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 q-analogs of the higher-order derivative and integral concerning another function ψ in a new space Lq;ψ¹_q;ψ [a,b], by the q-shifting operator _aφ_q (y) = qy + (1 - q)a. Then, we introduce the q-analogs of the left and right sided ψ-Riemann-Liouville (RL) fractional integral and derivative on a finite interval [a,b]. Furthermore, we investigate their important characteristics such as boundedness, continuity, semi-group, and fundamentals of fractional q-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 (q; ψ)-RL fractional derivative by utilizing the Banach fixed point technique (FBT). The new operators cover the existing classical fractional and q-fractional operators; and we can deduce for the first time the q-analogs of the Katugampola, Hadamard, and conformable RL fractional operators.