{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,3]],"date-time":"2026-04-03T03:37:30Z","timestamp":1775187450230,"version":"3.50.1"},"reference-count":33,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2024,11,21]],"date-time":"2024-11-21T00:00:00Z","timestamp":1732147200000},"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>Special functions have been widely used in fractional calculus, particularly for addressing the symmetric behavior of the function. This paper provides improved delta Mittag\u2013Leffler and exponential functions to establish new types of fractional difference operators in the setting of Riemann\u2013Liouville and Liouville\u2013Caputo. We give some properties of these discrete functions and use them as the kernel of the new fractional operators. In detail, we propose the construction of the new fractional sums and differences. We also find the Laplace transform of them. Finally, the relationship between the Riemann\u2013Liouville and Liouville\u2013Caputo operators are examined to verify the feasibility and effectiveness of the new fractional operators.<\/jats:p>","DOI":"10.3390\/sym16121562","type":"journal-article","created":{"date-parts":[[2024,11,21]],"date-time":"2024-11-21T12:25:42Z","timestamp":1732191942000},"page":"1562","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Improved Fractional Differences with Kernels of Delta Mittag\u2013Leffler and Exponential Functions"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1567-0264","authenticated-orcid":false,"given":"Miguel","family":"Vivas-Cortez","sequence":"first","affiliation":[{"name":"Faculty of Exact and Natural Sciences, School of Physical Sciences and Mathematics, Pontifical Catholic University of Ecuador, Av. 12 de Octubre 1076 y Roca, Quito 17-01-2184, Ecuador"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6837-8075","authenticated-orcid":false,"given":"Pshtiwan Othman","family":"Mohammed","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Education, University of Sulaimani, Sulaimani 46001, Iraq"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2788-809X","authenticated-orcid":false,"given":"Juan L. G.","family":"Guirao","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics and Statistics, Technical University of Cartagena, Hospital de Marina, 30203 Cartagena, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0206-3828","authenticated-orcid":false,"given":"Majeed A.","family":"Yousif","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Education, University of Zakho, Zakho 42002, Iraq"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9927-2388","authenticated-orcid":false,"given":"Ibrahim S.","family":"Ibrahim","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Education, University of Zakho, Zakho 42002, Iraq"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7600-1729","authenticated-orcid":false,"given":"Nejmeddine","family":"Chorfi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,11,21]]},"reference":[{"key":"ref_1","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_2","doi-asserted-by":"crossref","first-page":"980","DOI":"10.1515\/fca-2020-0051","article-title":"Nontrivial solutions of non-autonomous Dirichlet fractional discrete problems","volume":"23","author":"Cabada","year":"2020","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1972","DOI":"10.1007\/s13540-024-00295-z","article-title":"Stability analysis of discrete-time tempered fractional-order neural networks with time delays","volume":"27","author":"Zhang","year":"2024","journal-title":"Fract. Calc. Appl. 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