Abstract Motivated mainly by certain interesting recent extensions of the generalized hypergeometric function [H. M. Srivastava, A. Çetinkaya and I. Onur Kıymaz, A certain generalized Pochhammer symbol and its applications to hypergeometric functions, Appl. Math. Comput. 226 2014, 484–491] by means of the generalized Pochhammer symbol, we introduce here a new extension of the generalized Mittag-Leffler function. We then systematically investigate several properties of the extended Mittag-Leffler function including some basic properties, Mellin, Euler-Beta, Laplace and Whittaker transforms. Furthermore, certain properties of the Riemann–Liouville fractional integrals and derivatives associated with the extended Mittag-Leffler function are also investigated. Some interesting special cases of our main results are pointed out.

中文翻译：

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扩展的 Mittag-Leffler 函数和相关的分数阶微积分算子

摘要 主要受广义超几何函数的某些有趣的近期扩展的启发 [HM Srivastava, A. Çetinkaya 和 I. Onur Kıymaz，某个广义 Pochhammer 符号及其在超几何函数中的应用，Appl。数学。计算。226 2014, 484–491] 通过广义 Pochhammer 符号，我们在这里引入了广义 Mittag-Leffler 函数的新扩展。然后，我们系统地研究了扩展 Mittag-Leffler 函数的几个属性，包括一些基本属性、Mellin、Euler-Beta、Laplace 和 Whittaker 变换。此外，还研究了与扩展 Mittag-Leffler 函数相关的 Riemann-Liouville 分数阶积分和导数的某些性质。指出了我们主要结果的一些有趣的特殊情况。