A Survey of Some Recent Developments on Higher Transcendental Functions of Analytic Number Theory and Applied Mathematics

Abstract

Often referred to as special functions or mathematical functions, the origin of many members of the remarkably vast family of higher transcendental functions can be traced back to such widespread areas as (for example) mathematical physics, analytic number theory and applied mathematical sciences. Here, in this survey-cum-expository review article, we aim at presenting a brief introductory overview and survey of some of the recent developments in the theory of several extensively studied higher transcendental functions and their potential applications. For further reading and researching by those who are interested in pursuing this subject, we have chosen to provide references to various useful monographs and textbooks on the theory and applications of higher transcendental functions. Some operators of fractional calculus, which are associated with higher transcendental functions, together with their applications, have also been considered. Many of the higher transcendental functions, especially those of the hypergeometric type, which we have investigated in this survey-cum-expository review article, are known to display a kind of symmetry in the sense that they remain invariant when the order of the numerator parameters or when the order of the denominator parameters is arbitrarily changed.

Geometric Properties of a Certain Class of Mittag–Leffler-Type Functions

“…Setting A = α and B = β in (25), we obtain the Mittag-Leffler-type function as defined in (9), which is also known as the Le Roy-type Mittag-Leffler function [33]. Similarly, for A = α, B = β and γ = 1, G…”

“…α,β,A,B (z) reduces to the Mittag-Leffler function. Moreover, the Bessel-Wright function J µ ν (z) given by (4) can be derived as a particular case if set A = B = 1, α = ν + 1, β = µ and γ = 2, and replace z by −z, in the definition (25). In particular, upon setting A = α, B = β, and γ = n in (25), a multi-index Mittag-Leffler function can be obtained.…”

“…Moreover, the Bessel-Wright function J µ ν (z) given by (4) can be derived as a particular case if set A = B = 1, α = ν + 1, β = µ and γ = 2, and replace z by −z, in the definition (25). In particular, upon setting A = α, B = β, and γ = n in (25), a multi-index Mittag-Leffler function can be obtained. These are important special functions, which have several applications in fractional calculus [53][54][55][56], mathematical physics and related branches of science and engineering [27,[57][58][59].…”

“…where φ(t) is a function of t satisfying appropriate conditions. For a reasonably detailed historical background and other details about the following interesting unification of the definition (6) and some multi-parameter extensions of several functions occurring in Analytic Number Theory, the reader is referred, for example, to the recent works [23][24][25]:…”

Some Integral Inequalities in 𝒱-Fractional Calculus and Their Applications

“…For more general Mittag-Leffler type functions that have been investigated rather systematically and extensively, see, for details, [9][10][11][12]).…”