A generalization of the Kannappan-sine addition law on semigroups

Given a semigroup S equipped with an involutive automorphic \(\sigma :S \rightarrow S\), we determine the complex-valued solutions of the following generalization of the Kannappan-sine addition law

$$f(x\sigma (y)z_0)=f(x)g(y)+f(y)g(x),\; x,y \in S. $$

As an application we obtain the solutions of the following functional equation

$$f(x\sigma (y)z_0)=f(x)f(z_1y)+f(z_1x)f(y),\; x,y \in S, $$

where \(z_0, z_1\) are two fixed elements in S such that \(z_0\ne z_1\). The continuous solutions on topological semigroups are given. We illustrate the main result with two examples.