{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,3]],"date-time":"2026-06-03T19:03:14Z","timestamp":1780513394874,"version":"3.54.1"},"reference-count":13,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2024,11,26]],"date-time":"2024-11-26T00:00:00Z","timestamp":1732579200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Scientific Research at Jouf University through the Fast-track Research Funding Program"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Many properties of special polynomials, such as recurrence relations, sum formulas, integral transforms and symmetric identities, have been studied in the literature with the help of generating functions and their functional equations. In this paper, we introduce hybrid forms of q-Mittag-Leffler functions. The q-Mittag-Leffler\u2013Bessel and q-Mittag-Leffler\u2013Tricomi functions are constructed using a q-symbolic operator. The generating functions, series definitions, q-derivative formulas and q-recurrence formulas for q-Mittag-Leffler\u2013Bessel and q-Mittag-Leffler\u2013Tricomi functions are obtained. The Nq-transforms and Nq-transforms of q-Mittag-Leffler\u2013Bessel and q-Mittag-Leffler\u2013Tricomi functions are obtained. These hybrid q-special functions are also studied by plotting their graphs for specific values of the indices and parameters.<\/jats:p>","DOI":"10.3390\/sym16121580","type":"journal-article","created":{"date-parts":[[2024,11,26]],"date-time":"2024-11-26T07:40:32Z","timestamp":1732606832000},"page":"1580","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["A Novel Family of q-Mittag-Leffler-Based Bessel and Tricomi Functions via Umbral Approach"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4681-9885","authenticated-orcid":false,"given":"Waseem Ahmad","family":"Khan","sequence":"first","affiliation":[{"name":"Department of Electrical Engineering, Prince Mohammad Bin Fahd University, P.O. Box 1664, Al Khobar 31952, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mofareh","family":"Alhazmi","sequence":"additional","affiliation":[{"name":"Mathematics Department, College of Science, Jouf University, P.O. Box 2014, Sakaka 72388, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8463-3611","authenticated-orcid":false,"given":"Tabinda","family":"Nahid","sequence":"additional","affiliation":[{"name":"Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore 632014, India"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2024,11,26]]},"reference":[{"key":"ref_1","unstructured":"Alatawi, M.S. (Department of Mathematics, Faculty of Science, University of Tabuk, Tabuk, Saudi Arabia); Raza, N. (Mathematics Section, Women\u2019s College, Aligarh Muslim University, Aligarh, India); Kumar, M. (Department of Mathematics, Aligarh Muslim University, Aligarh, India); Khan, W.A. (Department of Mathematics and Natural Sciences, Prince Mohammad Bin Fahd University, Al Khobar, Saudi Arabia) Insights into q-Mittag-Leffler-Laguerre Polynomials: Properties and Operational Identities. 2024, submitted."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"793","DOI":"10.1515\/gmj-2020-2079","article-title":"q-Tricomi functions and quantum algebra representations","volume":"28","author":"Riyasat","year":"2021","journal-title":"Georgian Math. J."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Kac, V.G., and Cheung, P. (2002). Quantum Calculus, Springer.","DOI":"10.1007\/978-1-4613-0071-7"},{"key":"ref_4","first-page":"11","article-title":"On integral representations of q-gamma and q-beta functions","volume":"9","author":"Kac","year":"2005","journal-title":"Rend. Math. Acc. Lincei"},{"key":"ref_5","first-page":"193","article-title":"On q-definite integrals","volume":"41","author":"Jackson","year":"1910","journal-title":"Quart. J. Pure Appl. Math."},{"key":"ref_6","first-page":"329","article-title":"q-Bessel functions: The point of view of the generating function method","volume":"17","author":"Dattoli","year":"1997","journal-title":"Rend. Mat. Appl."},{"key":"ref_7","first-page":"153","article-title":"Symmetric q-Bessel functions","volume":"51","author":"Dattoli","year":"1996","journal-title":"Le Matematiche"},{"key":"ref_8","first-page":"159","article-title":"Linear sequential q-difference equations of fractional order","volume":"12","author":"Mansour","year":"2009","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"413","DOI":"10.2298\/FIL1302411A","article-title":"On q-Sumudu transforms of certain q-polynomials","volume":"27","author":"Albayrak","year":"2013","journal-title":"Filomat"},{"key":"ref_10","first-page":"239","article-title":"On q-analogues of Sumudu transform","volume":"21","author":"Albayrak","year":"2013","journal-title":"Analele Stiintifice Ale Univ. Ovidius Constanta"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2587","DOI":"10.2298\/FIL1709587A","article-title":"On q-analogues of the Natural Transform of Certain q-Bessel Functions and Some Application","volume":"31","year":"2017","journal-title":"Filomat"},{"key":"ref_12","first-page":"189","article-title":"On q-Laplace transforms of the q-Bessel functions","volume":"10","author":"Purohit","year":"2007","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_13","unstructured":"Horwood, E. (1983). Basic Hypergeometric Functions and Applications, Chichester."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/12\/1580\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:39:45Z","timestamp":1760114385000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/12\/1580"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,11,26]]},"references-count":13,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2024,12]]}},"alternative-id":["sym16121580"],"URL":"https:\/\/doi.org\/10.3390\/sym16121580","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,11,26]]}}}