The Prabhakar fractional $q$-integral and $q$-differential operators, and their properties

Abstract: In this paper, we have introduced the Prabhakar fractional $q$-integral and $q$-differential operators. We first study the semi-group property of the Prabhakar fractional $q$-integral operator, which allowed us to introduce the corresponding $q$-differential operator. Formulas for compositions of $q$-integral and $q$-differential operators are also presented. We show the boundedness of the Prabhakar fractional $q$-integral operator in the class of $q$-integrable functions.