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.

我们使用涉及Hilfer分数阶导数算子的分数演算方法，得出\（_ pR_q（\ lambda，\ eta; z）\）函数的一些经典和分数性质。此外，我们引入了一个积分运算符，其中包含\（_ pR_q（\ lambda，\ eta; z）\）函数作为内核：\（\ begin {aligned} \ left（{\ mathcal {R}} _ {\ lambda， \ eta，\ mathbf {u} _ {\ mathbf {p}}，\ mathbf {v} _ {\ mathbf {q}}，\ omega; a +} f \ right）（x）= \ int _a ^ x \ left（xr \ right）^ {\ eta -1} {} _pR_q \！\ left [{\ left。{\ begin {array} {c} \ mathbf {u} _ {\ mathbf {p}} \\ \ mathbf {v} _ {\ mathbf {q}} \ end {array}} \ \ right | \ lambda，\ eta; \ omega（xr）^ \ lambda} \ right] f（r）\ \ mathrm {d} r，\; \; \;（x> a），\ end {aligned} \）在空间L（a，  b）。我们确定运算符\（{\ mathcal {R}} _ {\ lambda，\ eta，\ mathbf {u} _ {\ mathbf {p}}，\ mathbf {v} _ {{mathbf {q} }，\ omega; a +} \）和Riemann–Liouville（RL）分数阶积分和微分算子。积分算子\（{\ mathcal {R}} _ {\ lambda，\ eta，\ mathbf {u} _ {\ mathbf {p}}，\ mathbf {v} _ {{mathbf {q}} ，\ omega; a +} \）及其反转。