A survey of some recent developments on higher transcendental functions of analytic number theory and applied mathematics

Date

2021

Authors

Srivastava, H.M.

Journal Title

Journal ISSN

Volume Title

Publisher

Symmetry

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.

Description

Keywords

gamma, digamma and polygamma functions, hypergeometric functions and their generalizations and multivariate extensions, Riemann and Hurwitz (or generalized) Zeta functions, Lerch's transcendent and the Hurwitz-Lerch Zeta functions, Mittag-Leffler type functions, Fox-Wright hypergeometric function, Riemann-Liouville and related fractional derivative operators, Liouville-Caputo fractional derivative operator, Fox-Wright hypergeometric function, generalized Fox-Wright function, operators of fractional calculus, quantum or basic (or q-) analysis, fractional-order quantum or basic (or q-) analysis

Citation

Srivastava, H. M. (2021). “A survey of some recent developments on higher transcendental functions of analytic number theory and applied mathematics.” Symmetry, 13(12), 2294. https://doi.org/10.3390/sym13122294