This paper presents new developments in quantum fractional calculus by introducing the q-analogues of the left and right ( ${L}\&{R}$ ) sided ψ-Caputo fractional derivatives (CFDs), and also establishes some essential properties of these novel operators. As an implementation of these new operators, we investigate the existence and uniqueness (EaU) of solutions for a new class of impulsive differential equations involving the q-analogue of the left-side ψ-CFD, by employing Banach’s and Krasnoselskii’s fixed point theorems (FPTs), and supporting with illustrative examples. The introduced operators are more general than the existing operators, such as q-CFD and the classical CFD.
