Abstract
The sine addition formula on a semigroup S is the functional equation \(f(xy) = f(x)g(y) + g(x)f(y)\) for all \(x,y \in S\). For some time the solutions have been known on groups, regular semigroups, and semigroups which are generated by their squares. The obstacle to finding the solution on all semigroups arose in the special case that g is a multiplicative function. We overcome this obstacle and find the general solution on all semigroups using a transfinite induction argument. A new type of solution appears which is not seen on regular semigroups or semigroups generated by their squares.
We also give the general solution of the sine subtraction formula \(f(x\sigma(y)) = f(x)g(y) - g(x)f(y)\) on monoids, where \(\sigma\) is an automorphic involution. The solutions of both equations can be described in terms of additive and multiplicative functions, with a slight new twist. The general continuous solutions on topological semigroups are also found. A variety of examples are given to illustrate the results.
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Ebanks, B. The sine addition and subtraction formulas on semigroups. Acta Math. Hungar. 164, 533–555 (2021). https://doi.org/10.1007/s10474-021-01149-3
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DOI: https://doi.org/10.1007/s10474-021-01149-3