Van Vleck’s and Kannappan’s functional equations for division ring-valued functions

Let S be a semigroup, \(z_{0}\) $z_0$ be a fixed element in the center of S, and \(\tau :S\rightarrow S\) $\tau :S\to S$ be an involution (an involutive anti-automorphism). Furthermore we let \({\mathbb {F}}\) $\mathbb{F}$ be a division ring of characteristic different from 2 and \({\mathbb {H}}\) $\mathbb{H}$ be the skew field of quaternions. Our main objective is to solve each of the functional equations \(\begin{aligned} K(xyz_{0})+K(x\tau (y)z_{0})= & 2K(x)K(y),\,\,\,x,y\in S, \\ V(x\tau (y)z_{0})-V(xyz_{0})= & 2V(x)V(y),\,\,\,x,y\in S, \end{aligned}\) $$\begin{array}{cc}\hfill K(xy{z}_0)+K(x\tau (y)z_0)=& 2K(x)K(y),x,y\in S,\hfill \\ \hfill V(x\tau (y)z_0)-V(xy{z}_0)=& 2V(x)V(y),x,y\in S,\hfill \end{array}$$ in which \(K:S\rightarrow {\mathbb {F}}\) $K:S\to \mathbb{F}$ and \(V:S\rightarrow {\mathbb {H}}\) $V:S\to \mathbb{H}$ are the unknown functions.