Abstract
We study the functional equation 
for unknown functions \(f,g,h,k,\ell :S \rightarrow \mathbb {C}\), where S is a semigroup and \(m_1,m_2:S \rightarrow \mathbb {C}\) are multiplicative functions. The study is divided into two main parts: \(m_1 = m_2\) and \(m_1 \ne m_2\). In some cases we assume that one or more of the unknown functions is central and/or that S is a monoid. The solutions are found in all cases under the umbrella assumption that S is a commutative monoid. The continuous solutions on topological semigroups are also found.
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Ebanks, B. Extensions of the sine addition law with an extra term. Aequat. Math. 97, 597–618 (2023). https://doi.org/10.1007/s00010-022-00925-y
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DOI: https://doi.org/10.1007/s00010-022-00925-y