Use of quantum calculus approach in Mathematical Sciences and its role in Geometric Function Theory

Abstract

The study of 300 years old history of quantum calculus or q-calculus or q-disease, since the Bernoulli and Euler, is often considered to be one of the most di ffi cult subjects to engage in mathematics. Nowadays there is a rapid growth of activities in the area of q-calculus due to its applications in various fields such as mathematics, mechanics, and physics. The history of study of q-calculus may be illustrated by its wide variety of applications in quantum mechanics, analytic number theory, theta functions, hypergeometric functions, theory of finite di ff erences, gamma function theory, Bernoulli and Euler polynomials, Mock theta functions, combinatorics, umbral calculus, multiple hypergeometric functions, Sobolev spaces, operator theory, and more recently in the theory of analytic and harmonic univalent functions. In q-calculus, we are generally interested in q-analogues that arise naturally, rather than in arbitrarily contriving q-analogues of known results. While focusing on excitement and romance with development of q-calculus and its applications in certain fields of mathematical sciences and physics, we will also look at q-analogues of some of the recent results in geometric function theory and, in particular, theory of analytic and harmonic univalent functions in the unit disc.