Special Functions in Applied Mathematics

1991 ◽  
pp. 1-14
Author(s):  
James B. Seaborn
Keyword(s):  
Special Functions ◽  
Applied Mathematics

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

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2294
Author(s):  
Hari Mohan Srivastava
Keyword(s):  
Number Theory ◽  
Special Functions ◽  
Applied Mathematics ◽  
Analytic Number Theory ◽  
Review Article ◽  
Mathematical Functions ◽  
Analytic Number ◽  
Recent Developments ◽  
Transcendental Functions ◽  
Mathematical Sciences

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.

Special Functions of Applied Mathematics (B. C. Carlson)

SIAM Review ◽  
1979 ◽  
Vol 21 (3) ◽  
pp. 420-421 ◽  
Author(s):  
Richard Askey
Keyword(s):  
Special Functions ◽  
Applied Mathematics

A family of the incomplete hypergeometric functions and associated integral transform and fractional derivative formulas

Filomat ◽  
2017 ◽  
Vol 31 (1) ◽  
pp. 125-140 ◽  
Author(s):  
Rekha Srivastava ◽  
Ritu Agarwal ◽  
Sonal Jain
Keyword(s):  
Hypergeometric Functions ◽  
Fractional Derivatives ◽  
Integral Transform ◽  
Special Functions ◽  
Applied Mathematics ◽  
Integral Transforms ◽  
Mathematical Physics ◽  
Natural Generalization ◽  
Derivative Formulas ◽  
Derivatives Of

Recently, Srivastava et al. [Integral Transforms Spec. Funct. 23 (2012), 659-683] introduced the incomplete Pochhammer symbols that led to a natural generalization and decomposition of a class of hypergeometric and other related functions as well as to certain potentially useful closed-form representations of definite and improper integrals of various special functions of applied mathematics and mathematical physics. In the present paper, our aim is to establish several formulas involving integral transforms and fractional derivatives of this family of incomplete hypergeometric functions. As corollaries and consequences, many interesting results are shown to follow from our main results.

scholarly journals Fractional Hermite-Hadamard Integral Inequalities for a New Class of Convex Functions

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1485
Author(s):  
Pshtiwan Othman Mohammed ◽  
Thabet Abdeljawad ◽  
Shengda Zeng ◽  
Artion Kashuri
Keyword(s):  
Convex Functions ◽  
Fractional Integral ◽  
Special Functions ◽  
Applied Mathematics ◽  
Strong Relationship ◽  
Integral Inequalities ◽  
Mathematical Methods ◽  
Fractional Integral Operators ◽  
New Class ◽  
Definition Of

Fractional integral inequality plays a significant role in pure and applied mathematics fields. It aims to develop and extend various mathematical methods. Therefore, nowadays we need to seek accurate fractional integral inequalities in obtaining the existence and uniqueness of the fractional methods. Besides, the convexity theory plays a concrete role in the field of fractional integral inequalities due to the behavior of its definition and properties. There is also a strong relationship between convexity and symmetric theories. So, whichever one we work on, we can then apply it to the other one due to the strong correlation produced between them, specifically in the last few decades. First, we recall the definition of φ-Riemann–Liouville fractional integral operators and the recently defined class of convex functions, namely the σ˘-convex functions. Based on these, we will obtain few integral inequalities of Hermite–Hadamard’s type for a σ˘-convex function with respect to an increasing function involving the φ-Riemann–Liouville fractional integral operator. We can conclude that all derived inequalities in our study generalize numerous well-known inequalities involving both classical and Riemann–Liouville fractional integral inequalities. Finally, application to certain special functions are pointed out.

scholarly journals Numerical aspects of special functions

Acta Numerica ◽  
2007 ◽  
Vol 16 ◽  
pp. 379-478 ◽  
Author(s):  
Nico M. Temme
Keyword(s):  
Numerical Evaluation ◽  
Recurrence Relations ◽  
Special Functions ◽  
Applied Mathematics ◽  
Mathematical Physics ◽  
Numerical Quadrature ◽  
Mathematical Statistics ◽  
Series Expansions

This paper describes methods that are important for the numerical evaluation of certain functions that frequently occur in applied mathematics, physics and mathematical statistics. This includes what we consider to be the basic methods, such as recurrence relations, series expansions (both convergent and asymptotic), and numerical quadrature. Several other methods are available and some of these will be discussed in less detail. Examples will be given on the use of special functions in certain problems from mathematical physics and mathematical statistics (integrals and series with special functions).

scholarly journals Certain Matrix Riemann–Liouville Fractional Integrals Associated with Functions Involving Generalized Bessel Matrix Polynomials

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 622
Author(s):  
Mohamed Abdalla ◽  
Mohamed Akel ◽  
Junesang Choi
Keyword(s):  
Jacobi Polynomials ◽  
Special Functions ◽  
Applied Mathematics ◽  
Fractional Integrals ◽  
Elementary Functions ◽  
Matrix Polynomials ◽  
Matrix Version

The fractional integrals involving a number of special functions and polynomials have significant importance and applications in diverse areas of science; for example, statistics, applied mathematics, physics, and engineering. In this paper, we aim to introduce a slightly modified matrix of Riemann–Liouville fractional integrals and investigate this matrix of Riemann–Liouville fractional integrals associated with products of certain elementary functions and generalized Bessel matrix polynomials. We also consider this matrix of Riemann–Liouville fractional integrals with a matrix version of the Jacobi polynomials. Furthermore, we point out that a number of Riemann–Liouville fractional integrals associated with a variety of functions and polynomials can be presented, which are presented as problems for further investigations.

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