{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,21]],"date-time":"2026-02-21T02:39:37Z","timestamp":1771641577852,"version":"3.50.1"},"reference-count":41,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2024,10,15]],"date-time":"2024-10-15T00:00:00Z","timestamp":1728950400000},"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 aim of this work is to examine some q-analogs and differential properties of the gamma integral operator and its convolution products. The q-gamma integral operator is introduced in two versions in order to derive pertinent conclusions regarding the q-exponential functions. Also, new findings on the q-trigonometric, q-sine, and q-cosine functions are extracted. In addition, novel results for first and second-order q-differential operators are established and extended to Heaviside unit step functions. Lastly, three crucial convolution products and extensive convolution theorems for the q-analogs are also provided.<\/jats:p>","DOI":"10.3390\/sym16101368","type":"journal-article","created":{"date-parts":[[2024,10,15]],"date-time":"2024-10-15T08:09:08Z","timestamp":1728979748000},"page":"1368","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Some Estimates for Certain q-analogs of Gamma Integral Transform Operators"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8955-5552","authenticated-orcid":false,"given":"Shrideh","family":"Al-Omari","sequence":"first","affiliation":[{"name":"Department of Scientific Basic Sciences, Faculty of Engineering Technology, Al-Balqa Applied University, Amman 11134, Jordan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0007-3742-6178","authenticated-orcid":false,"given":"Wael","family":"Salameh","sequence":"additional","affiliation":[{"name":"Faculty of Information Technology, Abu Dhabi University, Abu Dhabi 59911, United Arab Emirates"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7761-4196","authenticated-orcid":false,"given":"Sharifah","family":"Alhazmi","sequence":"additional","affiliation":[{"name":"Mathematics Department, Al-Qunfudah University College, Umm Al-Qura University, Mecca 24382, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,10,15]]},"reference":[{"key":"ref_1","first-page":"193","article-title":"On q-definite integrals","volume":"41","author":"Jackson","year":"1910","journal-title":"Q. J. Pure Appl. Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"192","DOI":"10.1112\/plms\/s2-2.1.192","article-title":"The application of basic numbers to Bessel\u2019s and Legendre\u2019s functions","volume":"2","author":"Jackson","year":"1905","journal-title":"Proc. Lond. Math. Soc."},{"key":"ref_3","first-page":"21","article-title":"q-Laplace transform for product of general class of q-polynomials and q-analog of L-function","volume":"11","author":"Vyas","year":"2020","journal-title":"J. Inequ. Appl."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"095009","DOI":"10.1103\/PhysRevD.62.095009","article-title":"q-deformed conformal quantum mechanics","volume":"62","author":"Youm","year":"2000","journal-title":"Phys. Rev. D"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"310","DOI":"10.1016\/S0378-4371(01)00680-X","article-title":"q-Deformed structures and nonextensive statistics: A comparative study","volume":"305","author":"Lavagno","year":"2002","journal-title":"Physica A"},{"key":"ref_6","unstructured":"Miller, K., and Ross, B. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations, John Willy and Sons, Inc."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1007\/s40995-019-00815-0","article-title":"Operators of Basic (or q-)Calculus and fractional q-calculus and their applications in geometric function theory of complex analysis","volume":"44","author":"Srivastava","year":"2020","journal-title":"Iran. J. Sci. Tech. Trans. A Sci."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Zhao, K., Liu, J., and Lv, X. (2024). A unified approach to solvability and stability of multipoint BVPs for Langevin and Sturm\u2013Liouville equations with CH\u2013fractional derivatives and impulses via coincidence theory. Fractal Fract., 8.","DOI":"10.3390\/fractalfract8020111"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1186\/s13662-024-03801-y","article-title":"Study on the stability and its simulation algorithm of a nonlinear impulsive ABC-fractional coupled system with a Laplacian operator via F-contractive mapping","volume":"2024","author":"Zhao","year":"2024","journal-title":"Adv. Contin. Discret. Model."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Zhao, K. (2023). Solvability, Approximation and stability of periodic boundary value problem for a nonlinear Hadamard fractional differential equation with p-Laplacian. Axioms, 12.","DOI":"10.3390\/axioms12080733"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1053","DOI":"10.2298\/FIL2304053Z","article-title":"Existence and uh-stability of integral boundary problem for a class of nonlinear higher-order Hadamard fractional Langevin equation via Mittag-Leffler functions","volume":"37","author":"Zhao","year":"2023","journal-title":"Filomat"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1","DOI":"10.2140\/pjm.1975.60.1_","article-title":"A fractional Leibniz q-formula","volume":"60","author":"Verma","year":"1975","journal-title":"Pac. J. Math."},{"key":"ref_13","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. Discret. Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"340","DOI":"10.1002\/mana.19490020604","article-title":"Beitrage Zur Theorie Der Heineschen Reihen, die 24 Integrale der hypergeometrischen q-diferenzengleichung, das q-Analog on der Laplace transformation","volume":"2","author":"Hahn","year":"1949","journal-title":"Math. Nachrichten"},{"key":"ref_15","first-page":"1","article-title":"Some results for Laplace-type integral operator in quantum calculus","volume":"124","author":"Baleanu","year":"2018","journal-title":"Adv. Diff. Equ."},{"key":"ref_16","first-page":"825618","article-title":"On a q-analog of the Elzaki transform Called Mangontarum q-Transform","volume":"2014","author":"Mangontarum","year":"2014","journal-title":"Dis. Dyn. Nat. Soc."},{"key":"ref_17","first-page":"355","article-title":"On q-Laplace and q-Sumudu transforms of a product of generalized q-Bessel functions","volume":"11","author":"Albayrak","year":"2020","journal-title":"Math. Eng. Sci. Aerosp."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1016\/j.indag.2018.10.002","article-title":"On weighted q-Daehee polynomials with their applications","volume":"30","author":"Araci","year":"2019","journal-title":"Indag. Math."},{"key":"ref_19","first-page":"1001","article-title":"On q-Laplace type integral operators and their applications","volume":"18","author":"Albayrak","year":"2011","journal-title":"J. Diff. Equ. Appl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1016\/j.jksus.2015.04.008","article-title":"On q-analogs of the Mangontarum transform for certain q-Bessel functions and some application","volume":"28","year":"2016","journal-title":"J. King Saud Univ.-Sci."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Naik, S., and Haubold, H. (2016). On the q-Laplace Transform and Related Special Functions. Axioms, 5.","DOI":"10.3390\/axioms5030024"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1","DOI":"10.5269\/bspm.42087","article-title":"On the extremal solutions of superlinear Helmholtz problems","volume":"40","author":"Dammak","year":"2022","journal-title":"Bol. Soc. Parana. Mat."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"67","DOI":"10.5269\/bspm.41734","article-title":"A new class of Fredholm integral equations of the second kind with non symmetric kernel: Solving by wavelets method","volume":"39","author":"Mennouni","year":"2021","journal-title":"Bol. Soc. Paran. Mat."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Nasri, N., Aissaoui, F., Bouhali, K., Frioui, A., Meftah, B., and Zennir, K. (2023). Radwan, T. Fractional weighted midpoint-type inequalities for s-convex functions. Symmetry, 15.","DOI":"10.3390\/sym15030612"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1611","DOI":"10.2298\/FIL1706611K","article-title":"Some families of q-sums and q-products","volume":"31","author":"Kim","year":"2017","journal-title":"Filomat"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1","DOI":"10.4134\/BKMS.2014.51.4.1155","article-title":"The q-deformed gamma function and q-deformed polygamma function","volume":"51","author":"Chung","year":"2014","journal-title":"Bull. Korean Math. Soc."},{"key":"ref_27","unstructured":"Kim, T., Rim, S., and Park, S. (2007). Daehee formula associated with the q-extensions of trigonometric functions. arXiv."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2803","DOI":"10.12785\/amis\/080616","article-title":"New symmetric identities involving q-zeta type functions","volume":"8","author":"Araci","year":"2014","journal-title":"Appl. Math. Inf. Sci."},{"key":"ref_29","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_30","doi-asserted-by":"crossref","unstructured":"Kac, V., and Cheung, P. (2001). Quantum Calculus, Springer.","DOI":"10.1007\/978-1-4613-0071-7"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"98","DOI":"10.3906\/mat-1411-70","article-title":"The q-analog of the Ef2;1 g-transform and its applications","volume":"40","author":"Salem","year":"2016","journal-title":"Turk. J. Math."},{"key":"ref_32","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_33","doi-asserted-by":"crossref","first-page":"333","DOI":"10.2991\/jnmp.2007.14.3.4","article-title":"Fractional q-calculus on a time scale","volume":"14","author":"Atici","year":"2007","journal-title":"J. Nonlinear Math. Phys."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"327019","DOI":"10.1155\/2014\/327019","article-title":"q-Sumudu transforms of q-analogs of Bessel functions","volume":"2014","author":"Ucar","year":"2014","journal-title":"Sci. World J."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"10","DOI":"10.1186\/s13662-020-03147-1","article-title":"The q-Sumudu transform and its certain properties in a generalized q-calculus theory","volume":"2021","year":"2021","journal-title":"Adv. Diff. Equ."},{"key":"ref_36","first-page":"51","article-title":"On q-analogs of Mangontarum transform of some polynomials and certain class of H-functions","volume":"23","year":"2016","journal-title":"Nonlinear Stud."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"8322","DOI":"10.1002\/mma.6002","article-title":"On a q-Laplace-type integral operator and certain class of series expansion","volume":"44","year":"2021","journal-title":"Math. Meth. Appl. Sci."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"156","DOI":"10.1134\/S1061920814020034","article-title":"On the q-Analog of the Laplace transform","volume":"21","author":"Chung","year":"2014","journal-title":"Russ. J. Math. Phys."},{"key":"ref_39","first-page":"135","article-title":"Some fractional q-integrals and q-derivatives","volume":"15","year":"1969","journal-title":"Proc. Edinb. Math. Soc."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1016\/j.jmaa.2005.02.051","article-title":"approximation properties of Gamma operators","volume":"311","author":"Zeng","year":"2005","journal-title":"J. Math. Anal. Appl."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"388","DOI":"10.4208\/jms.v47n4.14.03","article-title":"properties of convergence for a class of generalized q-Gamma operators","volume":"47","year":"2014","journal-title":"J. Math. 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