The Integral Mittag-Leffler, Whittaker and Wright Functions

Alexander Apelblat,Juan González-Santander

doi:10.3390/math9243255

Alexander Apelblat, Juan González-Santander

Open Access

https://doi.org/10.3390/math9243255

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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.

Highlights

The appearance of special functions of mathematical physics was associated with solutions of particular ordinary differential equations, while the integral special functions arrived much later in mathematical literature after properties of these functions were investigated
Integral special functions were introduced as new special functions, which can be applied in many circumstances, especially in operational calculus, where they are frequently serving as direct and inverse integral transforms
Three new special functions are presented in this investigation: the integral Mittag-Leffler functions, the integral Whittaker functions, and the integral Wright functions

Summary

Introduction

The appearance of special functions of mathematical physics was associated with solutions of particular ordinary differential equations, while the integral special functions arrived much later in mathematical literature after properties of these functions were investigated. To the first group belong the exponential integral −Ei(− x ), the sine and cosine integrals, si( x ), Si( x ), ci( x ) and Ci( x ), and the corresponding integrals of hyperbolic trigonometric functions, Shi( x ) and Chi( x ) These functions are defined in the following way [1,2,3,4,5]. Integrands in the second group of integral special functions include special functions, the most well-known and applied of which are the integral Bessel functions (see, e.g., [3,7,9,10,11,12,13]). In analogy to the integral Bessel functions and with the possibility of extension to other special functions, this work introduces three new integral functions. Derived integral functions are tabulated and in some cases graphically presented (see [3])

The Integral Mittag-Leffler Functions

The Integral Whittaker Functions

The Integral Wright Functions

Conclusions