Generalized Sums Based On Transcendental Functions

In this work we are discussing the generalized sums that we can obtain from transcendental functions. The generalized sums are operations which widespread the addition of real numbers. Using these sums, we will see that we can form some Abelian groups. The study is based on the generalized sums proposed in his \kappa-calculus by Giorgio Kaniadakis, who used it in the framework of a generalized statistics, first applied to special relativity. Besides the investigation of some groups, the paper is also proposing examples which could be suitable for teaching purposes, in the framework of courses on theoretical physics, relativity and algebra applied to physics. Actually, the main aim of the paper is that of popularizing the existence of groups having as their operation a generalized sum.