An extension of Van Vleck's functional equation for the sine

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In St3 H. Stetkær obtained the solutions of Van Vleck's functional equation for the sine f(xτ(y)z₀)-f(xyz₀) =2f(x)f(y), x,y∈ G, where G is a semigroup, τ is an involution of G and z₀ is a fixed element in the center of G. The purpose of this paper is to determine the complex-valued solutions of the following extension of Van Vleck's functional equation for the sine μ(y)f(xτ(y)z₀)-f(xyz₀) =2f(x)f(y), x,y∈ G, where μ: G C is a multiplicative function such that μ(xτ(x))=1 for all x∈ G. Furthermore, we obtain the solutions of a variant of Van Vleck's functional equation for the sine μ(y)f(σ(y)xz₀)-f(xyz₀) = 2f(x)f(y), x,y∈ G on monoids, and where σ is an automorphism involutive of G.