An integral operator involving generalized Mittag-Leffler function and associated fractional calculus results

Abstract

In the present paper, we first introduce and investigate the generalized extended Mittag-Leffler (GEML) function which is represented in the following manner:

$$\begin{aligned}&E_{\delta , \kappa }^{\vartheta ; d}(z; q, \rho , \zeta )= \sum \limits _{n=0}^{\infty }\frac{B_{q}^{(\rho , \zeta )}(\vartheta +n, d-\vartheta )}{B(\vartheta , d-\vartheta )}\frac{(d)_{ n}}{\Gamma (\delta n+ \kappa )} \frac{z^{n}}{n!}\\ &\left( \begin{array}{cc} {\mathfrak{R}}(d)> {\mathfrak{R}}(\vartheta )> 0, {\mathfrak{R}}(\delta )> 0,{\mathfrak{R}}(\kappa )> 0, \\ {\mathfrak{R}}(q) \ge 0,\, \text{min}\left\{ {\mathfrak{R}}(\vartheta +n), {\mathfrak{R}}(d-\vartheta ), {\mathfrak{R}}(\rho ), {\mathfrak{R}}(\zeta )\right\} > 0 \end{array}\right) \end{aligned}$$

and propose some of it’s integral representations. Next, we present fractional calculus of function of our study. Further, we introduce and study an integral operator whose kernel is generalized extended Mittag-Leffler (GEML) function and point out it’s known special cases. Next, we derive some properties of aforementioned integral operator which includes it’s composition relationship with right-sided Riemann–Liouville fractional integral operator $I^{\gamma }_{a+}$ and boundedness. Finally, we obtain image of $(\tau -a)^{\alpha -1}\Phi _{l_{j};\upsilon _{j}Q}^{k_{j};\varrho _{j}P}(\beta \tau ,s,a)$ under integral operator of our study. The results derived in this paper generalizes the results obtained by Özarslan and Yilmaz (J Inequal Appl 85:1–10, 2014) and Rahman et al. (Sociedad Matemática Mexican. https://doi.org/10.1007/s40590-017-0167-5, 2017).