In this paper, a beta operator is used with Caputo Marichev-Saigo-Maeda (MSM) fractional differentiation of extended Mittag-Leffler function in terms of beta function. Further in this paper, some corollaries and consequences are shown that are the special cases of our main findings. We apply the beta operator on the right-sided MSM fractional differential operator and on the left-sided MSM fractional differential operator. We also apply beta operator on the right-sided MSM fractional differential operator with Mittag-Leffler function and the left-sided MSM fractional differential operator with Mittag-Leffler function.

中文翻译：

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具有扩展Mittag-Leffler函数的Caputo Marichev-Saigo-Maeda分数微分算子的Beta算子

在本文中，将Beta算子与Caputo Marichev-Saigo-Maeda（MSM）结合使用，扩展了Mittag-Leffler函数在β功能方面的分数区分。此外，本文还显示了一些推论和后果，这些是我们主要发现的特例。我们将beta运算符应用于右侧MSM分数微分运算符和左侧MSM分数微分运算符。我们还将beta运算符应用于具有Mittag-Leffler函数的右侧MSM分数微分运算符和具有Mittag-Leffler函数的左侧MSM分数微分运算符。