The extended Mittag-Leffler function via fractional calculus

Volume 10, Issue 8, pp 4244--4253 http://dx.doi.org/10.22436/jnsa.010.08.19

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Authors

Gauhar Rahman - Department of Mathematics, International Islamic University, Islamabad, Pakistan. Dumitru Baleanu - Department of Mathematics, Cankaya University, Ankara, Turkey. - Institute of Space Sciences, Magurele-Bucharest, Romania. Maysaa Al Qurashi - Department of Mathematics, College of Science, King Saud University, Riyadh, Saudia Arabia. Sunil Dutt Purohit - Department of HEAS (Mathematics), Rajasthan Technical University, Kota, India. Shahid Mubeen - Department of Mathematics, University of Sargodha, Sargodha, Pakistan. Muhammad Arshad - Department of Mathematics, International Islamic University, Islamabad, Pakistan.

Abstract

In this study, our main attempt is to introduce fractional calculus (fractional integral and differential) operators which contain the following new family of extended Mittag-Leffler function: \[ E_{\alpha,\beta}^{\gamma;q, c}(z)=\sum\limits_{n=0}^{\infty}\frac{B_p(\gamma+nq, c-\gamma)(c)_{nq}}{B(\gamma, c-\gamma)\Gamma(\alpha n+\beta)}\frac{z^n}{n!},~~~ (z,\beta, \gamma\in\mathbb{C}), \] as its kernel. We also investigate a certain number of their consequences containing the said function in their kernels.

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Keywords

• Fractional integration
• differential operator
• Mittag-Leffler function
• Lebesgue measurable function
• extended Mittag-Leffler function.

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