Can we solve the follwing functional equation $$f(xy)=g(x)h(y)+g(y)h(x)$$ on semigroups for unknown complex valued functions $f,g,h$ ?
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5$\begingroup$ for the clueless like me on what "pexiderization" refers to: eqworld.ipmnet.ru/en/solutions/fe/fe3108.pdf $\endgroup$– Carlo BeenakkerCommented Nov 6, 2022 at 17:04
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I corrected a typo on the right-hand-side of the equation in the OP, I'm unsure whether the left-hand-side has a typo, but the generalized sine addition law on semigroups is known in the form $$g(xy)=g(x)h(y)+h(x)g(y),$$ so in terms of two unknown functions $g$ and $h$, generalizing $$\sin(x+y)=\sin x\cos y+\cos x\sin y,$$ see Ebanks - The sine addition and subtraction formulas on semigroups (2021).
answered Nov 6, 2022 at 15:35
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4$\begingroup$ Ebanks also has a later paper on the more general version the OP discusses: Ebanks - Around the sine addition law and d'Alembert's equation on semigroups. $\endgroup$– LSpiceCommented Nov 6, 2022 at 16:23