The multinomial Mittag-Leffler function plays a crucial role in the study of multi-term time-fractional evolution equations. In this work we establish basic properties of the Prabhakar type generalization of this function with the main emphasis on complete monotonicity. As particular examples, the relaxation functions for equations with multiple time-derivatives in the so-called “natural” and “modified” forms are studied in detail and useful estimates are derived. The obtained results extend known properties of the classical Mittag-Leffler function. The main tools used in this work are Laplace transform and Bernstein functions’ technique.

中文翻译：

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具有多个时间导数的完全单调多项式米塔格勒夫勒型函数和扩散方程

多项式Mittag-Leffler函数在多项式时间分数演化方程的研究中起着至关重要的作用。在这项工作中，我们建立了该函数的Prabhakar类型概括的基本属性，主要强调完全单调性。作为特定示例，详细研究了具有多个时间导数的所谓“自然”和“修改”形式的方程的松弛函数，并得出了有用的估计值。获得的结果扩展了经典Mittag-Leffler函数的已知性质。这项工作中使用的主要工具是拉普拉斯变换和伯恩斯坦函数的技术。