Let S be a semigroup and K be a field. In a recent article we introduced a new cosine functional equation g(xyz)−g(x)g(yz)−g(y)g(xz)−g(z)g(xy)+2g(x)g(y)g(z)=0 for an unknown function g:S→K. It was shown that this equation is closely connected to the sine addition formula, and for K=C its solutions are expressible in terms of multiplicative functions. Here we solve the more general functional equation f(xyz)+g(x)g(yz)+g(y)g(xz)+g(z)g(xy)+h(x)h(y)h(z)=0 for three unknown functions f,g,h:S→C, where S is a monoid. The solutions are linear combinations of two multiplicative functions.
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2025
Published: 2025-11-02
10.2478/amsil