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Let $S$ be a semigroup, $Z(S)$ the center of $S$ and $\sigma:S\rightarrow S$ is an involutive automorphism. Our main results is that we describe the solutions of the Kannappan-Wilson functional equation \[\displaystyle \int_{S} f(xyt)d\mu(t) +\displaystyle \int_{S} f(\sigma(y)xt)d\mu(t)= 2f(x)g(y),\ x,y\in S,\] and the Van Vleck-Wilson functional equation \[\displaystyle \int_{S} f(xyt)d\mu(t) -\displaystyle \int_{S} f(\sigma(y)xt)d\mu(t)= 2f(x)g(y),\ x,y\in S,\] where $\mu$ is a measure that is a linear combination of Dirac measures $(\delta_{z_i})_{i\in I}$, such that $z_i\in Z(S)$ for all $i\in I$. Interesting consequences of these results are presented., Comment: Acta Mathematica Hungarica

Let $S$ be a semigroup. Our main results is that we describe the complex-valued solutions of the following functional equations \[g(x\sigma (y)) = g(x)g(y)+f(x)f(y),\ x,y\in S,\] \[f(x\sigma (y)) = f(x)g(y)+f(y)g(x),\ x,y\in S,\] and \[f(x\sigma (y)) = f(x)g(y)-f(y)g(x),\ x,y\in S,\] where $\sigma :S\rightarrow S$ is an automorphism that need not be involutive. As a consequence we show that the first two equations are equivalent to their variants. We also give some applications., Comment: arXiv admin note: substantial text overlap with arXiv:2210.06181

We determine the complex-valued solutions of the following functional equation \[f(xy)+\mu (y)f(\sigma (y)x) = 2f(x)g(y),\quad x,y\in S,\] where $S$ is a semigroup and $\sigma$ an automorphism, $\mu :S\rightarrow \mathbb{C}$ is a multiplicative function such that $\mu (x\sigma (x))=1$ for all $x\in S$.

We treat two related trigonometric functional equations on semigroups. First we solve the $\mu$-sine subtraction law \[\mu(y) k(x \sigma(y))=k(x) l(y)-k(y) l(x), \quad x, y \in S,\] for $k, l : S\rightarrow \mathbb{C}$, where $S$ is a semigroup and $\sigma$ an involutive automorphism, $\mu :S\rightarrow \mathbb{C}$ is a multiplicative function such that $\mu (x\sigma (x))=1$ for all $x\in S$, then we determine the complex-valued solutions of the following functional equation \[f(xy) - \mu (y)f(\sigma (y)x) = g(x)h(y),\quad x,y\in S,\] on a larger class of semigroups.

Our main result is that we describe the solutions $g,f:S\rightarrow\mathbb{C}$ of the functional equation \[g(x\sigma(y))=g(x)g(y)-f(x)f(y)+\alpha f(x\sigma(y)),\quad x,y\in S,\] where $S$ is a semigroup, $\alpha \in \mathbb{C}$ is a fixed constant and $\sigma :S\rightarrow S$ an involutive automorphism.

In this paper, we determine the complex-valued solutions of the following functional equations \[g(x\sigma (y)) = g(x)g(y)+f(x)f(y),\quad x,y\in S,\]\[f(x\sigma (y)) = f(x)g(y)+f(y)g(x),\quad x,y\in S,\]\[f(x\sigma (y)) = f(x)g(y)+f(y)g(x)-g(x)g(y),\quad x,y\in S,\]\[f(x\sigma(y))=f(x)g(y)+f(y)g(x)+\alpha g(x\sigma(y)),\quad x,y\in S,\]\[f(x\sigma(y))=f(x)g(y)-f(y)g(x)+\alpha g(x\sigma(y)),\quad x,y\in S,\] where $S$ is a semigroup, $\alpha \in \mathbb{C}\backslash \lbrace 0\rbrace$ is a fixed constant and $\sigma :S\rightarrow S$ an involutive automorphism.