A generalization of the Kannappan-sine addition law on semigroups

Given a semigroup S equipped with an involutive automorphic \(\sigma :S \rightarrow S\) $\sigma :S\to S$ , we determine the complex-valued solutions of the following generalization of the Kannappan-sine addition law \(f(x\sigma (y)z_0)=f(x)g(y)+f(y)g(x),\; x,y \in S. \) $$f(x\sigma (y)z_0)=f(x)g(y)+f(y)g(x),x,y\in S.$$ As an application we obtain the solutions of the following functional equation \(f(x\sigma (y)z_0)=f(x)f(z_1y)+f(z_1x)f(y),\; x,y \in S, \) $$f(x\sigma (y)z_0)=f(x)f(z_{1}y)+f(z_{1}x)f(y),x,y\in S,$$ where \(z_0, z_1\) $z_0,z_1$ are two fixed elements in S such that \(z_0\ne z_1\) $z_0\ne z_1$ . The continuous solutions on topological semigroups are given. We illustrate the main result with two examples.