{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,24]],"date-time":"2026-01-24T18:27:38Z","timestamp":1769279258632,"version":"3.49.0"},"reference-count":42,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2022,4,23]],"date-time":"2022-04-23T00:00:00Z","timestamp":1650672000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this article, we obtain certain finite integrals concerning generalized Mittag\u2013Leffler functions, which are evaluated in terms of the generalized Fox\u2013Wright function. The integrals of concern are unified in nature and thereby yield some new integral formulas as special cases. Moreover, we numerically compute some integrals using the Gaussian quadrature formula and draw a comparison with the main integrals by using graphical numerical investigation.<\/jats:p>","DOI":"10.3390\/sym14050869","type":"journal-article","created":{"date-parts":[[2022,4,24]],"date-time":"2022-04-24T00:45:21Z","timestamp":1650761121000},"page":"869","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Unified Integrals of Generalized Mittag\u2013Leffler Functions and Their Graphical Numerical Investigation"],"prefix":"10.3390","volume":"14","author":[{"given":"Nabiullah","family":"Khan","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics, Aligarh Muslim University, Aligarh 202002, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mohammad Iqbal","family":"Khan","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Aligarh Muslim University, Aligarh 202002, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4208-6784","authenticated-orcid":false,"given":"Talha","family":"Usman","sequence":"additional","affiliation":[{"name":"Department of General Requirements, University of Technology and Applied Sciences-Sur, Muscat 133, Oman"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7469-5402","authenticated-orcid":false,"given":"Kamsing","family":"Nonlaopon","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8955-5552","authenticated-orcid":false,"given":"Shrideh","family":"Al-Omari","sequence":"additional","affiliation":[{"name":"Department of Scientific Basic Sciences, Faculty of Engineering Technology, Al-Balqa Applied University, Amman 11134, Jordan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,4,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1016\/j.jksus.2015.04.002","article-title":"Certain recent fractional integral inequalities associated with the hypergeometric operators","volume":"28","author":"Jain","year":"2016","journal-title":"J. King Saud Univ.-Sci."},{"key":"ref_2","first-page":"108","article-title":"On the Generalized Stieltjes transform of Fox\u2019s kernel function and its properties in the space of generalized functions","volume":"23","author":"Baleanu","year":"2017","journal-title":"J. Comput. Anal. Appl."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"45","DOI":"10.36753\/mathenot.421329","article-title":"Certain sequence of functions involving generalized hypergeometric functions","volume":"3","author":"Agarwal","year":"2015","journal-title":"Math. Sci. Appl. E-Notes"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2021","DOI":"10.1515\/gmj-2021-2105","article-title":"Integral transforms and probality distributions involving generalized hypergeometric function","volume":"28","author":"Khan","year":"2021","journal-title":"Georgian J. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"560","DOI":"10.1186\/s13662-020-03019-8","article-title":"Unified integral associated with the generalized V-function","volume":"2020","author":"Chandak","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_6","first-page":"1","article-title":"A note on generalized integral operator associated with multiindex Mittag-Leffler function, Filomat 30, 1931\u20131939","volume":"448","author":"Choi","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Gorenflo, R., Kilbas, A.A., Mainardi, F., and Rogosin, S.V. (2020). Mittag-Leffler Functions, Related Topics and Applications, Springer.","DOI":"10.1007\/978-3-662-61550-8"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"298628","DOI":"10.1155\/2011\/298628","article-title":"Mittag-Leffler Functions and Their Applications","volume":"2011","author":"Haubold","year":"2011","journal-title":"J. Appl. Math."},{"key":"ref_9","unstructured":"Kiryakova, V. (1993). Generalized Fractional Calculus and Applications, CRC Press."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1885","DOI":"10.1016\/j.camwa.2009.08.025","article-title":"The multi-index Mittag-Leffler functions as an important class of special functions of fractional calculus","volume":"59","author":"Kiryakova","year":"2010","journal-title":"Comput. Math. Appl."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Kochubei, A., and Luchko, Y. (2019). Fractional Differential Equations. Handbook of Fractional Calculus with Applications, De Gruyter.","DOI":"10.1515\/9783110571660"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Mainardi, F. (2020). Why the Mittag-Leffler Function Can Be Considered the Queen Function of the Fractional Calculus?. Entropy, 22.","DOI":"10.3390\/e22121359"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"29","DOI":"10.4134\/CKMS.c150247","article-title":"Certain integrals associated with generalized mittag-leffler function","volume":"32","author":"Agarwal","year":"2017","journal-title":"Commun. Korean Math. Soc."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Almalahi, M.A., Ghanim, F., Botmart, T., Bazighifan, O., and Askar, S. (2021). Qualitative Analysis of Langevin Integro-Fractional Differential Equation under Mittag\u2013Leffler Functions Power Law. Fractal Fract., 5.","DOI":"10.3390\/fractalfract5040266"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"206","DOI":"10.26637\/MJM0702\/0011","article-title":"Estimation of certain integrals with extended multi-index Bessel function","volume":"7","author":"Kamarujjama","year":"2019","journal-title":"Malaya J. Mat."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"491","DOI":"10.1186\/s13662-020-02948-8","article-title":"Computation of certain integral formulas involving generalized Wright function","volume":"2020","author":"Khan","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"112840","DOI":"10.1016\/j.cam.2020.112840","article-title":"Some properties concerning the analysis of generalized Wright function","volume":"376","author":"Khan","year":"2020","journal-title":"J. Comput. Appl. Math."},{"key":"ref_18","first-page":"13","article-title":"Integral transform of generalized K-Mittag-Lefller function","volume":"9","author":"Khan","year":"2018","journal-title":"J. Fract. Calc. Appl."},{"key":"ref_19","first-page":"325","article-title":"Some Generating Relations of Extended Mittag-Leffler Functions","volume":"59","author":"Khan","year":"2019","journal-title":"Kyungpook Math. J."},{"key":"ref_20","first-page":"71","article-title":"A note on extended beta function involving generalized Mittag-Leffler function and its applications","volume":"12","author":"Khan","year":"2022","journal-title":"TWMS J. App. Eng. Math."},{"key":"ref_21","first-page":"98","article-title":"Unified approach to the certain integrals of k-Mittag-Leffler type function of two variables","volume":"39","author":"Khan","year":"2019","journal-title":"Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. Math."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Mihai, M.V., Awan, M.U., Noor, M.A., Du, T., Kashuri, A., and Noor, K.I. (2019). On Extended General Mittag\u2013Leffler Functions and Certain Inequalities. Fractal Fract., 3.","DOI":"10.3390\/fractalfract3020032"},{"key":"ref_23","first-page":"7","article-title":"A Singular Integral Equation with a Generalized Mittag-Leffler Function in the Kernel","volume":"19","author":"Prabhakar","year":"1971","journal-title":"Yokohama Math. J."},{"key":"ref_24","unstructured":"Prudnikov, A.P., Brychkov, Y.A., and Marichev, O.I. (1992). Integral and Series V.1. More Special Functio, Gordon and Breach."},{"key":"ref_25","unstructured":"Rainville, E.D. (1960). Special Functions, The Macmillan Company."},{"key":"ref_26","first-page":"21","article-title":"Some properties relating to the generalized Mittag-Leffler function","volume":"4","author":"Salim","year":"2009","journal-title":"Adv. Appl. Math. Anal."},{"key":"ref_27","first-page":"1","article-title":"A generalization of Mittag-Leffler function and integral operator associated with fractional calculus","volume":"3","author":"Salim","year":"2012","journal-title":"J. Fract. Calc. Appl."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"797","DOI":"10.1016\/j.jmaa.2007.03.018","article-title":"On a generalization of Mittag-Leffler function and its properties","volume":"336","author":"Shukla","year":"2007","journal-title":"J. Math. Anal. Appl."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Singh, P., Jain, S., and Cattani, C. (2021). Some Unified Integrals for Generalized Mittag-Leffler Functions. Axioms, 10.","DOI":"10.3390\/axioms10040261"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"208","DOI":"10.1186\/s13660-019-2162-z","article-title":"Certain integrals involving multivariate Mittag-Leffler function","volume":"2019","author":"Suthar","year":"2019","journal-title":"J. Inequalities Appl."},{"key":"ref_31","first-page":"554","article-title":"Sur la nouvelle fonction E\u03b1(x)","volume":"137","year":"1903","journal-title":"CR Acad. Sci. Paris"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"12757","DOI":"10.3934\/math.2021736","article-title":"A basic study of a fractional integral operator with extended Mittag-Leffler kernel","volume":"6","author":"Rahman","year":"2021","journal-title":"AIMS Math."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1007\/BF02403202","article-title":"Uber den fundamental Satz in der Theories der Funktionen E\u03b1(z)","volume":"29","author":"Wiman","year":"1905","journal-title":"Acta Math."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1186\/2193-1801-2-337","article-title":"On some properties of the generalized Mittag-Leffler function","volume":"2","author":"Khan","year":"2013","journal-title":"SpringerPlus"},{"key":"ref_35","first-page":"423","article-title":"The asymptotic expansion of integral functions defined by Taylor series","volume":"238","author":"Wright","year":"1940","journal-title":"Philos. Trans. R. Soc. London. Ser. A Math. Phys. Sci."},{"key":"ref_36","first-page":"125","article-title":"Estimation of a modified integral associated with a special function kernel of Fox\u2019s H-function type","volume":"35","year":"2020","journal-title":"Commun. Korean Math. Soc."},{"key":"ref_37","unstructured":"Abramowitz, M., and Stegun, I.A. (1948). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1112\/plms\/s2-27.1.389","article-title":"The Asymptotic Expansion of Generalized Hypergeometric Functions","volume":"2","author":"Fox","year":"1928","journal-title":"Proc. Lond. Math. Soc."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Al-Omari, S. (2018). A revised version of the generalized Kr\u00e4tzel-Fox integral operators. Mathematics, 6.","DOI":"10.3390\/math6110222"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1515\/gmj-2016-0056","article-title":"On a Class of Generalized Meijer-Laplace Transforms of Fox Function Type Kernels and Their Extension to a Class of Boehmians","volume":"25","year":"2018","journal-title":"Georgian Math. J."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1112\/plms\/s2-46.1.389","article-title":"The asymptotic expansion of the generalized hypergeometric function","volume":"2","author":"Wright","year":"1940","journal-title":"Proc. Lond. Math. Soc."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"286","DOI":"10.1112\/jlms\/s1-10.40.286","article-title":"The asymptotic expansion of the generalized hypergeometric function","volume":"1","author":"Wright","year":"1935","journal-title":"J. Lond. Math. Soc."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/5\/869\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:59:25Z","timestamp":1760137165000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/5\/869"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,4,23]]},"references-count":42,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2022,5]]}},"alternative-id":["sym14050869"],"URL":"https:\/\/doi.org\/10.3390\/sym14050869","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,4,23]]}}}