{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:24:44Z","timestamp":1760145884609,"version":"build-2065373602"},"reference-count":21,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2024,9,4]],"date-time":"2024-09-04T00:00:00Z","timestamp":1725408000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we introduce the concept of generalized Fourier series, generated by the p-trigonometric functions, namely cosp and sinp, recently introduced related to the generalized complex numbers systems. The aim of this study is to represent a periodic signal as a sum of p-sine and p-cosine functions. In order to achieve this, we first present the integrals of the product of the same or different family of p-trigonometric functions over the full period of these functions to understand the orthogonality properties. Next, we use these integrals to derive the coefficients of the generalized p-Fourier series along with a few examples. The generalized Fourier series can be used to expand an arbitrary forcing function in the solution of a non-homogeneous linear ordinary differential equation (ODE) with constant coefficients.<\/jats:p>","DOI":"10.3390\/axioms13090600","type":"journal-article","created":{"date-parts":[[2024,9,4]],"date-time":"2024-09-04T05:54:47Z","timestamp":1725429287000},"page":"600","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Fourier Series Related to p-Trigonometric Functions"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0009-0002-8333-6097","authenticated-orcid":false,"given":"Ali Hamzah","family":"Alibrahim","sequence":"first","affiliation":[{"name":"Centre for Environmental Mathematics, Faculty of Environment, Science and Economy, University of Exeter, Penryn Campus, Penryn TR10 9FE, UK"},{"name":"Mathematics Department, College of Science, Jouf University, Sakaka P.O. Box 2014, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8394-5303","authenticated-orcid":false,"given":"Saptarshi","family":"Das","sequence":"additional","affiliation":[{"name":"Centre for Environmental Mathematics, Faculty of Environment, Science and Economy, University of Exeter, Penryn Campus, Penryn TR10 9FE, UK"},{"name":"Institute for Data Science and Artificial Intelligence, University of Exeter, North Park Road, Exeter EX4 4QE, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,9,4]]},"reference":[{"key":"ref_1","unstructured":"Yaglom, I.M. (2012). A Simple Non-Euclidean Geometry and Its Physical Basis: An Elementary Account of Galilean Geometry and the Galilean Principle of Relativity, Springer Science & Business Media."},{"key":"ref_2","unstructured":"Yaglom, I.M. (2014). Complex Numbers in Geometry, Academic Press."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"118","DOI":"10.1080\/0025570X.2004.11953236","article-title":"Geometry of generalized complex numbers","volume":"77","author":"Harkin","year":"2004","journal-title":"Math. Mag."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"62","DOI":"10.1007\/s00006-018-0878-3","article-title":"p-Trigonometric approach to elliptic biquaternions","volume":"28","author":"Tosun","year":"2018","journal-title":"Adv. Appl. Clifford Algebr."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"96","DOI":"10.36890\/iejg.545136","article-title":"Elliptic matrix representations of elliptic biquaternions and their applications","volume":"11","author":"Tosun","year":"2018","journal-title":"Int. Electron. J. Geom."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1179","DOI":"10.1007\/s00006-016-0642-5","article-title":"The Holditch-type theorem for the polar moment of inertia of the orbit curve in the generalized complex plane","volume":"26","year":"2016","journal-title":"Adv. Appl. Clifford Algebr."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"239","DOI":"10.32323\/ujma.430853","article-title":"Holditch-Type Theorem for Non-Linear Points in Generalized Complex Plane Cp","volume":"1","year":"2018","journal-title":"Univers. J. Math. Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"889","DOI":"10.1007\/s00006-015-0530-4","article-title":"One-parameter planar motions in generalized complex number plane CJ","volume":"25","year":"2015","journal-title":"Adv. Appl. Clifford Algebr."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1007\/s00022-018-0450-2","article-title":"Burmester theory in Cayley\u2013Klein planes with affine base","volume":"109","author":"Eren","year":"2018","journal-title":"J. Geom."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Alibrahim, A.H., and Das, S. (2024). Bridging the p-Special Functions between the Generalized Hyperbolic and Trigonometric Families. Mathematics, 12.","DOI":"10.3390\/math12081242"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"143","DOI":"10.33434\/cams.789085","article-title":"On the trigonometric and p-trigonometric functions of elliptical complex variables","volume":"3","year":"2020","journal-title":"Commun. Adv. Math. Sci."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Zygmund, A. (2002). Trigonometric Series, Cambridge University Press.","DOI":"10.1017\/CBO9781316036587"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"838","DOI":"10.1070\/IM2015v079n04ABEH002763","article-title":"Summability of trigonometric Fourier series at-points and a generalization of the Abel\u2013Poisson method","volume":"79","author":"Trigub","year":"2015","journal-title":"Izv. Math."},{"key":"ref_14","unstructured":"Bary, N.K. (2014). A Treatise on Trigonometric Series: Volume 1, Elsevier."},{"key":"ref_15","unstructured":"Folland, G.B. (2009). Fourier Analysis and Its Applications, American Mathematical Society."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1007\/s40753-021-00134-z","article-title":"The role of Fourier series in mathematics and in signal theory","volume":"7","year":"2021","journal-title":"Int. J. Res. Undergrad. Math. Educ."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Grafakos, L. (2008). Classical and Modern Fourier Analysis, Springer.","DOI":"10.1007\/978-0-387-09434-2"},{"key":"ref_18","unstructured":"Hardy, G.H., and Rogosinski, W. (1999). Fourier Series, Courier Corporation."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"81","DOI":"10.2307\/1967238","article-title":"Introduction to the theory of Fourier\u2019s series","volume":"7","year":"1906","journal-title":"Ann. Math."},{"key":"ref_20","unstructured":"Trigub, R.M., and Belinsky, E.S. (2012). Fourier Analysis and Approximation of Functions, Springer Science & Business Media."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Alibrahim, A.H., and Das, S. (2024). The forgotten p-versine and p-coversine family of functions revisited. PLoS ONE, 19.","DOI":"10.1371\/journal.pone.0308529"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/9\/600\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:48:27Z","timestamp":1760111307000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/9\/600"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,9,4]]},"references-count":21,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2024,9]]}},"alternative-id":["axioms13090600"],"URL":"https:\/\/doi.org\/10.3390\/axioms13090600","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2024,9,4]]}}}