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

  • M. K. Bansal
  • N. Jolly
  • R. Jain
  • Devendra KumarEmail author
Original Research Paper


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., 2017).


Mittag-Leffler function Generalized Beta function Hilfer derivative Integral operator 

Mathematics Subject Classification

33E12 33B15 36A33 47G10 


Compliance with ethical standards

Conflict of interest

The author declares that there is no conflict of interest regarding the publication of this article.

Human/animals participation

The authors declare that there is no research involving human participants and/or animals in the content this paper.


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Copyright information

© Forum D'Analystes, Chennai 2018

Authors and Affiliations

  1. 1.Department of Applied SciencesGovernment Engineering CollegeBanswaraIndia
  2. 2.Department of MathematicsMalaviya National Institute of TechnologyJaipurIndia
  3. 3.Department of MathematicsUniversity of RajasthanJaipurIndia

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