{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:29:43Z","timestamp":1760146183019,"version":"build-2065373602"},"reference-count":15,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2024,10,16]],"date-time":"2024-10-16T00:00:00Z","timestamp":1729036800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"King Saud University, Riyadh, Saudi Arabia","award":["RSPD2024R974"],"award-info":[{"award-number":["RSPD2024R974"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The aim of this paper is to extend the concept of the orthogonal derivative to provide a new integral representation of the fractional Riesz derivative. Specifically, we investigate the orthogonal derivative associated with Gegenbauer polynomials\u00a0Cn(\u03bd)(x), where\u00a0\u03bd&gt;\u221212. Building on the work of Diekema and Koornwinder, the n-th derivative is obtained as the limit of an integral involving Gegenbauer polynomials as the kernel. When this limit is omitted, it results in the approximate Gegenbauer orthogonal derivative, which serves as an effective approximation of the n-th order derivative. Using this operator, we introduce a novel extension of the fractional Riesz derivative, denoted as\u00a0D\u03b1x, providing an alternative framework for fractional calculus.<\/jats:p>","DOI":"10.3390\/axioms13100715","type":"journal-article","created":{"date-parts":[[2024,10,16]],"date-time":"2024-10-16T10:11:04Z","timestamp":1729073464000},"page":"715","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["The Orthogonal Riesz Fractional Derivative"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2743-2036","authenticated-orcid":false,"given":"Fethi","family":"Bouzeffour","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Sciences, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,10,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"637","DOI":"10.1016\/j.jat.2012.01.003","article-title":"Differentiation by integration using orthogonal polynomials, a survey","volume":"164","author":"Diekema","year":"2012","journal-title":"J. Approx. Theory"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"273","DOI":"10.3390\/math3020273","article-title":"The fractional orthogonal derivative","volume":"3","author":"Diekema","year":"2015","journal-title":"Mathematics"},{"key":"ref_3","unstructured":"Samko, S.G., Kilbas, A.A., and Marichev, O.I. (1993). Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Bucur, C., and Valdinoci, E. (2016). Nonlocal Diffusion and Applications, Springer.","DOI":"10.1007\/978-3-319-28739-3"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"109009","DOI":"10.1016\/j.jcp.2019.109009","article-title":"What is the fractional Laplacian? A comparative review with new results","volume":"404","author":"Lischke","year":"2020","journal-title":"J. Comput. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1007\/s11464-019-0774-8","article-title":"Limiting weak-type behaviors for Riesz transforms and maximal operators in Bessel setting","volume":"14","author":"Hou","year":"2019","journal-title":"Front. Math. China"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1016\/j.acha.2023.05.003","article-title":"Riesz transform associated with the fractional Fourier transform and applications in image edge detection","volume":"66","author":"Fu","year":"2023","journal-title":"Appl. Comput. Harmon. Anal."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"476","DOI":"10.1137\/23M1555442","article-title":"Fractional Fourier Transforms Meet Riesz Potentials and Image Processing","volume":"17","author":"Fu","year":"2024","journal-title":"SIAM J. Imaging Sci."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1515\/fca-2019-0019","article-title":"On Riesz derivative","volume":"22","author":"Cai","year":"2019","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_10","unstructured":"Feller, W. (1952). On a generalization of Marcel Riesz potentials and the semi-groups generated by them. Meddelanden Lunds Universitets Matematiska Seminarium (Comm. S\u00e9m. Math\u00e9m. Universit\u00e9 de Lund), Tome suppl. d\u00e9di\u00e9 \u00e0 M. Riesz."},{"key":"ref_11","first-page":"153","article-title":"The fundamental solution of the space-time fractional diffusion equation","volume":"4","author":"Mainardi","year":"2001","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1716","DOI":"10.1515\/fca-2021-0074","article-title":"Difference Between Riesz Derivative and Fractional Laplacian on the Proper Subset of \u211d","volume":"24","author":"Jiao","year":"2021","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Koekoek, R., Lesky, P.A., and Swarttouw, R.F. (2010). Hypergeometric Orthogonal Polynomials. Hypergeometric Orthogonal Polynomials and Their q-Analogues, Springer.","DOI":"10.1007\/978-3-642-05014-5"},{"key":"ref_14","unstructured":"Watson, G.N. (1990). A Treatise on the Theory of Bessel Functions, Cambridge University Press."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"230","DOI":"10.1080\/10652469.2021.1925268","article-title":"On the fractional Bessel operator","volume":"33","author":"Bouzeffour","year":"2022","journal-title":"Integral Transform. Spec. Funct."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/10\/715\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:14:21Z","timestamp":1760112861000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/10\/715"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,10,16]]},"references-count":15,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2024,10]]}},"alternative-id":["axioms13100715"],"URL":"https:\/\/doi.org\/10.3390\/axioms13100715","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2024,10,16]]}}}