On the κ-Deformed Cyclic Functions and the Generalized Fourier Series in the Framework of the κ-Algebra

We explore two possible generalizations of the Euler formula for the complex κ-exponential, which give two different sets of κ-deformed cyclic functions endowed with different analytical properties. In a case, the κ-sine and κ-cosine functions take real values on ℜ and are characterized by an asymptotic log-periodic behavior. In the other case, the κ-cyclic functions take real values only in the region |x| ≤ 1/|κ|, while, for |x| > 1/|κ|, they assume purely imaginary values with an increasing modulus. However, the main mathematical properties of the standard cyclic functions, opportunely reformulated in the formalism of the κ-mathematics, are fulfilled by the two sets of the κ-trigonometric functions. In both cases, we study the orthogonality and the completeness relations and introduce their respective generalized Fourier series for square integrable functions.