Another cosine-sine functional equation on semigroups
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Let $S$ be a semigroup. In this paper we find the solutions $f,g,h:\;S\longrightarrow \mathbb{C}$ of the Cosine-Sine other functional equation $f(xy)=f(x)f(y)+g(x)g(y)+h(x)h(y),\,x,y\in S,$ in terms of multiplicative functions and the solutions of special cases of the sine and the cosine-sine functional equations. As an application, we give the solutions $f,g,h:\;S\longrightarrow \mathbb{C}$ of the functional equation $ f(xy)=g(x)h(y)+h(x)g(y)+f(x)f(y),\;\;x,y\in S.$
Abstract
Let $S$ be a semigroup. In this paper we find the solutions $f,g,h:\;S\longrightarrow \mathbb{C}$ of the Cosine-Sine other functional equation $f(xy)=f(x)f(y)+g(x)g(y)+h(x)h(y),\,x,y\in S,$ in terms of multiplicative functions and the solutions of special cases of the sine and the cosine-sine functional equations. As an application, we give the solutions $f,g,h:\;S\longrightarrow \mathbb{C}$ of the functional equation $ f(xy)=g(x)h(y)+h(x)g(y)+f(x)f(y),\;\;x,y\in S.$