The quantum (or q-) calculus is a vital area of study in the field of traditional mathematical analysis. It concentrates on a theoretically valuable generalization of integration and differentiation operations. Quantum calculus is a wide area of mathematical science with historical origins, as well as a revived focus in the modern era. Notably, quantum calculus has a long tradition that can be traced back to Bernoulli and Euler's function. However, it has piqued the interest of contemporary mathematicians in recent decades, owing to its wide range of applications. It entails complex calculations and computations, making it more difficult than other mathematical topics.

In recent years, there has been a dramatic increase of activity in the region of q-calculus and its applications in diverse disciplines such as physics, mathematics, and mechanics. The development of q-calculus can be demonstrated by its widespread applications in the theory of finite differences, quantum mechanics, theta and mock theta functions, analytic number theory, hypergeometric functions, combinatorics, multiple hypergeometric functions, gamma function theory, Sobolev spaces, Bernoulli and Euler polynomials, operator theory, and, more recently in the geometric theory of analytic and harmonic univalent functions. The above-mentioned areas of application have made q-calculus research essential. Apart from these applications, simple (or q-) series and basic (or q-) polynomials, especially basic (or q-) hypergeometric functions and basic (or q-) hypergeometric polynomials, have found widespread use in number theory and partition theory. Indeed, basic (or q-) hypergeometric functions have been used in a wide range of fields, namely, combinatorial analysis, finite vector spaces, Lie theory, particle physics, nonlinear electric circuit theory, mechanical engineering, heat conduction theory, quantum mechanics, cosmology, and statistics.

The goal of this Special Issue is to invite original research and reviews focusing on the q-calculus and fractional q- calculus, particularly in the sense of geometric function theory of complex analysis. The key motivation for proposing this Special Issue is for study in geometric function theory of complex analysis using the classical q-calculus and the fractional q-calculus to have implementation potential and result in significant contributions and advancements on these and other similar topics.

Potential topics include but are not limited to the following:

• Analytic functions and univalent functions in q-analogue
• q-Harmonic functions and q-Meromorphic functions
• q-Differential subordination
• Applications of q-analogue of special functions in geometric function theory