Generalized Fractional Calculus Operators and the $_pR_q(\lambda ,\eta ;z)$ Function

Abstract

We derive some classical and fractional properties of $_pR_q(\lambda ,\eta ;z)$ function using fractional calculus approach involving Hilfer fractional derivative operator. Further, we introduce an integral operator which contains $_pR_q(\lambda ,\eta ;z)$ function as a kernel: $\begin{aligned} \left( {\mathcal{R}}_{\lambda ,\eta ,\mathbf{u}_{\mathbf{p}},\mathbf{v}_{\mathbf{q}},\omega ;a+} f\right) (x) = \int _a^x \left( x-r\right) ^{\eta -1} {}_pR_q\!\left[ {\left. {\begin{array}{c} \mathbf{u}_{\mathbf{p}}\\ \mathbf{v}_{\mathbf{q}}\end{array}} \ \right| \lambda ,\eta ; \omega (x-r)^\lambda }\right] f(r) \ \mathrm{d}r, \;\;\; (x>a), \end{aligned}$in space L(a, b). We establish the composition of the operator ${\mathcal{R}}_{\lambda ,\eta ,\mathbf{u}_{\mathbf{p}},\mathbf{v}_{\mathbf{q}},\omega ;a+}$ with the Riemann–Liouville (R–L) fractional integral and differential operators. Some composition properties of integral operator ${\mathcal{R}}_{\lambda ,\eta ,\mathbf{u}_{\mathbf{p}},\mathbf{v}_{\mathbf{q}},\omega ;a+}$ and their inversion have also been exhibited.