标题： 余弦正弦函数方程在半群上 显示英文标题

标题： Cosine-sine functional equation on semigroups

摘要： 设$S$是一个半群。 我们根据乘法函数、正弦加法定律的特殊情形$$\varphi(xy)=\varphi(x)\chi(y)+\chi(x)\varphi(y), x,y\in S$$的解来确定泛函方程\begin{equation*}f(xy)=f(x)g(y)+g(x)f(y)+h(x)h(y), x,y\in S,\end{equation*}的复值解$f,g,h$，其中$\chi:S\to\mathbb{C}$是一个乘法函数，同时也根据余弦-正弦泛函方程的特殊情况$$\psi(xy)=\psi(x)\chi(y)+\chi(x)\psi(y)+\varphi(x)\varphi(y), x,y\in S$$的解来确定，其中$\chi:S\to\mathbb{C}$是一个乘法函数，$\varphi:S\to\mathbb{C}$满足对$(\varphi,\chi)$满足正弦加法定律。 显示英文摘要

摘要： Let $S$ be a semigroup. We determine the complex-valued solutions $f,g,h$ of the functional equation \begin{equation*}f(xy)=f(x)g(y)+g(x)f(y)+h(x)h(y), x,y\in S,\end{equation*} in terms of multiplicative functions, solutions of the special case $$\varphi(xy)=\varphi(x)\chi(y)+\chi(x)\varphi(y), x,y\in S$$ of the sine addition law, where $\chi:S\to\mathbb{C}$ is a multiplicative function, and also in terms of solutions of the particular case $$\psi(xy)=\psi(x)\chi(y)+\chi(x)\psi(y)+\varphi(x)\varphi(y), x,y\in S$$ of the cosine-sine functional equation where $\chi:S\to\mathbb{C}$ is a multiplicative function and $\varphi:S\to\mathbb{C}$ such that the pair $(\varphi,\chi)$ satisfies the sine addition law.