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 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.

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 EXCEL-orientated procedure for calculating the values of special functions with interval argument assigned on the hyperbolic form

2021 ◽  
Vol 5 (4) ◽  
pp. 116-123
Author(s):  
Valeriy Dubnitskiy ◽  
Anatolii Kobylin ◽  
Oleg Kobylin ◽  
Yuriy Kushneruk
Keyword(s):  
Exponential Function ◽  
Gamma Function ◽  
Special Functions ◽  
Beta Function ◽  
Digamma Function ◽  
Hyperbolic Sine ◽  
Hyperbolic Cosine ◽  
Hyperbolic Form ◽  
Hyperbolic Trigonometry ◽  
Integral Exponential Function

Aim of the work is to propose the main terms of the EXCEL-orientated procedures for calculating the values of elementary and special functions with interval argument that is assigned on the hyperbolic form. The results of the work. The methods of presenting the interval values in the hyperbolic form and the rules of addition, subtraction, multiplication, and division of this values were considered. The procedures of calculating the function values, whose arguments can be degenerate or interval values were described. Namely, the direct and the reverse functions of the linear trigonometry, the direct and the reverse functions of the hyperbolic trigonometry, exponential function, arbitrary exponential function and power function, Gamma-function, incomplete Gamma-function, digamma-function, trigamma-function, tetragamma-function, pentagamma-function, Beta-function and its partial derivatives, integral exponential function, integral logarithm, dilogarithm, Frenel integrals, sine integral, cosine integral, hyperbolic sine integral, hyperbolic cosine integral. The basic terms of the EXCEL-orientated procedures for calculating the values of elementary and special functions with interval argument that is assigned on the hyperbolic form were proposed. The numerical examples were provided, that illustrate the application of the proposed methods.

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 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 A Class of Functions satisfying certain boundary value problems

2021 ◽  
Vol 16 (3) ◽  
Author(s):  
Hemant Kumar
Keyword(s):  
Boundary Value Problems ◽  
Fractional Integral ◽  
Special Functions ◽  
Integral Formula ◽  
Boundary Value ◽  
Integral Representations ◽  
Double Integral ◽  
Fractal Strings ◽  
Weyl Fractional Integral ◽  
Class Of Functions

In this paper for constructing of a class of functions consisting of integral representations, we introduce a double integral, a formula pertaining to extended fractal strings, consisting of separate variables functions in the integrand. Further by this double integral formula, we determine various functions, integrals and contour integral representations on introducing different special functions for these integrand functions connecting to RiemannLiouville and Weyl fractional integral functions. Finally, we apply our obtained results to find certain boundary value problems and present precise examples.

scholarly journals A (p,ν)-extension of Srivastava's triple hypergeometric function H_ B and its properties

Analysis ◽  
2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Showkat Ahmad Dar ◽  
R. B. Paris
Keyword(s):  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Mellin Transform ◽  
Beta Function ◽  
Integral Representations ◽  
Recursion Formulas ◽  
Extended Beta Function ◽  
Extended Function

Abstract In this paper, we obtain a ( p , ν ) {(p,\nu)} -extension of Srivastava’s triple hypergeometric function H B ⁢ ( ⋅ ) {H_{B}(\,\cdot\,)} , by using the extended beta function B p , ν ⁢ ( x , y ) {B_{p,\nu}(x,y)} introduced in [R. K. Parmar, P. Chopra and R. B. Paris, On an extension of extended beta and hypergeometric functions, J. Class. Anal. 11 2017, 2, 91–106]. We give some of the main properties of this extended function, which include several integral representations involving Exton’s hypergeometric function, the Mellin transform, a differential formula, recursion formulas and a bounded inequality.

scholarly journals A New Extension of Beta and Hypergeometric Functions

2018 ◽  
Author(s):  
Gauhar Rahman ◽  
Gulnaz Kanwal ◽  
Kottakkaran Sooppy Nisar ◽  
Abdul Ghaffar
Keyword(s):  
Beta Distribution ◽  
Hypergeometric Functions ◽  
Mellin Transform ◽  
Beta Function ◽  
Moment Generating Function ◽  
Integral Representations ◽  
Confluent Hypergeometric Functions ◽  
Summation Formulas ◽  
Differentiation Formulas ◽  
Mean Variance

The main objective of this paper is to introduce a further extension of extended (p, q)-beta function by considering two Mittag-Leffler function in the kernel. We investigate various properties of this newly defined beta function such as integral representations, summation formulas and Mellin transform. We define extended beta distribution and its mean, variance and moment generating function with the help of extension of beta function. Also, we establish an extension of extended (p, q)-hypergeometric and (p, q)-confluent hypergeometric functions by using the extension of beta function. Various properties of newly defined extended hypergeometric and confluent hypergeometric functions such as integral representations, Mellin transformations, differentiation formulas, transformation and summation formulas are investigated.

scholarly journals Certain investigations in the field of special functions

2020 ◽  
Vol 1 (1) ◽  
pp. 87-98
Author(s):  
Maisoon A. Kulib ◽  
Ahmed A. Al-Gonah ◽  
Salem S. Barahmah
Keyword(s):  
Hypergeometric Function ◽  
Special Functions ◽  
Beta Function ◽  
Confluent Hypergeometric Function ◽  
Integral Representations ◽  
Confluent Hypergeometric Functions ◽  
Research Fields ◽  
Summation Formulas ◽  
Physical Problems ◽  
Wide Range

Motivated mainly by a variety of applications of Euler's Beta, hypergeometric, and confluent hypergeometric functions together with their extensions in a wide range of research fields such asengineering, chemical, and physical problems. In this paper, we introduce modified forms of some extended special functions such as Gamma function, Beta function, hypergeometric function and confluent hypergeometric function by making use of the idea given in reference \cite{9}. Also, certain investigations including summation formulas, integral representations and Mellin transform of these modified functions are derived. Further, many known results are obtained asspecial cases of our main results.

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.

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