{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,23]],"date-time":"2025-10-23T05:29:37Z","timestamp":1761197377898},"reference-count":29,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2016,6,13]],"date-time":"2016-06-13T00:00:00Z","timestamp":1465776000000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Proceedings of the Edinburgh Mathematical Society"],"published-print":{"date-parts":[[2017,2]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this paper we present the basic tools of a fractional function theory in higher dimensions by means of a fractional correspondence to the Weyl relations via fractional Riemann\u2013Liouville derivatives. A Fischer decomposition, Almansi decomposition, fractional Euler and Gamma operators, monogenic projection, and basic fractional homogeneous powers are constructed. Moreover, we establish the fractional Cauchy\u2013Kovalevskaya extension (FCK extension) theorem for fractional monogenic functions defined on \u211d<jats:sup><jats:italic>d<\/jats:italic><\/jats:sup>. Based on this extension principle, fractional Fueter polynomials, forming a basis of the space of fractional spherical monogenics, i.e. fractional homogeneous polynomials, are introduced. We study the connection between the FCK extension of functions of the form<jats:italic><jats:bold>x<\/jats:bold>P<jats:sub>l<\/jats:sub><\/jats:italic>and the classical Gegenbauer polynomials. Finally, we present an example of an FCK extension.<\/jats:p>","DOI":"10.1017\/s0013091516000109","type":"journal-article","created":{"date-parts":[[2016,6,13]],"date-time":"2016-06-13T10:50:27Z","timestamp":1465815027000},"page":"251-272","source":"Crossref","is-referenced-by-count":16,"title":["Fischer Decomposition and Cauchy\u2013Kovalevskaya Extension in Fractional Clifford Analysis: The Riemann\u2013Liouville Case"],"prefix":"10.1017","volume":"60","author":[{"given":"N.","family":"Vieira","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2016,6,13]]},"reference":[{"key":"S0013091516000109_ref022","volume-title":"Theory and applications of fractional differential equations","volume":"204","year":"2006"},{"key":"S0013091516000109_ref021","first-page":"191","volume-title":"Hypercomplex Analysis: new perspectives and applications","year":"2014"},{"key":"S0013091516000109_ref020","doi-asserted-by":"publisher","DOI":"10.1016\/0001-8708(77)90017-2"},{"key":"S0013091516000109_ref009","first-page":"115","volume-title":"Hypercomplex analysis","year":"2009"},{"key":"S0013091516000109_ref008","doi-asserted-by":"publisher","DOI":"10.1016\/j.geomphys.2013.09.005"},{"key":"S0013091516000109_ref007","first-page":"170","volume-title":"Function spaces in complex and Clifford analysis","year":"2008"},{"key":"S0013091516000109_ref006","first-page":"17","volume-title":"Oeuvres compl\u00e8tes","volume":"VII","year":"1892"},{"key":"S0013091516000109_ref005","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2011.03.021"},{"key":"S0013091516000109_ref004","doi-asserted-by":"crossref","first-page":"449","DOI":"10.4171\/rmi\/606","volume":"26","year":"2010","journal-title":"Rev. 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