Extended q-Difference Operator of the Second Kind: Analytical Insights and Diverse Applications

The q-difference operator is a basic tool in q-calculus, widely used in many mathematical and scientific fields, such as statistical physics, fractal geometry, quantum mechanics, number theory, combinatorics, and orthogonal polynomials. Its applications are also found in advanced sciences, like quantum theory, mechanics, and the theory of relativity. In this paper, we define the extended q-difference operator of the second kind and its inverse. We further derive the Leibniz theorem, Montmorte’s theorem, and several properties associated with the extended q-difference operator of the second kind. Additionally, a formula for the sum of partial sums of higher powers of real numbers in an arithmetic progression is constructed using its inverse. Numerical examples are given to illustrate the results.