Fractional calculus with an integral operator containing a generalized Mittag–Leffler function in the kernel

In this paper, we introduce and investigate a fractional calculus with an integral operator which contains the following family of generalized Mittag–Leffler functions:Eα,βγ,κ(z)=∑n=0∞(γ)κnΓ(αn+β)znn!(z,β,γ∈C;R(α)>max{0,R(κ)-1};R(κ)>0)in its kernel, (λ)ν(λ)ν being the familiar Pochhammer symbol. A number of corollaries and consequences of the main results presented here are also considered.