平方生成半群上的余弦-正弦函数方程组
A system of cosine-sine functional equations on a semigroup generated by its squares
摘要：
给定一个由其平方生成的半群$S$，我们确定下列余弦-正弦函数方程组\begin{align*} f(xy)=f(x)g_{1}(y)+g_{1}(x)f(y)+λ_{1}^{2}\,h(x)h(y),\; x,y\in S,\\ h(xy)=h(x)g_{2}(y)+g_{2}(x)h(y)+λ_{2}^{2}\,f(x)f(y),\; x,y\in S, \end{align*}的复值解，其中$λ_{1},λ_{2}\in\mathbb{C}$是给定常数，$f,g_{1},g_{2},h:S\to\mathbb{C}$是未知函数.
英文摘要：
Given a semigroup $S$ generated by its squares, we determine the complex-valued solutions of the following system of cosine-sine functional equations \begin{align*} f(xy)=f(x)g_{1}(y)+g_{1}(x)f(y)+λ_{1}^{2}\,h(x)h(y),\; x,y\in S,\\ h(xy)=h(x)g_{2}(y)+g_{2}(x)h(y)+λ_{2}^{2}\,f(x)f(y),\; x,y\in S, \end{align*} where $λ_{1},λ_{2}\in\mathbb{C}$ are given constants and $f,g_{1},g_{2},h:S\to\mathbb{C}$ are unknown functions.
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