The Integral Mittag-Leffler, Whittaker and Wright Functions

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.

Differentiation of integral Mittag-Leffler and integral Wright functions with respect to parameters

“…already mentioned in the text (see [2]), derivatives of the Mittag-Leffler function (1.1) with respect to α and β, i.e.…”

“…In our recent paper [2], two new special functions were introduced, namely the integral Mittag-Leffler function:…”

“…It is possible to obtain these integral functions and their Laplace transforms in terms of elementary and special functions in closed-form for many particular cases of the parameters α and β with the help of MATHEMATICA program [2].…”

Finite and Infinite Hypergeometric Sums Involving the Digamma Function

“…In [3], we found some reduction formulas for the first derivative of the Wright function with respect to the parameters for particular values of α and β. Next, we extend these reduction formulas.…”

Finite and infinite hypergeometric sums involving the digamma function

“…Some authors have contributed to enhance this compilation, such as Miller [3], who used reduction formulas of the Kampé de Fériet function; and Cvijović [4], who used the derivative of the Pochhammer symbol. Sums involving the digamma function occur in the expressions of the derivatives of the Mittag-Leffler function and the Wright function with respect to parameters [5,6]. Also, they occur in the derivation of asymptotic expansions for Mellin-Barnes integrals [7].…”