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

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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.

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Notes

  1. Mainardi, F.: On some properties of the Mittag–Leffler function, \(E_{\alpha }(tz)\), completely monotone for \(t>0\) with \(0<{\alpha }<1\). arXiv:1305.0161

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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.

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

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Authors and Affiliations

  1. Department of Mathematics, International Islamic University, Islamabad, Pakistan

    Gauhar Rahman & Muhammad Arshad

  2. Department of Mathematics, Anand International College of Engineering, Near Kanota, Agra Road, Jaipur, Rajasthan, 303012, India

    Praveen Agarwal

  3. Department of Mathematics, University of Sargodha, Sargodha, Pakistan

    Shahid Mubeen

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  1. Gauhar Rahman
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  2. Praveen Agarwal
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  3. Shahid Mubeen
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  4. Muhammad Arshad
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Correspondence to Praveen Agarwal.

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Rahman, G., Agarwal, P., Mubeen, S. et al. Fractional integral operators involving extended Mittag–Leffler function as its Kernel. Bol. Soc. Mat. Mex. 24, 381–392 (2018). https://doi.org/10.1007/s40590-017-0167-5

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