Citation: Sherine, V.R.; Gerly, T.G.; Chellamani, P.; Al-Sabri, E.H.A.; Ismail, R.; Xavier, G.B.A.; Avinash, N. A Method for Performing the Symmetric Anti-Difference Equations in Quantum Fractional Calculus. Symmetry 2022, 14, 2604. https:// doi.org/10.3390/sym14122604 Academic Editor: Hüseyin Budak Received: 10 November 2022 Accepted: 1 December 2022 Published: 8 December 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). symmetry S S Article A Method for Performing the Symmetric Anti-Difference Equations in Quantum Fractional Calculus V. Rexma Sherine 1, * , T. G. Gerly 1 , P. Chellamani 2 , Esmail Hassan Abdullatif Al-Sabri 3,4 , Rashad Ismail 3,4 , G. Britto Antony Xavier 1 , N. Avinash 1 1 Department of Mathematics, Sacred Heart College (Autonomous), Tirupattur 635601, Tamil Nadu, India 2 Department of Mathematics, St. Joseph’s College of Engineering, OMR, Chennai 600119, Tamil Nadu, India 3 Department of Mathematics, College of Science and Arts (Muhayl Assir), King Khalid Universiy, Abha 62529, Saudi Arabia 4 Department of Mathematics and Computer, Faculty of Science, IBB University, Ibb 70270, Yemen * Correspondence: rexmaprabu123@gmail.com Abstract: 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 (α) and (q, h) α difference operators. Several integer order theorems based on q and q (α) difference operator have been published, which gave us the idea to derive the fractional order anti-difference equations for q and q (α) difference operators. In order to develop the fractional order anti-difference equations for q and q (α) 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 (α) 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) α 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) α difference operators for verification. Keywords: q and (q, h) difference operators; quantum geometric function; alpha quantum geometric function; gamma functions and fractional order sum 1. Introduction The study of calculus without limits is nowadays known as quantum calculus. Jack- son’s work [1] sheds light on the invention of q-calculus, often known as quantum calculus, while in 1908, Euler and Jacobi had already developed this type of calculus. The field of q-calculus emerged as a link between mathematics and physics. Numerous mathematical fields, including combinatorics, orthogonal polynomials, number theory, fundamental hyper-geometric functions, as well as other sciences, including mechanics, quantum theory, and the theory of relativity, make extensive use of it. Most of the basic facts of quantum calculus are covered in the book by Kac and Cheung [2]. Quantum calculus is a branch within the mathematical topic of time scales calculus. The q-differential equations are typically defined on a time scale set T q , where q is the scale index. Time scales offer a unifying framework for investigating the dynamic equations. The majority of the fundamental theory in the calculus of time scales was compiled in the text by Bohner and Peterson [3]. Though quantum calculus plays a major role in physics, engineers and mathematics also show interest in fractional q-difference equations and q-calculus. The main focus of developing q-difference equations is to characterize some unique physical processes and other areas. Some of the topics that have been developed and investigated in conjunction Symmetry 2022, 14, 2604. https://doi.org/10.3390/sym14122604 https://www.mdpi.com/journal/symmetry