{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:37:17Z","timestamp":1760233037184,"version":"build-2065373602"},"reference-count":30,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2022,12,8]],"date-time":"2022-12-08T00:00:00Z","timestamp":1670457600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Scientific Research at King Khalid University","award":["R.G.P.1\/383\/43"],"award-info":[{"award-number":["R.G.P.1\/383\/43"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we develop theorems on finite and infinite summation formulas by utilizing the q and (q,h) anti-difference operators, and also we extend these core theorems to q(\u03b1) and (q,h)\u03b1 difference operators. Several integer order theorems based on q and q(\u03b1) difference operator have been published, which gave us the idea to derive the fractional order anti-difference equations for q and q(\u03b1) difference operators. In order to develop the fractional order anti-difference equations for q and q(\u03b1) difference operators, we construct a function known as the quantum geometric and alpha-quantum geometric function, which behaves as the class of geometric series. We can use this function to convert an infinite summation to a limited summation. Using this concept and by the gamma function, we derive the fractional order anti-difference equations for q and q(\u03b1) difference operators for polynomials, polynomial factorials, and logarithmic functions that provide solutions for symmetric difference operator. We provide appropriate examples to support our results. In addition, we extend these concepts to the (q,h) and (q,h)\u03b1 difference operators, and we derive several integer and fractional order theorems that give solutions for the mixed symmetric difference operator. Finally, we plot the diagrams to analyze the (q,h) and (q,h)\u03b1 difference operators for verification.<\/jats:p>","DOI":"10.3390\/sym14122604","type":"journal-article","created":{"date-parts":[[2022,12,9]],"date-time":"2022-12-09T02:20:31Z","timestamp":1670552431000},"page":"2604","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Method for Performing the Symmetric Anti-Difference Equations in Quantum Fractional Calculus"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8787-2711","authenticated-orcid":false,"given":"V. Rexma","family":"Sherine","sequence":"first","affiliation":[{"name":"Department of Mathematics, Sacred Heart College (Autonomous), Tirupattur 635601, Tamil Nadu, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9409-6896","authenticated-orcid":false,"given":"T. G.","family":"Gerly","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Sacred Heart College (Autonomous), Tirupattur 635601, Tamil Nadu, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9735-1004","authenticated-orcid":false,"given":"P.","family":"Chellamani","sequence":"additional","affiliation":[{"name":"Department of Mathematics, St. Joseph\u2019s College of Engineering, OMR, Chennai 600119, Tamil Nadu, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0418-193X","authenticated-orcid":false,"given":"Esmail Hassan Abdullatif","family":"Al-Sabri","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science and Arts (Muhayl Assir), King Khalid Universiy, Abha 62529, Saudi Arabia"},{"name":"Department of Mathematics and Computer, Faculty of Science, IBB University, Ibb 70270, Yemen"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7080-3824","authenticated-orcid":false,"given":"Rashad","family":"Ismail","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science and Arts (Muhayl Assir), King Khalid Universiy, Abha 62529, Saudi Arabia"},{"name":"Department of Mathematics and Computer, Faculty of Science, IBB University, Ibb 70270, Yemen"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1911-1395","authenticated-orcid":false,"given":"G. Britto Antony","family":"Xavier","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Sacred Heart College (Autonomous), Tirupattur 635601, Tamil Nadu, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9558-260X","authenticated-orcid":false,"given":"N.","family":"Avinash","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Sacred Heart College (Autonomous), Tirupattur 635601, Tamil Nadu, India"}]}],"member":"1968","published-online":{"date-parts":[[2022,12,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1017\/S0080456800002751","article-title":"XI.\u2014On q-functions and a certain difference operator","volume":"46","author":"Jackson","year":"1909","journal-title":"Earth Environ. Sci. Trans. R. Soc. Edinb."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Kac, V.G., and Cheung, P. (2002). 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