$q$-ANALOGUE OF HILFER-KATUGAMPOLA FRACTIONAL DERIVATIVES AND APPLICATIONS

A novel $q^{p}$-variant of the $q-$Mittag-Leffler function and a quantum analogue $^p\mathcal{D}^{\alpha,\beta}_{a\pm,q}$ of the Hilfer-Katugampola fractional derivative are defined. Then, generalizations of the $q-$Taylor's formula and the $q-$differential transform and its inverse are obtained using the operator $^p\mathcal{D}^{\alpha,\beta}_{a\pm,q}$. Additionally, a few properties of the newly defined $q$-differential transform are established. Finally, three proposed fractional $q$-difference equations are solved to show the effectiveness of the transform.