{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,15]],"date-time":"2026-06-15T04:10:35Z","timestamp":1781496635699,"version":"3.54.1"},"reference-count":42,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2015,5,4]],"date-time":"2015-05-04T00:00:00Z","timestamp":1430697600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>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|&gt;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.<\/jats:p>","DOI":"10.3390\/e17052812","type":"journal-article","created":{"date-parts":[[2015,5,4]],"date-time":"2015-05-04T10:33:04Z","timestamp":1430735584000},"page":"2812-2833","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":16,"title":["On the \u03ba-Deformed Cyclic Functions and the Generalized  Fourier Series in the Framework of the \u03ba-Algebra"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6699-5889","authenticated-orcid":false,"given":"Antonio","family":"Scarfone","sequence":"first","affiliation":[{"name":"Istituto Sistemi Complessi (ISC\u2013CNR) c\/o Politecnico di Torino, Corso Duca degli Abruzzi 24, Torino I-10129, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2015,5,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1016\/S0378-4371(01)00184-4","article-title":"Non-linear kinetics underlying generalized statistics","volume":"296","author":"Kaniadakis","year":"2001","journal-title":"Physica A"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"056125","DOI":"10.1103\/PhysRevE.66.056125","article-title":"Statistical mechanics in the context of special relativity","volume":"66","author":"Kaniadakis","year":"2002","journal-title":"Phys. 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