Further analytic/numeric development for the new Mittag-Leffler-type function Ea

Develop additional formulas, integral representations (including improper or contour integrals), applications, and numerical algorithms for the new Mittag-Leffler-type function Ea(s) defined by the series in equation (3.6), thereby extending its theoretical and computational toolkit beyond what is currently available.

Background

The function Ea(s) is introduced as a new Mittag-Leffler-type function central to solutions of L-fractional linear equations. Unlike the classical Mittag-Leffler function, its theory and computational methods are less developed, prompting the need for further analytic and numerical advancements.