x)f(y),x,y\in G$$ on monoids, and where $\sigma$ is an automorphism involutive of $G$.