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Boletín de la Sociedad Matemática Mexicana

, Volume 24, Issue 2, pp 381–392 | Cite as

Fractional integral operators involving extended Mittag–Leffler function as its Kernel

  • Gauhar Rahman
  • Praveen Agarwal
  • Shahid Mubeen
  • Muhammad Arshad
Original Article

Abstract

This paper is devoted to the study of fractional calculus with an integral and differential operators containing the following family of extended Mittag–Leffler function:
$$\begin{aligned} E_{\alpha ,\beta }^{\gamma ;c}(z; p)=\sum \limits _{n=0}^{\infty }\frac{B_p(\gamma +n, c-\gamma )(c)_{n}}{B(\gamma , c-\gamma )\Gamma (\alpha n+\beta )}\frac{z^n}{n!}, (z,\beta , \gamma \in \mathbb {C}), \end{aligned}$$
in its kernel. Also, we further introduce a certain number of consequences of fractional integral and differential operators containing the said function in their kernels.

Keywords

Fractional integral operator Fractional differential operator Mittag–Leffler function Lebesgue measurable function Extended Mittag–Leffler function 

Mathematics Subject Classification

33C20 33E20 26A33 26A99 

Notes

Acknowledgements

The authors would like to express profound gratitude to referees for deeper review of this paper and the referee’s useful suggestions that led to an improved presentation of the paper.

Compliance with ethical standards

Conflict of interest The authors declare that there is no conflict of interests.

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Copyright information

© Sociedad Matemática Mexicana 2017

Authors and Affiliations

  • Gauhar Rahman
    • 1
  • Praveen Agarwal
    • 2
  • Shahid Mubeen
    • 3
  • Muhammad Arshad
    • 1
  1. 1.Department of MathematicsInternational Islamic UniversityIslamabadPakistan
  2. 2.Department of MathematicsAnand International College of EngineeringJaipurIndia
  3. 3.Department of MathematicsUniversity of SargodhaSargodhaPakistan

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