A Comprehensive Exploration of Quantum Calculus: Extending Classical Calculus to Discrete Frameworks

Abstract

This manuscript explores quantum calculus (q-calculus), a framework that extends classical calculus to discrete settings without the use of limits. Beginning with a historical overview, it outlines the motivations and benefits of   q-calculus across various fields. Key concepts, such as q-analogs and q-integers, are discussed with a focus on their application to Mersenne numbers in number theory. The paper also examines the q-derivative, its relevance to financial models, and how q-analogs enhance predictive capabilities. Additionally, it delves into q-integrals and their connection to Riemann-Stieltjes functions, extending the integral framework to discrete cases. The study concludes by linking q-calculus to quantum computing, highlighting recent advancements and identifying gaps in current research, aiming to underscore its potential role in shaping future quantum technologies.