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 Variants of Wilson's functional equation on semigroups Commun. Korean Math. Soc. 2020 Vol. 35, No. 3, 711-722 https://doi.org/10.4134/CKMS.c190326Published online July 5, 2020 Omar Ajebbar, Elhoucien Elqorachi Ibn Zohr University; Ibn Zohr University Abstract : Given a semigroup $S$ generated by its squares equipped with an involutive automorphism $\sigma$ and a multiplicative function $\mu:S\to\mathbb{C}$ such that $\mu(x\sigma(x))=1$ for all $x\in S$, we determine the complex-valued solutions of the following functional equations \begin{equation*}f(xy)+\mu(y)f(\sigma(y)x)=2f(x)g(y),\, x,y\in S\end{equation*} and \begin{equation*}f(xy)+\mu(y)f(\sigma(y)x)=2f(y)g(x),\, x,y\in S.\end{equation*} Keywords : Semigroup, involutive automorphism, multiplicative function, d'Alembert's equation, Wilson's equation MSC numbers : Primary 39B52; Secondary 39B32 Downloads: Full-text PDF   Full-text HTML