{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,1]],"date-time":"2026-06-01T17:10:42Z","timestamp":1780333842742,"version":"3.54.1"},"reference-count":43,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2022,7,25]],"date-time":"2022-07-25T00:00:00Z","timestamp":1658707200000},"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>The discrete fractional operators of Riemann\u2013Liouville and Liouville\u2013Caputo are omnipresent due to the singularity of the kernels. Therefore, convexity analysis of discrete fractional differences of these types plays a vital role in maintaining the safe operation of kernels and symmetry of discrete delta and nabla distribution. In their discrete version, the generalized or modified forms of various operators of fractional calculus are becoming increasingly important from the viewpoints of both pure and applied mathematical sciences. In this paper, we present the discrete version of the recently modified fractional calculus operator with the Mittag-Leffler-type kernel. Here, in this article, the expressions of both the discrete nabla derivative and its counterpart nabla integral are obtained. Some applications and illustrative examples are given to support the theoretical results.<\/jats:p>","DOI":"10.3390\/sym14081519","type":"journal-article","created":{"date-parts":[[2022,7,26]],"date-time":"2022-07-26T00:17:27Z","timestamp":1658794647000},"page":"1519","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":20,"title":["Modified Fractional Difference Operators Defined Using Mittag-Leffler Kernels"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6837-8075","authenticated-orcid":false,"given":"Pshtiwan Othman","family":"Mohammed","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Education, University of Sulaimani, Sulaimani 46001, Iraq"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9277-8092","authenticated-orcid":false,"given":"Hari Mohan","family":"Srivastava","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada"},{"name":"Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan"},{"name":"Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, Baku AZ1007, Azerbaijan"},{"name":"Section of Mathematics, International Telematic University Uninettuno, I-00186 Rome, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0286-7244","authenticated-orcid":false,"given":"Dumitru","family":"Baleanu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Cankaya University, Ankara 06530, Turkey"},{"name":"Institute of Space Sciences, R76900 Magurele-Bucharest, Romania"},{"name":"Department of Natural Sciences, School of Arts and Sciences, Lebanese American University, Beirut 11022801, Lebanon"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2908-1807","authenticated-orcid":false,"given":"Khadijah M.","family":"Abualnaja","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,7,25]]},"reference":[{"key":"ref_1","unstructured":"Leibniz, G.W. (1849). Letter from Hanover, Germany to G.F.A. L\u2019Hospital, September 30, 1695. Math. Schriften, reprinted in Olns Verl.\u00a01962, 2, 301\u2013302."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"111694","DOI":"10.1016\/j.chaos.2021.111694","article-title":"On the modified Gardner type equation and its time fractional form","volume":"155","author":"Wang","year":"2022","journal-title":"Chaos Solit. Fract."},{"key":"ref_3","unstructured":"Miller, K.S., and Ross, B. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons."},{"key":"ref_4","unstructured":"Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2250081","DOI":"10.1142\/S0218348X22500815","article-title":"A new (3 + 1)-dimensional KDV equation and MKDV equation with their corresponding fractional forms","volume":"30","author":"Wang","year":"2022","journal-title":"Fractals"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"2150101","DOI":"10.1142\/S0218348X21501012","article-title":"Symmetry analysis, analytical solutions and conservation laws of a generalized KdV-Burgers\u2013Kuramoto equation and its fractional version","volume":"29","author":"Wang","year":"2021","journal-title":"Fractals"},{"key":"ref_7","unstructured":"Podlubny, I. (1999). Fractional Differential Equations, Academic Press."},{"key":"ref_8","first-page":"73","article-title":"Fractional-order derivatives and integrals: Introductory overview and recent developments","volume":"60","author":"Srivastava","year":"2020","journal-title":"Kyungpook Math. J."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Baleanu, D., Diethelm, K., Scalas, E., and Trujillo, J.J. (2017). Fractional Calculus: Models and Numerical Methods, World Scientific Publishing Company.","DOI":"10.1142\/10044"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1595","DOI":"10.1007\/s11071-021-06359-6","article-title":"A new (3 + 1)-dimensional Schr\u00f6dinger equation: Derivation, soliton solutions and conservation laws","volume":"104","author":"Wang","year":"2021","journal-title":"Nonlinear Dyn."},{"key":"ref_11","unstructured":"Abel, N.H. (2022, June 26). Oplosning af et Par Opgaver ved Hjelp af Bestemte Integraler. Magazin for Aturvidenskaberne. Aargang I, Bind 2, Christiania. Available online: https:\/\/abelprisen.no\/sites\/default\/files\/2021-04\/Magazin_for_Naturvidenskaberne_oplosning_av_et_par1_opt.pdf."},{"key":"ref_12","first-page":"1","article-title":"Memoire sur quelques questions de geometries et de mecanique, et sur un nouveau genre de calcul pourr esoundre ces questions","volume":"13","author":"Liouville","year":"1832","journal-title":"J. \u00c9col. Polytech."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"529","DOI":"10.1111\/j.1365-246X.1967.tb02303.x","article-title":"Linear models of dissipation whose Q is almost frequency independent\u2014II","volume":"13","author":"Caputo","year":"1967","journal-title":"Geophys. J. Int."},{"key":"ref_14","first-page":"1","article-title":"Fractional derivatives and Cauchy problem for differential equations of fractional order","volume":"3","author":"Dzherbashian","year":"1968","journal-title":"Izv. AN Armenian SSR Ser. Math."},{"key":"ref_15","first-page":"275","article-title":"Zur Theorie der elastischen Nachwirkung","volume":"70","author":"Boltzmann","year":"1874","journal-title":"Sitzber. Akad. Wiss. Wien Math. Naturw. Kl."},{"key":"ref_16","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_17","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1080\/10652460310001600717","article-title":"Generalized Mittag-Leffler function and generalized fractional calculus operators","volume":"15","author":"Kilbas","year":"2004","journal-title":"Integral Transform. Spec. Funct."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"517","DOI":"10.1016\/j.cnsns.2018.07.035","article-title":"Series representations for fractional-calculus operators involving generalised Mittag-Leffler functions","volume":"67","author":"Fernandez","year":"2019","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2240129","DOI":"10.1142\/S0218348X22401296","article-title":"On an extension of the operator with Mittag-Leffler kernel","volume":"30","author":"Baleanu","year":"2022","journal-title":"Fractals"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"763","DOI":"10.2298\/TSCI160111018A","article-title":"New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model","volume":"20","author":"Atangana","year":"2016","journal-title":"Thermal Sci."},{"key":"ref_21","first-page":"423","article-title":"The asymptotic expansion of integral functions defined by Taylor series\u2014I","volume":"238","author":"Wright","year":"1940","journal-title":"Philos. Trans. Ro. Soc. Lond. Ser. A Math. Phys. Sci."},{"key":"ref_22","first-page":"1501","article-title":"Some parametric and argument variations of the operators of fractional calculus and related special functions and integral transformations","volume":"22","author":"Srivastava","year":"2021","journal-title":"J. Nonlinear Convex Anal."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"135","DOI":"10.55579\/jaec.202153.340","article-title":"An introductory overview of fractional-calculus operators based upon the Fox-Wright and related higher transcendental functions","volume":"5","author":"Srivastava","year":"2021","journal-title":"J. Adv. Eng. Comput."},{"key":"ref_24","first-page":"123","article-title":"Some families of Mittag-Leffler type functions and associated operators of fractional calculus","volume":"7","author":"Srivastava","year":"2016","journal-title":"TWMS J. Pure Appl. Math."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Srivastava, H.M. (2021). A survey of some recent developments on higher transcendental functions of analytic number theory and applied mathematics. Symmetry, 13.","DOI":"10.3390\/sym13122294"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Goodrich, C., and Peterson, A.C. (2015). Discrete Fractional Calculus, Springer.","DOI":"10.1007\/978-3-319-25562-0"},{"key":"ref_27","first-page":"165","article-title":"A transform method in discrete fractional calculus","volume":"2","author":"Atici","year":"2007","journal-title":"Int. J. Differ. Equ."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jmaa.2010.02.009","article-title":"Modeling with fractional difference equations","volume":"369","author":"Atici","year":"2010","journal-title":"J. Math. Anal. Appl."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"084308","DOI":"10.1063\/1.4958920","article-title":"Riesz Riemann-Liouville difference on discrete domains","volume":"26","author":"Wu","year":"2016","journal-title":"Chaos"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"232","DOI":"10.1186\/s13662-016-0949-5","article-title":"Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels","volume":"2016","author":"Abdeljawad","year":"2016","journal-title":"Adv. Differ. Equ."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"6391","DOI":"10.1002\/mma.8176","article-title":"On positivity and monotonicity analysis for discrete fractional operators with discrete Mittag-Leffler kernel","volume":"45","author":"Mohammed","year":"2022","journal-title":"Math. Meth. Appl. Sci."},{"key":"ref_32","first-page":"36","article-title":"Dual identities in fractional difference calculus within Riemann","volume":"2017","author":"Abdeljawad","year":"2017","journal-title":"Adv. Differ. Equ."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"78","DOI":"10.1186\/s13662-017-1126-1","article-title":"Monotonicity results for fractional difference operators with discrete exponential kernels","volume":"2017","author":"Abdeljawad","year":"2017","journal-title":"Adv. Differ. Equ."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Mohammed, P.O., Abdeljawad, T., and Hamasalh, F.K. (2021). On Riemann-Liouville and Caputo fractional forward difference monotonicity analysis. Mathematics, 9.","DOI":"10.3390\/math9111303"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1515\/anly-2019-0050","article-title":"Positivity and monotonicity results for triple sequential fractional differences via convolution","volume":"40","author":"Goodrich","year":"2020","journal-title":"Analysis"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"4961","DOI":"10.3934\/dcds.2020207","article-title":"Positivity, monotonicity, and convexity for convolution operators","volume":"40","author":"Goodrich","year":"2020","journal-title":"Discrete Contin. Dyn. Syst."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1155\/2013\/406910","article-title":"On delta and nabla Caputo fractional differences and dual identities","volume":"2013","author":"Abdeljawad","year":"2013","journal-title":"Discret. Dyn. Nat. Soc."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"023102","DOI":"10.1063\/1.5085726","article-title":"Fractional operators with generalized Mittag-Leffler kernels and their differintegrals","volume":"29","author":"Abdeljawad","year":"2019","journal-title":"Chaos"},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Abdeljawad, T., and Fernandez, A. (2019). On a new class of fractional difference-sum operators with discrete Mittag-Leffler kernels. Mathematics, 7.","DOI":"10.3390\/math7090772"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"146","DOI":"10.1016\/j.chaos.2018.09.022","article-title":"Different type kernel h-fractional differences and their fractional h\u2013sums","volume":"116","author":"Abdeljawad","year":"2018","journal-title":"Chaos Solit. Fract."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"4062","DOI":"10.3934\/mbe.2022186","article-title":"New classifications of monotonicity investigation for discrete operators with Mittag-Leffler kernel","volume":"19","author":"Mohammed","year":"2022","journal-title":"Math. Biosci. Eng."},{"key":"ref_42","first-page":"1","article-title":"Monotonicity analysis of a nabla discrete fractional operator with discrete Mittag-Leffler kernel","volume":"116","author":"Abdeljawad","year":"2017","journal-title":"Chaos Solitons Fract."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1016\/j.chaos.2019.06.012","article-title":"Fractional difference operators with discrete generalized Mittag-Leffler kernels","volume":"126","author":"Abdeljawad","year":"2019","journal-title":"Chaos Solitons Fract."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/8\/1519\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:56:07Z","timestamp":1760140567000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/8\/1519"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7,25]]},"references-count":43,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2022,8]]}},"alternative-id":["sym14081519"],"URL":"https:\/\/doi.org\/10.3390\/sym14081519","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,7,25]]}}}