scholarly journals Integral Representations and Algebraic Decompositions of the Fox-Wright Type of Special Functions

2018 ◽  
Author(s):  
Dimiter Prodanov
Keyword(s):  
Hypergeometric Functions ◽  
Special Functions ◽  
Integral Operators ◽  
Integral Representations ◽  
Wright Function ◽  
Fractional Differential

The manuscript surveys the special functions of the Fox-Wright type. These functions are generalizations of the hypergeometric functions. Notable representatives of the type are the Mittag-Leffler functions and the Wright function. The integral representations of such functions are given and the conditions under which these function can be represented by simpler functions are demonstrated. The connection with generalized fractional differential and integral operators is demonstrated and discussed.

scholarly journals Integral Representations and Algebraic Decompositions of the Fox-Wright Type of Special Functions

2019 ◽  
Vol 3 (1) ◽  
pp. 4 ◽  
Author(s):  
Dimiter Prodanov
Keyword(s):  
Hypergeometric Functions ◽  
Special Functions ◽  
Integral Operators ◽  
Integral Representations ◽  
Wright Function ◽  
Fractional Differential

The manuscript surveys the special functions of the Fox-Wright type. These functions are generalizations of the hypergeometric functions. Notable representatives of the type are the Mittag-Leffler functions and the Wright function. The integral representations of such functions are given and the conditions under which these function can be represented by simpler functions are demonstrated. The connection with generalized Erdélyi-Kober fractional differential and integral operators is demonstrated and discussed.

scholarly journals ON A NEW GENERALIZED BETA FUNCTION DEFINED BY THE GENERALIZED WRIGHT FUNCTION AND ITS APPLICATIONS

2021 ◽  
Vol 6 (2) ◽  
pp. 852
Author(s):  
UMAR MUHAMMAD ABUBAKAR ◽  
Soraj Patel
Keyword(s):  
Hypergeometric Functions ◽  
Special Functions ◽  
Beta Function ◽  
Integral Representations ◽  
Wright Function ◽  
Confluent Hypergeometric Functions ◽  
Summation Formulas ◽  
Generalized Wright Function ◽  
Beta Error ◽  
Extended Beta Function

Various extensions of classical gamma, beta, Gauss hypergeometric and confluent hypergeometric functions have been proposed recently by many researchers. In this paper, we further generalized extended beta function with some of its properties such as symmetric properties, summation formulas, integral representations, connection with some other special functions such as classical beta, error, Mittag – Leffler, incomplete gamma, hypergeometric, classical Wright, Fox – Wright, Fox H and Meijer G – functions. Furthermore, the generalized beta function is used to generalize classical and other extended Gauss hypergeometric, confluent hypergeometric, Appell’s and Lauricella’s functions.

scholarly journals Investigation of Extended k-Hypergeometric Functions and Associated Fractional Integrals

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Mohamed Abdalla ◽  
Muajebah Hidan ◽  
Salah Mahmoud Boulaaras ◽  
Bahri-Belkacem Cherif
Keyword(s):  
Probability Theory ◽  
Hypergeometric Functions ◽  
Fractional Integral ◽  
Special Functions ◽  
Integral Operators ◽  
Integral Representations ◽  
Fractional Integrals ◽  
Convergence Properties ◽  
Fractional Integral Operators ◽  
Derivative Formulas

Hypergeometric functions have many applications in various areas of mathematical analysis, probability theory, physics, and engineering. Very recently, Hidan et al. (Math. Probl. Eng., ID 5535962, 2021) introduced the (p, k)-extended hypergeometric functions and their various applications. In this line of research, we present an expansion of the k-Gauss hypergeometric functions and investigate its several properties, including, its convergence properties, derivative formulas, integral representations, contiguous function relations, differential equations, and fractional integral operators. Furthermore, the current results contain several of the familiar special functions as particular cases, and this extension may enrich the theory of special functions.

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 Some Incomplete Hypergeometric Functions and Incomplete Riemann-Liouville Fractional Integral Operators

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 483 ◽  
Author(s):  
Mehmet Ali Özarslan ◽  
Ceren Ustaoğlu
Keyword(s):  
Hypergeometric Functions ◽  
Fractional Integral ◽  
Recurrence Relations ◽  
Beta Function ◽  
Integral Operators ◽  
Integral Representations ◽  
Fractional Integral Operators ◽  
Derivative Formulas ◽  
Generating Relations ◽  
Transformation Formulas

Very recently, the incomplete Pochhammer ratios were defined in terms of the incomplete beta function B y ( x , z ) . With the help of these incomplete Pochhammer ratios, we introduce new incomplete Gauss, confluent hypergeometric, and Appell’s functions and investigate several properties of them such as integral representations, derivative formulas, transformation formulas, and recurrence relations. Furthermore, incomplete Riemann-Liouville fractional integral operators are introduced. This definition helps us to obtain linear and bilinear generating relations for the new incomplete Gauss hypergeometric functions.

scholarly journals Extended k-Gamma and k-Beta Functions of Matrix Arguments

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1715
Author(s):  
Ghazi S. Khammash ◽  
Praveen Agarwal ◽  
Junesang Choi
Keyword(s):  
Gamma Function ◽  
Hypergeometric Functions ◽  
Special Functions ◽  
Integral Formula ◽  
Beta Function ◽  
Integral Representations ◽  
Matrix Argument ◽  
Functional Relations

Various k-special functions such as k-gamma function, k-beta function and k-hypergeometric functions have been introduced and investigated. Recently, the k-gamma function of a matrix argument and k-beta function of matrix arguments have been presented and studied. In this paper, we aim to introduce an extended k-gamma function of a matrix argument and an extended k-beta function of matrix arguments and investigate some of their properties such as functional relations, inequality, integral formula, and integral representations. Also an application of the extended k-beta function of matrix arguments to statistics is considered.

scholarly journals Investigation of the k-Analogue of Gauss Hypergeometric Functions Constructed by the Hadamard Product

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 714
Author(s):  
Mohamed Abdalla ◽  
Muajebah Hidan
Keyword(s):  
Hypergeometric Functions ◽  
Hadamard Product ◽  
Special Function ◽  
Function Theory ◽  
Special Functions ◽  
Integral Transforms ◽  
Integral Representations ◽  
Convergence Properties ◽  
Pochhammer Symbol ◽  
Definition Of

Traditionally, the special function theory has many applications in various areas of mathematical physics, economics, statistics, engineering, and many other branches of science. Inspired by certain recent extensions of the k-analogue of gamma, the Pochhammer symbol, and hypergeometric functions, this work is devoted to the study of the k-analogue of Gauss hypergeometric functions by the Hadamard product. We give a definition of the Hadamard product of k-Gauss hypergeometric functions (HPkGHF) associated with the fourth numerator and two denominator parameters. In addition, convergence properties are derived from this function. We also discuss interesting properties such as derivative formulae, integral representations, and integral transforms including beta transform and Laplace transform. Furthermore, we investigate some contiguous function relations and differential equations connecting the HPkGHF. The current results are more general than previous ones. Moreover, the proposed results are useful in the theory of k-special functions where the hypergeometric function naturally occurs.

scholarly journals On a generalized three-parameter wright function of Le Roy type

2017 ◽  
Vol 20 (5) ◽  
Author(s):  
Roberto Garrappa ◽  
Sergei Rogosin ◽  
Francesco Mainardi
Keyword(s):  
Asymptotic Expansion ◽  
Power Series ◽  
Laplace Transform ◽  
Analytic Continuation ◽  
Special Function ◽  
Special Functions ◽  
Integral Representations ◽  
Radius Of Convergence ◽  
Wright Function ◽  
Finite Radius

AbstractRecently S. Gerhold and R. Garra – F. Polito independently introduced a new function related to the special functions of the Mittag-Leffler family. This function is a generalization of the function studied by É. Le Roy in the period 1895-1905 in connection with the problem of analytic continuation of power series with a finite radius of convergence. In our note we obtain two integral representations of this special function, calculate its Laplace transform, determine an asymptotic expansion of this function on the negative semi-axis (in the case of an integer third parameter

scholarly journals A study on certain properties of generalized special functions defined by Fox-Wright function

2020 ◽  
Vol 5 (1) ◽  
pp. 147-162
Author(s):  
Enes Ata ◽  
İ. Onur Kıymaz
Keyword(s):  
Hypergeometric Function ◽  
Special Functions ◽  
Confluent Hypergeometric Function ◽  
Integral Representations ◽  
Gauss Hypergeometric Function ◽  
Wright Function ◽  
Summation Formulas ◽  
Frequent Use ◽  
Derivative Formulas ◽  
Transformation Formulas

AbstractIn this study, motivated by the frequent use of Fox-Wright function in the theory of special functions, we first introduced new generalizations of gamma and beta functions with the help of Fox-Wright function. Then by using these functions, we defined generalized Gauss hypergeometric function and generalized confluent hypergeometric function. For all the generalized functions we have defined, we obtained their integral representations, summation formulas, transformation formulas, derivative formulas and difference formulas. Also, we calculated the Mellin transformations of these functions.

scholarly journals Degenerate Analogues of Euler Zeta, Digamma, and Polygamma Functions

2020 ◽  
Vol 2020 ◽  
pp. 1-9 ◽  
Author(s):  
Fuli He ◽  
Ahmed Bakhet ◽  
Mohamed Akel ◽  
Mohamed Abdalla
Keyword(s):  
Zeta Function ◽  
Infinite Series ◽  
Hypergeometric Functions ◽  
Recurrence Relations ◽  
Special Functions ◽  
Mathematical Physics ◽  
Integral Representations ◽  
Digamma Function ◽  
Polygamma Function

In recent years, much attention has been paid to the role of degenerate versions of special functions and polynomials in mathematical physics and engineering. In the present paper, we introduce a degenerate Euler zeta function, a degenerate digamma function, and a degenerate polygamma function. We present several properties, recurrence relations, infinite series, and integral representations for these functions. Furthermore, we establish identities involving hypergeometric functions in terms of degenerate digamma function.

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