THE q-ANALOGS OF FRACTIONAL OPERATORS CONCERNING ANOTHER FUNCTION IN THE POWER-LAW

This paper presents new concepts of fractional quantum operators by connecting fractional and quantum calculus. First, we define the [Formula: see text]-analogs of the higher-order derivative and integral concerning another function [Formula: see text] in a new space [Formula: see text], by the [Formula: see text]-shifting operator [Formula: see text]. Then, we introduce the [Formula: see text]-analogs of the left and right sided [Formula: see text]-Riemann–Liouville (RL) fractional integral and derivative on a finite interval [Formula: see text]. Furthermore, we investigate their important characteristics such as boundedness, continuity, semi-group, and fundamentals of fractional [Formula: see text]-calculus theorem. Finally, to demonstrate the application of these new operators, we establish the existence and uniqueness (EaU) of the solution for a new class of nonlocal implicit differential equation involving [Formula: see text]-RL fractional derivative by utilizing the Banach fixed point technique (FBT). The new operators cover the existing classical fractional and [Formula: see text]-fractional operators; and we can deduce for the first time the [Formula: see text]-analogs of the Katugampola, Hadamard, and conformable RL fractional operators.