scholarly journals The Integral Mittag-Leffler, Whittaker and Wright Functions

Mathematics ◽  
2021 ◽  
Vol 9 (24) ◽  
pp. 3255
Author(s):  
Alexander Apelblat ◽  
Juan Luis González-Santander
Keyword(s):  
Closed Form ◽  
Hypergeometric Functions ◽  
Special Functions ◽  
Laplace Transforms ◽  
Graphical Form ◽  
Generalized Hypergeometric Functions ◽  
Infinite Sums ◽  
First Time ◽  
Mathematical Literature

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.

scholarly journals Two dimensional Laplace transforms of generalized hypergeometric functions

1988 ◽  
Vol 11 (1) ◽  
pp. 167-175 ◽  
Author(s):  
R. S. Dahiya ◽  
I. H. Jowhar
Keyword(s):  
Hypergeometric Functions ◽  
Laplace Transforms ◽  
Two Dimensional ◽  
Generalized Hypergeometric Functions

The object of this paper is to obtain new operational relations between the original and the image functions that involve generalized hypergeometricG-functions.

scholarly journals Distributions of the Ratio and Product of Two Independent Weibull and Lindley Random Variables

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
N. J. Hassan ◽  
A. Hawad Nasar ◽  
J. Mahdi Hadad
Keyword(s):  
Hypergeometric Functions ◽  
Special Functions ◽  
Random Variables ◽  
Distribution Functions ◽  
Cumulative Distribution ◽  
Parabolic Cylinder ◽  
Generalized Hypergeometric Functions ◽  
Confluent Hypergeometric Functions ◽  
Parabolic Cylinder Functions ◽  
The Moment

In this paper, we derive the cumulative distribution functions (CDF) and probability density functions (PDF) of the ratio and product of two independent Weibull and Lindley random variables. The moment generating functions (MGF) and the k-moment are driven from the ratio and product cases. In these derivations, we use some special functions, for instance, generalized hypergeometric functions, confluent hypergeometric functions, and the parabolic cylinder functions. Finally, we draw the PDF and CDF in many values of the parameters.

scholarly journals Spherical curves and quadratic relationships for special functions

1985 ◽  
Vol 27 (1) ◽  
pp. 111-124
Author(s):  
Even Mehlum ◽  
Jet Wimp
Keyword(s):  
Differential Equation ◽  
Hypergeometric Functions ◽  
Bessel Functions ◽  
Special Functions ◽  
Position Vector ◽  
Third Order ◽  
Generalized Hypergeometric Functions ◽  
Transcendental Functions ◽  
Horn Functions ◽  
Vector Differential Equation

AbstractWe show that the position vector of any 3-space curve lying on a sphere satisfies a third-order linear (vector) differential equation whose coefficients involve a single arbitrary function A(s). By making various identifications of A(s), we are led to nonlinear identities for a number of higher transcendental functions: Bessel functions, Horn functions, generalized hypergeometric functions, etc. These can be considered natural geometrical generalizations of sin2t + cos2t = 1. We conclude with some applications to the theory of splines.

Multi-parametric mittag-leffler functions and their extension

2013 ◽  
Vol 16 (2) ◽  
Author(s):  
Anatoly Kilbas ◽  
Anna Koroleva ◽  
Sergei Rogosin
Keyword(s):  
Fractional Calculus ◽  
Historical Development ◽  
Hypergeometric Functions ◽  
Special Functions ◽  
Integral Representations ◽  
Generalized Hypergeometric Functions ◽  
Late Professor ◽  
Wright Generalized Hypergeometric Functions

AbstractThis paper surveys one of the last contributions by the late Professor Anatoly Kilbas (1948–2010) and research made under his advisorship. We briefly describe the historical development of the theory of the discussed multi-parametric Mittag-Leffler functions as a class of the Wright generalized hypergeometric functions. The method of the Mellin-Barnes integral representations allows us to extend the considered functions to the case of arbitrary values of parameters. Thus, the extended Mittag-Leffler-type functions appear. The properties of these special functions and their relations to the fractional calculus are considered. Our results are based mainly on the properties of the Fox H-functions, as one of the widest class of special functions.

scholarly journals On the Mellin transform of a product of hypergeometric functions

1998 ◽  
Vol 40 (2) ◽  
pp. 222-237 ◽  
Author(s):  
Allen R. Miller ◽  
H. M. Srivastava
Keyword(s):  
Hypergeometric Functions ◽  
Mellin Transform ◽  
Fundamental Relation ◽  
Generalized Hypergeometric Functions ◽  
Lommel Functions ◽  
Mathematical Literature

AbstractWe obtain representations for the Mellin transform of the product of generalized hypergeometric functions0F1[−a2x2]1F2[−b2x2]fora, b > 0. The later transform is a generalization of the discontinuous integral of Weber and Schafheitlin; in addition to reducing to other known integrals (for example, integrals involving products of powers, Bessel and Lommel functions), it contains numerous integrals of interest that are not readily available in the mathematical literature. As a by-product of the present investigation, we deduce the second fundamental relation for3F2[1]. Furthermore, we give the sine and cosine transforms of1F2[−b2x2].

scholarly journals A Guide to Special Functions in Fractional Calculus

Mathematics ◽  
2021 ◽  
Vol 9 (1) ◽  
pp. 106
Author(s):  
Virginia Kiryakova
Keyword(s):  
Applied Sciences ◽  
Fractional Calculus ◽  
Hypergeometric Functions ◽  
Special Functions ◽  
Generalized Hypergeometric Functions ◽  
Social Events ◽  
Transcendental Functions ◽  
Meijer G Function ◽  
Wright Generalized Hypergeometric Functions

Dedicated to the memory of Professor Richard Askey (1933–2019) and to pay tribute to the Bateman Project. Harry Bateman planned his “shoe-boxes” project (accomplished after his death as Higher Transcendental Functions, Vols. 1–3, 1953–1955, under the editorship by A. Erdélyi) as a “Guide to the Functions”. This inspired the author to use the modified title of the present survey. Most of the standard (classical) Special Functions are representable in terms of the Meijer G-function and, specially, of the generalized hypergeometric functions pFq. These appeared as solutions of differential equations in mathematical physics and other applied sciences that are of integer order, usually of second order. However, recently, mathematical models of fractional order are preferred because they reflect more adequately the nature and various social events, and these needs attracted attention to “new” classes of special functions as their solutions, the so-called Special Functions of Fractional Calculus (SF of FC). Generally, under this notion, we have in mind the Fox H-functions, their most widely used cases of the Wright generalized hypergeometric functions pΨq and, in particular, the Mittag–Leffler type functions, among them the “Queen function of fractional calculus”, the Mittag–Leffler function. These fractional indices/parameters extensions of the classical special functions became an unavoidable tool when fractalized models of phenomena and events are treated. Here, we try to review some of the basic results on the theory of the SF of FC, obtained in the author’s works for more than 30 years, and support the wide spreading and important role of these functions by several examples.

scholarly journals Summation Formulas Obtained by Means of the Generalized Chain Rule for Fractional Derivatives

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
S. Gaboury ◽  
R. Tremblay
Keyword(s):  
General Formula ◽  
Hypergeometric Functions ◽  
Fractional Derivatives ◽  
Special Functions ◽  
Mathematical Physics ◽  
Chain Rule ◽  
Generalized Hypergeometric Functions ◽  
Summation Formulas ◽  
Several Variables ◽  
Special Cases

In 1970, several interesting new summation formulas were obtained by using a generalized chain rule for fractional derivatives. The main object of this paper is to obtain a presumably new general formula. Many special cases involving special functions of mathematical physics such as the generalized hypergeometric functions, the Appell F1 function, and the Lauricella functions of several variables FD(n) are given.

scholarly journals Exact solutions involving special functions for unsteady convective flow of magnetohydrodynamic second grade fluid with ramped conditions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Muhammad Bilal Riaz ◽  
Kashif Ali Abro ◽  
Khadijah M. Abualnaja ◽  
Ali Akgül ◽  
Aziz Ur Rehman ◽  
...  
Keyword(s):  
Convective Flow ◽  
Special Functions ◽  
Laplace Transforms ◽  
Second Grade ◽  
Second Grade Fluid ◽  
Mathematical Methods ◽  
Grade Fluid ◽  
Temperature And Velocity Fields ◽  
The Mathematical Model ◽  
First Time

AbstractA number of mathematical methods have been developed to determine the complex rheological behavior of fluid’s models. Such mathematical models are investigated using statistical, empirical, analytical, and iterative (numerical) methods. Due to this fact, this manuscript proposes an analytical analysis and comparison between Sumudu and Laplace transforms for the prediction of unsteady convective flow of magnetized second grade fluid. The mathematical model, say, unsteady convective flow of magnetized second grade fluid, is based on nonfractional approach consisting of ramped conditions. In order to investigate the heat transfer and velocity field profile, we invoked Sumudu and Laplace transforms for finding the hidden aspects of unsteady convective flow of magnetized second grade fluid. For the sake of the comparative analysis, the graphical illustration is depicted that reflects effective results for the first time in the open literature. In short, the obtained profiles of temperature and velocity fields with Laplace and Sumudu transforms are in good agreement on the basis of numerical simulations.

scholarly journals A Note on Certain Laplace Transforms of Convolution-Type Integrals Involving Product of Two Generalized Hypergeometric Functions

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Arjun Kumar Rathie ◽  
Young Hee Geum ◽  
Hwajoon Kim
Keyword(s):  
Hypergeometric Functions ◽  
Research Result ◽  
Research Paper ◽  
Laplace Transforms ◽  
Convolution Type ◽  
Generalized Hypergeometric Functions

The aim of this research paper is to provide as many as forty-five attractive Laplace transforms of convolution type related to the product of generalized hypergeometric functions. These are achieved by employing summation theorems for the series pFp−1 (for p = 2,3,4 , and 5) available in the literature. The obtained research result is to provide an easier method than the existing method.

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