A trigonometric functional equation with an automorphism

Let S be a semigroup. In the present paper, we determine the complex-valued solutions (f,g) of the functional equationg(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. ۶, ۹۰, ۱۱۴۷-۱۱۶۸ (۲۰۱۶), and also the ones by Aserrar and Elqorachi in A generalization of the cosine addition law on semigroups. Aequat Math. ۹۷, ۷۸۷–۸۰۴ (۲۰۲۳). Some consequences of our results are presented.

کلیدواژه ها:

functional equation ، semigroup ، Addition law ، Automorphism