The integral Mittag-Leffler, Whittaker and Wright functions

TL;DR

This paper introduces new integral functions related to Mittag-Leffler, Whittaker, and Wright functions, providing their representations, properties, and graphical behavior, with applications to Laplace transforms and special cases.

Contribution

It presents the first introduction of these integral functions, their closed-form expressions, and their representations as hypergeometric and elementary functions.

Findings

New integral functions are introduced and characterized.

Closed-form and hypergeometric representations are provided.

Graphical analysis illustrates the functions' behavior.

Abstract

Integral Mittag-Leffler, Whittaker and Wright functions with integrands similar to those which already exist in mathematical literature are introduced for the first time. For particular values of parameters, they can be presented in closed-form. In most reported cases, these new integral functions are expressed as generalized hypergeometric functions but also in terms of elementary and special functions. The behavior of some of the new integral functions is presented in graphical form. By using the MATHEMATICA program to obtain infinite sums that define the Mittag-Leffler, Whittaker, and Wright functions and also their corresponding integral functions, these functions and many new Laplace transforms of them are also reported in the Appendices for integral and fractional values of parameters.