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Entropy 2015, 17(5), 2812-2833; doi:10.3390/e17052812

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

Istituto Sistemi Complessi (ISC–CNR) c/o Politecnico di Torino, Corso Duca degli Abruzzi 24, Torino I-10129, Italy
Academic Editor: Giorgio Kaniadakis
Received: 30 March 2015 / Revised: 26 April 2015 / Accepted: 27 April 2015 / Published: 4 May 2015
(This article belongs to the Special Issue Entropic Aspects in Statistical Physics of Complex Systems)
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Abstract

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. View Full-Text
Keywords: complex κ-exponential; κ-cyclic functions; κ-algebra; generalized Fourier series complex κ-exponential; κ-cyclic functions; κ-algebra; generalized Fourier series
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Scarfone, A.M. On the κ-Deformed Cyclic Functions and the Generalized Fourier Series in the Framework of the κ-Algebra. Entropy 2015, 17, 2812-2833.

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