### Wilson's functional equations on groups.

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Given a (not necessarily unitary) character μ:G → (ℂ∖0,·) of a group G we find the solutions g: G → ℂ of the following version of d’Alembert’s functional equation $g\left(xy\right)+\mu \left(y\right)g\left(x{y}^{-1}\right)=2g\left(x\right)g\left(y\right)$, x,y ∈ G. (*) The classical equation is the case of μ = 1 and G = ℝ. The non-zero solutions of (*) are the normalized traces of certain representations of G on ℂ². Davison proved this via his work [20] on the pre-d’Alembert functional equation on monoids. The present paper presents a detailed exposition of these results working directly...

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