A trigonometric functional equation with an automorphism

Document Type : Research Paper


1 Department of Mathematics, Faculty of Sciences, Ibn Zohr University, Agadir, Morocco

2 Department of Mathematics, National Technical University of Athens, Athens, Greece



Let $S$ be a semigroup. In the present paper, we determine the complex-valued solutions $(f,g)$ of the functional equation
$$g(x\sigma (y)) = g(x)g(y)-f(x)f(y)+\alpha f(x\sigma(y)),\  x,y\in S,$$
where $\sigma :S\rightarrow S$ is an automorphism that need not be involutive, and $\alpha \in \mathbb{C}$ is a fixed constant. Our results generalize and extend the ones by Stetk\ae r in The cosine addition law with an additional term. Aequat Math., no. 6, 90, 1147-1168 (2016), and also the ones by Aserrar and Elqorachi in A generalization of the cosine addition law on semigroups. Aequat Math. 97, 787–804 (2023). Some consequences of our results are presented.


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Articles in Press, Corrected Proof
Available Online from 25 December 2023
  • Receive Date: 05 September 2023
  • Revise Date: 01 November 2023
  • Accept Date: 09 November 2023