On the delta Mittag-Leffler functions and its application in monotonic analysis

In this paper, we first introduce a discrete Mittag-Leffler function of delta type. Using the Laplace transformation, some properties of the new special function are obtained. Second, we use this function to define new discrete fractional operators, namely AB fractional differences and sums, based on the Riemann-Liouville operators. We also applied the Laplace transformation on the new special functions and the related discrete operators. Finally, we propose and implement the mean value technique of discrete fractional calculus and demonstrate the advantages in terms of AB fractional differences.