{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,3]],"date-time":"2026-06-03T21:39:01Z","timestamp":1780522741280,"version":"3.54.1"},"reference-count":15,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2012,10,5]],"date-time":"2012-10-05T00:00:00Z","timestamp":1349395200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Recently, an extended operator of fractional derivative related to a generalized Beta function was used in order to obtain some generating relations involving the extended hypergeometric functions [1]. The main object of this paper is to present a further generalization of the extended fractional derivative operator and apply the generalized extended fractional derivative operator to derive linear and bilinear generating relations for the generalized extended Gauss, Appell and Lauricella hypergeometric functions in one, two and more variables. Some other properties and relationships involving the Mellin transforms and the generalized extended fractional derivative operator are also given.<\/jats:p>","DOI":"10.3390\/axioms1030238","type":"journal-article","created":{"date-parts":[[2012,10,5]],"date-time":"2012-10-05T13:51:07Z","timestamp":1349445067000},"page":"238-258","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":43,"title":["A Class of Extended Fractional Derivative Operators and Associated Generating Relations Involving Hypergeometric Functions"],"prefix":"10.3390","volume":"1","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9277-8092","authenticated-orcid":false,"given":"H. M.","family":"Srivastava","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Rakesh K.","family":"Parmar","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Government College of Engineering and Technology, Bikaner 334004, Rajasthan, India"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Purnima","family":"Chopra","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Marudhar Engineering College, Raisar, NH-11 Jaipur Road, Bikaner 334001, Rajasthan, India"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2012,10,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1825","DOI":"10.1016\/j.mcm.2010.07.011","article-title":"Some generating relations for extended hypergeometric function via generalized fractional derivative operator","volume":"52","year":"2010","journal-title":"Math. Comput. Model."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1016\/0377-0427(95)00018-6","article-title":"Asymptotic and closed form of a generalized incomplete gamma function","volume":"67","author":"Chaudhry","year":"1996","journal-title":"J. Comput. Appl. Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/S0377-0427(96)00102-1","article-title":"Extension of Euler\u2019s beta function","volume":"78","author":"Chaudhry","year":"1997","journal-title":"J. Comput. Appl. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1016\/0377-0427(94)90187-2","article-title":"Generalized incomplete gamma functions with applications","volume":"55","author":"Chaudhry","year":"1994","journal-title":"J. Comput. Appl. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1016\/0377-0427(94)00026-W","article-title":"On the decomposition of generalized incomplete gamma functions with applications of Fourier transforms","volume":"59","author":"Chaudhry","year":"1995","journal-title":"J. Comput. Appl. Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"703","DOI":"10.1216\/rmjm\/1022009290","article-title":"Reduction of a generalized incomplete gamma function, related Kamp\u00e9 de F\u00e9riet functions, and incomplete Weber integrals","volume":"30","author":"Miller","year":"2000","journal-title":"Rocky Mountain J. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"4601","DOI":"10.1016\/j.cam.2010.04.019","article-title":"Extension of gamma, beta and hypergeometric functions","volume":"235","author":"Altn","year":"2011","journal-title":"J. Comput. Appl. Math."},{"key":"ref_8","first-page":"589","article-title":"Extended hypergeometric and confluent hypergeometric functions","volume":"159","author":"Chaudhry","year":"2004","journal-title":"Appl. Math. Comput."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Chaudhry, M.A., and Zubair, S.M. (2002). On a Class of Incomplete Gamma Functions with Applications, CRC Press (Chapman and Hall).","DOI":"10.1201\/9781420036046"},{"key":"ref_10","unstructured":"Erd\u00e9lyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F.G. (1954). Tables of Integral Transforms, Volume II, McGraw-Hill Book Company."},{"key":"ref_11","unstructured":"Samko, S.G., Kilbas, A.A., and Marichev, O.I. (1993). Fractional Integrals and Derivatives: Theory and Applications, Translated from the Russian:  Integrals and Derivatives of Fractional Order and Some of Their Applications (\u201cNauka i Tekhnika\", Minsk, 1987), Gordon and Breach Science Publishers."},{"key":"ref_12","unstructured":"Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations. North-Holland Mathematical Studies, Elsevier (North-Holland) Science Publishers."},{"key":"ref_13","unstructured":"Appell, P., and Kamp\u00e9 de F\u00e9riet, J. (1926). Fonctions Hyperg\u00e9om\u00e9triques et Hypersph\u00e9riques: Polyn\u00f4mes d\u2019Hermite, Gauthier-Villars."},{"key":"ref_14","unstructured":"Srivastava, H.M., and Manocha, H.L. (1984). A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons."},{"key":"ref_15","first-page":"1","article-title":"Operators of fractional integration and their applications","volume":"118","author":"Srivastava","year":"2001","journal-title":"Appl. Math. Comput."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/1\/3\/238\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:52:41Z","timestamp":1760219561000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/1\/3\/238"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,10,5]]},"references-count":15,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2012,12]]}},"alternative-id":["axioms1030238"],"URL":"https:\/\/doi.org\/10.3390\/axioms1030238","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,10,5]]}}}