On the k-deformed cyclic functions and the generalized Fourier series in the framework of the k-algebra

We explore two possible generalizations of the Euler formula for the complex k-exponential, which give two different sets of k-deformed cyclic functions endowed with different analytical properties. In a case, the k-sine and k -cosine functions take real values on R and are characterized by an asymptotic log-periodic behavior. In the other case, the k-cyclic functions take real values only in the region x=<1/k, while, for x>1/k , 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 k-mathematics, are fulfilled by the two sets of the k-trigonometric functions. In both cases, we study the orthogonality and the completeness relations and introduce their respective generalized Fourier series for square integrable functions.