COMPLETELY MONOTONE MULTINOMIAL MITTAG-LEFFLER TYPE FUNCTIONS AND DIFFUSION EQUATIONS WITH MULTIPLE TIME-DERIVATIVES

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.

Automated summary

This work establishes basic properties of the Prabhakar type generalization of the multinomial Mittag-Leffler function, focusing on complete monotonicity. As particular examples, relaxation functions for equations with multiple time-derivatives in 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, with Laplace transform and Bernstein functions' technique being the main tools used in this work.