{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,21]],"date-time":"2026-02-21T02:39:39Z","timestamp":1771641579633,"version":"3.50.1"},"reference-count":38,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2024,1,24]],"date-time":"2024-01-24T00:00:00Z","timestamp":1706054400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this article, the authors introduce the q-analogue of the M-function, and establish four theorems related to the Riemann\u2013Liouville fractional q-calculus operators pertaining to the newly defined q-analogue of M-functions. In addition, to establish the solution of fractional q-kinetic equations involving the q-analogue of the M-function, we apply the technique of the q-Laplace transform and the q-Sumudu transform and its inverse to obtain the solution in closed form. Due to the general nature of the q-calculus operators and defined functions, a variety of results involving special functions can only be obtained by setting the parameters appropriately.<\/jats:p>","DOI":"10.3390\/axioms13020078","type":"journal-article","created":{"date-parts":[[2024,1,25]],"date-time":"2024-01-25T08:44:07Z","timestamp":1706172247000},"page":"78","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Fractional q-Calculus Operators Pertaining to the q-Analogue of \r\nM-Function and Its Application to Fractional q-Kinetic Equations"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8654-2545","authenticated-orcid":false,"given":"Biniyam","family":"Shimelis","sequence":"first","affiliation":[{"name":"Department of Mathematics, Wollo University, Dessie P.O. Box 1145, Ethiopia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9978-2177","authenticated-orcid":false,"given":"Daya Lal","family":"Suthar","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Wollo University, Dessie P.O. Box 1145, Ethiopia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5415-1777","authenticated-orcid":false,"given":"Dinesh","family":"Kumar","sequence":"additional","affiliation":[{"name":"Department of Applied Sciences, College of Agriculture, Agriculture University Jodhpur, Jodhpur 342304, India"}]}],"member":"1968","published-online":{"date-parts":[[2024,1,24]]},"reference":[{"key":"ref_1","first-page":"449","article-title":"A note on a generalized M-series as a special function of fractional calculus","volume":"12","author":"Sharma","year":"2009","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_2","first-page":"187","article-title":"Fractional integration and fractional differentiation of the M-series","volume":"11","author":"Sharma","year":"2008","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_3","first-page":"554","article-title":"Sur la nouvelle function E\u03b1 (x), C. R","volume":"137","year":"1903","journal-title":"Acad. Sci. Paris Ser. II"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1007\/BF02403202","article-title":"Uber den Fundamentalsatz in der Teorie der Funktionen E\u03b1(x)","volume":"29","author":"Wiman","year":"1905","journal-title":"Acta Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1080\/10652469308819007","article-title":"The Mittag-Leffler and related functions","volume":"1","author":"Miller","year":"1993","journal-title":"Integral Transform. Spec. Funct."},{"key":"ref_6","first-page":"165","article-title":"Fractional calculus of the generalized Mittag\u2013Leffler functions","volume":"31","author":"Saxena","year":"2009","journal-title":"J. Indian Acad. Math."},{"key":"ref_7","first-page":"225","article-title":"Some results on fractional calculus operators associated with the M-function","volume":"33","author":"Purohit","year":"2010","journal-title":"Hadronic J."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1320830","DOI":"10.1080\/23311835.2017.1320830","article-title":"Marichev-Saigo-Maeda fractional calculus operators, Srivastava polynomials and generalized Mittag-Leffler function","volume":"4","author":"Mishra","year":"2017","journal-title":"Cogent Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1080\/10652469808819189","article-title":"On Mittag-Leffler type function and applications","volume":"7","author":"Saigo","year":"1998","journal-title":"Integral Transform. Spec. Funct."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1023\/A:1002695807970","article-title":"The fractional kinetic equation and thermonuclear functions","volume":"327","author":"Haubold","year":"2000","journal-title":"Astrophys. Space Sci."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1023\/A:1021175108964","article-title":"On fractional kinetic equations","volume":"282","author":"Saxena","year":"2002","journal-title":"Astrophys. Space Sci."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1023\/B:ASTR.0000032531.46639.a7","article-title":"Unified fractional kinetic equation and a fractional diffusion equation","volume":"290","author":"Saxena","year":"2004","journal-title":"Astrophys. Space Sci."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"657","DOI":"10.1016\/j.physa.2004.06.048","article-title":"On generalized fractional kinetic equations","volume":"34","author":"Saxena","year":"2004","journal-title":"Physica A"},{"key":"ref_14","first-page":"455","article-title":"Solution of a general family of kinetic fractional equations associated with the generalized Mittag-Leffler function","volume":"23","author":"Kumar","year":"2018","journal-title":"Nonlinear Funct. Anal. Appl."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"289387","DOI":"10.1155\/2015\/289387","article-title":"On generalized fractional kinetic equations involving generalized Bessel function of the first kind","volume":"2015","author":"Kumar","year":"2015","journal-title":"Math. Probl. Eng."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"8645161","DOI":"10.1155\/2020\/8645161","article-title":"Solution of fractional kinetic equations associated with the (p,q)-Mathieu-type series","volume":"2020","author":"Suthar","year":"2020","journal-title":"Discrete Dyn. Nat. Soc."},{"key":"ref_17","first-page":"3803","article-title":"Class of integrals and applications of fractional kinetic equation with the generalized multi-index Bessel function","volume":"14","author":"Suthar","year":"2021","journal-title":"Discrete Contin. Dyn. Syst. Ser. S"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"5074039","DOI":"10.1155\/2019\/5074039","article-title":"Application of Laplace transform on fractional kinetic equation pertaining to the generalized Galu\u00e9 type Struve function","volume":"2019","author":"Habenom","year":"2019","journal-title":"Adv. Math. Phys."},{"key":"ref_19","first-page":"13","article-title":"Computable solution of fractional kinetic equations using Mathieu-type series","volume":"234","author":"Khan","year":"2019","journal-title":"Adv. Differ. Equ."},{"key":"ref_20","first-page":"183","article-title":"On solutions of generalized kinetic equations of fractional order","volume":"32","author":"Gupta","year":"2014","journal-title":"Bol. Soc. Parana. Mat."},{"key":"ref_21","unstructured":"Annaby, M.H., and Mansour, Z.S. (2012). With a Foreword by Mourad Ismail Lecture Notes in Math. 2056, Springer."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Kac, V.G., and Cheung, P. (2002). Quantum Calculus, Universitext Springer.","DOI":"10.1007\/978-1-4613-0071-7"},{"key":"ref_23","unstructured":"Gasper, G., and Rahman, M. (1990). With a Foreword by Richard Askey Encyclopedia Math. Appl. 35, Cambridge University Press."},{"key":"ref_24","first-page":"1","article-title":"On certain q-difference equations and q-Laplace transform","volume":"28","author":"Abdi","year":"1962","journal-title":"Proc. Nat. Inst. Sci. India Part A"},{"key":"ref_25","first-page":"10","article-title":"Some results for Laplace-type integral operator in quantum calculus","volume":"124","author":"Baleanu","year":"2018","journal-title":"Adv. Differ. Equ."},{"key":"ref_26","first-page":"239","article-title":"On q-analogues of Sumudu transform","volume":"21","author":"Albayrak","year":"2013","journal-title":"An. \u015etiin\u0163. Univ. Ovidius Constan\u0163a Ser. Mat."},{"key":"ref_27","first-page":"33","article-title":"On fractional q-kinetic equation","volume":"36","author":"Garg","year":"2012","journal-title":"Mat. Bilten"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"726","DOI":"10.3906\/mat-1703-7","article-title":"An application of q-Sumudu transform for fractional q -kinetic equation","volume":"42","author":"Purohit","year":"2018","journal-title":"Turk. J. Math."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"311","DOI":"10.2298\/AADM0701311R","article-title":"Fractional integrals and derivatives in q-calculus","volume":"1","author":"Rajkovic","year":"2007","journal-title":"Appl. Anal. Discrete Math."},{"key":"ref_30","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_31","first-page":"1","article-title":"Generalization of Mittag-Leffler function and its application in quantum-calculus","volume":"2","author":"Jain","year":"2018","journal-title":"Int. J. Innov. Res. Technol. Manag."},{"key":"ref_32","first-page":"15","article-title":"A generalization of q-Mittag-Leffler function","volume":"35","author":"Purohit","year":"2011","journal-title":"Mat. Bilten"},{"key":"ref_33","first-page":"791","article-title":"On some recurrence relation of generalized q-Mittag-Leffler function","volume":"6","author":"Sharma","year":"2016","journal-title":"Math. Aeterna"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1017\/S0305004100045060","article-title":"Certain fractional q-integrals and q-derivatives","volume":"66","author":"Agarwal","year":"1969","journal-title":"Proc. Camb. Philos. Soc."},{"key":"ref_35","first-page":"31","article-title":"Variational methods for fractional q-Sturm-Liouville problems","volume":"150","author":"Mansour","year":"2016","journal-title":"Bound. Value Probl."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"340","DOI":"10.1002\/mana.19490020604","article-title":"Beitrage Zur Theorie der Heineschen Reihen, die 24 Integrale der hypergeo-metrischen q-Differenzengleichung, das q-Analogon der Laplace Transformation","volume":"2","author":"Hahn","year":"1949","journal-title":"Math. Nachr."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1017\/S0013091500011469","article-title":"Some fractional q-integrals and q-derivatives","volume":"15","year":"1966","journal-title":"Proc. Edinb. Math. Soc."},{"key":"ref_38","first-page":"141","article-title":"Certain properties of fractional calculus operators associated with generalized Mittag-Leffler function","volume":"8","author":"Saxena","year":"2005","journal-title":"Fract. Calc. Appl. Anal."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/2\/78\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T13:48:49Z","timestamp":1760104129000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/2\/78"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,1,24]]},"references-count":38,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2024,2]]}},"alternative-id":["axioms13020078"],"URL":"https:\/\/doi.org\/10.3390\/axioms13020078","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,1,24]]}}}