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We explore two possible generalizations of the Euler formula for the complex \(\kappa\)-exponential, which give two different sets of \(\kappa\)-deformed cyclic functions endowed with different analytical properties. In a case, the \(\kappa\)-sine and \(\kappa\)-cosine functions take real values on \(\Re\) and are characterized by an asymptotic log-periodic behavior. In the other case, the \(\kappa\)-cyclic functions take real values only in the region \(|x|\leq1/|\kappa|\), while, for \(|x|>1/|\kappa|\), 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 \(\kappa\)-mathematics, are fulfilled by the two sets of the \(\kappa\)-trigonometric functions. In both cases, we study the orthogonality and the completeness relations and introduce their respective generalized Fourier series for square integrable functions

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