Abstract
The cosine addition formula on a semigroup S is the functional equation \(g(xy) = g(x)g(y) - f(x)f(y)\) for all \(x,y \in S\). We find its general solution for \(g,f \colon S \to \mathbb{C}\), using the recently found general solution of the sine addition formula \(f(xy) = f(x)g(y) + g(x)f(y)\) on semigroups. A simpler proof of this latter result is also included, with some details added to the solution.
We also solve the cosine subtraction formula \(g(x\sigma(y)) = g(x)g(y) + f(x)f(y)\) on monoids, where \(\sigma\) is an automorphic involution. The solutions of these functional equations are described mostly in terms of additive and multiplicative functions, but for some semigroups there exist points where f and/or g can take arbitrary values.
The continuous solutions on topological semigroups are also found.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J. Aczél, Lectures on Functional Equations and their Applications, Academic Press (New York, 1966)
J. Aczél and J. Dhombres, Functional Equations in Several Variables, with applications to mathematics, information theory and to the natural and social sciences, Encyclopedia of Mathematics and its Applications, vol. 31, Cambridge University Press (Cambridge, 1989)
Ajebbar, O., Elqorachi, E.: Solutions and stability of trigonometric functional equations on an amenable group with an involutive automorphism. Commun. Korean Math. Soc. 34, 55-82 (2019)
J.K. Chung, Pl. Kannappan, and C.T. Ng, A generalization of the cosine-sine functional equation on groups, Linear Algebra Appl., 66 (1985), 259-277
B. Ebanks, The sine addition and subtraction formulas on semigroups, Acta Math. Hungar. (to appear)
B. Ebanks, Generalized sine and cosine addition laws and a Levi-Civita functional equation on monoids, Results Math., 76 (2021), paper no. 16
Ebanks, B., Stetkær, H.: d’Alembert’s other functional equation on monoids with an involution. Aequationes Math. 89, 187-206 (2015)
H. Stetkær, Functional Equations on Groups, World Scientific (Singapore, 2013)
Stetkær, H.: The cosine addition law with an additional term. Aequationes Math. 90, 1147-1168 (2016)
E. Vincze, Eine allgemeinere Methode in der Theorie der Funktionalgleichungen. II, Publ. Math. Debrecen, 9 (1962), 314-323
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ebanks, B. The cosine and sine addition and subtraction formulas on semigroups. Acta Math. Hungar. 165, 337–354 (2021). https://doi.org/10.1007/s10474-021-01167-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10474-021-01167-1