{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,24]],"date-time":"2026-04-24T16:36:54Z","timestamp":1777048614585,"version":"3.51.4"},"reference-count":35,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2022,9,5]],"date-time":"2022-09-05T00:00:00Z","timestamp":1662336000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12001095"],"award-info":[{"award-number":["12001095"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper investigates the composition structures of certain fractional integral operators whose kernels are certain types of generalized hypergeometric functions. It is shown how composition formulas of these operators can be closely related to the various Erd\u00e9lyi-type hypergeometric integrals. We also derive a derivative formula for the fractional integral operator and some applications of the operator are considered for a certain Volterra-type integral equation, which provide two generalizations to Khudozhnikov\u2019s integral equation (see below). Some specific relationships, examples, and some future research problems are also discussed.<\/jats:p>","DOI":"10.3390\/sym14091845","type":"journal-article","created":{"date-parts":[[2022,9,8]],"date-time":"2022-09-08T09:51:09Z","timestamp":1662630669000},"page":"1845","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["On the Composition Structures of Certain Fractional Integral Operators"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7433-4490","authenticated-orcid":false,"given":"Min-Jie","family":"Luo","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, Donghua University, Shanghai 201620, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ravinder Krishna","family":"Raina","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Technology & Engineering, Maharana Pratap University of Agriculture and Technology, Udaipur 313001, India"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,9,5]]},"reference":[{"key":"ref_1","first-page":"135","article-title":"A remark on integral operators involving the Gauss hypergeometric functions","volume":"11","author":"Saigo","year":"1978","journal-title":"Math. Rep. Coll. Gen. Educ. Kyushu Univ."},{"key":"ref_2","first-page":"55","article-title":"On the H\u00f6lder continuity of the generalized fractional integrals and derivatives","volume":"12","author":"Saigo","year":"1980","journal-title":"Math. Rep. Coll. Gen. Educ. Kyushu Univ."},{"key":"ref_3","first-page":"33","article-title":"A generalization of fractional calculus and its applications to Euler-Darboux equation","volume":"412","author":"Saigo","year":"1981","journal-title":"RIMS Kokyuroku"},{"key":"ref_4","unstructured":"Kiryakova, V. (1994). Generalized Fractional Calculus and Applications, Longman Scientific and Technical."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Naheed, S., Mubeen, S., Rahman, G., Khan, A.Z., and Nisar, K.S. (2022). Certain integral and differential formulas involving the product of Srivastava\u2019s polynomials and extended Wright function. Fractal Fract., 6.","DOI":"10.3390\/fractalfract6020093"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1017\/S0308210517000257","article-title":"Applications of differential subordinations for norm estimates of an integral operator","volume":"148","author":"Dziok","year":"2018","journal-title":"Proc. R. Soc. Edinb. Sect. A Math."},{"key":"ref_7","first-page":"159","article-title":"On two Saigo\u2019s fractional integral operators in the class of univalent functions","volume":"9","author":"Kiryakova","year":"2006","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_8","first-page":"53","article-title":"On the fractional calculus operator involving Gauss\u2019s series and its application to certain statistical distributions","volume":"14","author":"Saigo","year":"1991","journal-title":"Rev. T\u00e9c. Ing. Univ. Zulia"},{"key":"ref_9","first-page":"251","article-title":"Fractional integral operators and the generalized hypergeometric functions","volume":"18","author":"Goyal","year":"1987","journal-title":"Indian J. Pure Appl. Math."},{"key":"ref_10","first-page":"403","article-title":"Fractional integral operators involving a product of generalized hypergeometric functions and a general class of polynomials","volume":"22","author":"Goyal","year":"1991","journal-title":"Indian J. Pure Appl. Math."},{"key":"ref_11","first-page":"121","article-title":"Fractional integral operators involving a product of generalized hypergeometric functions and a general class of polynomials. II","volume":"23","author":"Goyal","year":"1992","journal-title":"Indian J. Pure Appl. Math."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Srivastava, H.M., and Owa, S. (1992). Certain properties of operators of fractional integration associated with Mellin and Laplace transformations. Current Topics in Analytic Function Theory, World Scientific.","DOI":"10.1142\/1628"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Araci, S., Rahman, G., Ghaffar, A., and Nisar, K.S. (2019). Fractional calculus of extended Mittag-Leffler function and its applications to statistical distribution. Mathematics, 7.","DOI":"10.3390\/math7030248"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"422","DOI":"10.1515\/fca-2017-0022","article-title":"Fractional integral operators characterized by some new hypergeometric summation formulas","volume":"20","author":"Luo","year":"2017","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_15","first-page":"161","article-title":"On a multiple \u010ceby\u0161ev type functional defined by a generalized fractional integral operator","volume":"10","author":"Luo","year":"2017","journal-title":"Tbil. Math. J."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"611","DOI":"10.14492\/hokmj\/1573722020","article-title":"The decompositional structure of certain fractional integral operators","volume":"48","author":"Luo","year":"2019","journal-title":"Hokkaido Math. J."},{"key":"ref_17","first-page":"79","article-title":"Integration of Volterra-type integral equations of the first kind with kernels containing some generalized hypergeometric functions","volume":"7","author":"Khudozhnikov","year":"1995","journal-title":"Matem. Mod."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Andrews, G.E., Askey, R., and Roy, R. (1999). Special Functions, Cambridge University Press.","DOI":"10.1017\/CBO9781107325937"},{"key":"ref_19","unstructured":"Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier Science B.V.. North-Holland Mathematics Studies."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1216\/RMJ-2013-43-1-291","article-title":"Transformation formulas for the generalized hypergeometric function with integral parameter differences","volume":"43","author":"Miller","year":"2013","journal-title":"Rocky Mt. J. Math."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"476","DOI":"10.1080\/10652469.2017.1312367","article-title":"Erd\u00e9lyi-type integrals for generalized hypergeometric functions with integral parameter differences","volume":"28","author":"Luo","year":"2017","journal-title":"Integral Transform. Spec. Funct."},{"key":"ref_22","first-page":"165","article-title":"On compositions of generalized fractional integrals","volume":"11","author":"Grinko","year":"1991","journal-title":"J. Math. Res. Expo."},{"key":"ref_23","unstructured":"Samko, S.G., Kilbas, A.A., and Marichev, O.I. (1993). Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"050007","DOI":"10.1063\/1.4936737","article-title":"On the origins of generalized fractional calculus","volume":"1690","author":"Kiryakova","year":"2015","journal-title":"AIP Conf. Proc."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1016\/j.chaos.2017.03.006","article-title":"Fractional calculus operators of special functions? The result is well predictable!","volume":"102","author":"Kiryakova","year":"2017","journal-title":"Chaos Solitons Fractals"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"630840","DOI":"10.1155\/2014\/630840","article-title":"On generalized fractional integral operators and the generalized Gauss hypergeometric functions","volume":"2014","author":"Baleanu","year":"2014","journal-title":"Abstr. Appl. Anal."},{"key":"ref_27","first-page":"433","article-title":"A study of Marichev-Saigo-Maeda fractional integral operators associated with the S-generalized Gauss hypergeometric function","volume":"59","author":"Bansal","year":"2019","journal-title":"Kyungpook Math. J."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2071","DOI":"10.1007\/BF02105396","article-title":"Factorization of integral transformations of convolution type","volume":"30","author":"Brychkov","year":"1985","journal-title":"J. Math. Sci."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Luchko, Y. (2021). General fractional integrals and derivatives with the Sonine kernels. Mathematics, 9.","DOI":"10.3390\/math9060594"},{"key":"ref_30","unstructured":"Olver, F.W.J., Lozier, D.W., Boisvert, R.F., and Clark, C.W. (2010). NIST Handbook of Mathematical Functions, Cambridge University Press."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1090\/S0002-9939-1992-1068116-2","article-title":"Generalized hypergeometric functions at unit argument","volume":"114","year":"1992","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_32","first-page":"443","article-title":"On compositions of generalized fractional integrals and evaluation of definite integrals with Gauss hypergeometric functions","volume":"11","author":"Grinko","year":"1991","journal-title":"J. Math. Res. Expo."},{"key":"ref_33","unstructured":"Gould, G.G. (1990). Integrals and Series. Volume 3: More Special Functions, Gordon and Breach Science Publishers."},{"key":"ref_34","first-page":"239","article-title":"Disjoint convolution sums of Stirling numbers","volume":"26","author":"Chu","year":"2021","journal-title":"Math. Commun."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Hilfer, R., and Luchko, Y. (2019). Desiderata for Fractional Derivatives and Integrals. Mathematics, 7.","DOI":"10.3390\/math7020149"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/9\/1845\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:23:45Z","timestamp":1760142225000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/9\/1845"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,9,5]]},"references-count":35,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2022,9]]}},"alternative-id":["sym14091845"],"URL":"https:\/\/doi.org\/10.3390\/sym14091845","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,9,5]]}}}