A new Generalization of Extended Beta and Hypergeometric Functions

Beta Function ◽

Summation Formulas ◽

A new generalization of extended beta function and its various properties,integral representations and distribution are given in this paper. In addition, we establishthe generalization of extended hypergeometric and con uent hypergeometric functionsusing the newly extended beta function. The properties these extended and con uenthypergeometric functions such as integral representations, Mellin transformations, dif-ferentiation formulas, transformation and summation formulas are also investigated.

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

MALAYSIAN JOURNAL OF COMPUTING ◽

10.24191/mjoc.v6i2.12018 ◽

2021 ◽

Vol 6 (2) ◽

pp. 852

Author(s):

UMAR MUHAMMAD ABUBAKAR ◽

Soraj Patel

Keyword(s):

Special Functions ◽

Beta Function ◽

Wright Function ◽

Summation Formulas ◽

Generalized Wright Function ◽

Beta Error ◽

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.

Further Generalized Beta Function with Three Parameters Mittag- Leffler Function

Earthline Journal of Mathematical Sciences ◽

10.34198/ejms.1119.4149 ◽

2018 ◽

pp. 41-49

Author(s):

Salem Saleh Barahmah

Keyword(s):

Mellin Transform ◽

Beta Function ◽

Summation Formulas ◽

The purpose of the present paper is to introduce a new extension of extended Beta function by product of two Mittag-Leffler functions. Further, we present certain results including summation formulas, integral representations and Mellin transform.

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

Analysis ◽

10.1515/anly-2018-0070 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Showkat Ahmad Dar ◽

R. B. Paris

Keyword(s):

Hypergeometric Function ◽

Mellin Transform ◽

Beta Function ◽

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.

A New Extension of Beta and Hypergeometric Functions

10.20944/preprints201801.0074.v1 ◽

2018 ◽

Author(s):

Gauhar Rahman ◽

Gulnaz Kanwal ◽

Kottakkaran Sooppy Nisar ◽

Abdul Ghaffar

Keyword(s):

Beta Distribution ◽

Mellin Transform ◽

Beta Function ◽

Moment Generating Function ◽

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.

A Note on the New Extended Beta Function with Its Application

Journal of Mathematics ◽

10.1155/2021/8353719 ◽

2021 ◽

Vol 2021 ◽

pp. 1-12

Author(s):

Vandana Palsaniya ◽

Ekta Mittal ◽

Sunil Joshi ◽

D. L. Suthar

Keyword(s):

Hypergeometric Function ◽

Beta Function ◽

Moment Generating Function ◽

Confluent Hypergeometric Function ◽

Summation Formulas ◽

Derivative Formulas ◽

New Type ◽

Extended Beta Function ◽

Transformation Formulas

The purpose of this research is to provide a systematic review of a new type of extended beta function and hypergeometric function using a confluent hypergeometric function, as well as to examine various belongings and formulas of the new type of extended beta function, such as integral representations, derivative formulas, transformation formulas, and summation formulas. In addition, we also investigate extended Riemann–Liouville (R-L) fractional integral operator with associated properties. Furthermore, we develop new beta distribution and present graphically the relation between moment generating function and ℓ .

A (p,ν)-extension of Srivastava’s triple hypergeometric function HC

Publications de l Institut Mathematique ◽

10.2298/pim2022033d ◽

2020 ◽

Vol 108 (122) ◽

pp. 33-45

Author(s):

S.A. Dar ◽

R.B. Paris

Keyword(s):

Hypergeometric Function ◽

Mellin Transform ◽

Beta Function ◽

Recursion Formulas ◽

Extended Beta Function ◽

Extended Function

We obtain a (p,?)-extension of Srivastava?s triple hypergeometric function HC(?) by employing the extended Beta function Bp,?(x, y) introduced in Parmar et al. [J. Class. Anal. 11 (2017), 91-106]. We give some of the main properties of this extended function, which include several integral representations, the Mellin transform, a differential formula, recursion formulas and a bounded inequality.

GENERALIZATION OF EXTENDED BETA FUNCTION, HYPERGEOMETRIC AND CONFLUENT HYPERGEOMETRIC FUNCTIONS

Honam Mathematical Journal ◽

10.5831/hmj.2011.33.2.187 ◽

2011 ◽

Vol 33 (2) ◽

pp. 187-206 ◽

Cited By ~ 16

Author(s):

Dong-Myung Lee ◽

Arjun K. Rathie ◽

Rakesh K. Parmar ◽

Yong-Sup Kim

Keyword(s):

Beta Function ◽

An extension of beta function, its statistical distribution, and associated fractional operator

Advances in Difference Equations ◽

10.1186/s13662-020-03142-6 ◽

2020 ◽

Vol 2020 (1) ◽

Author(s):

Ankita Chandola ◽

Rupakshi Mishra Pandey ◽

Ritu Agarwal ◽

Sunil Dutt Purohit

Keyword(s):

Beta Function ◽

Summation Formula ◽

Moment Generating Function ◽

Statistical Distribution ◽

Cumulative Distribution ◽

Fractional Operator ◽

Lauricella Function ◽

AbstractRecently, various forms of extended beta function have been proposed and presented by many researchers. The principal goal of this paper is to present another expansion of beta function using Appell series and Lauricella function and examine various properties like integral representation and summation formula. Statistical distribution for the above extension of beta function has been defined, and the mean, variance, moment generating function and cumulative distribution function have been obtained. Using the newly defined extension of beta function, we build up the extension of hypergeometric and confluent hypergeometric functions and discuss their integral representations and differentiation formulas. Further, we define a new extension of Riemann–Liouville fractional operator using Appell series and Lauricella function and derive its various properties using the new extension of beta function.

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

Mathematics ◽

10.3390/math7050483 ◽

2019 ◽

Vol 7 (5) ◽

pp. 483 ◽

Cited By ~ 6

Author(s):

Mehmet Ali Özarslan ◽

Ceren Ustaoğlu

Keyword(s):

Fractional Integral ◽

Recurrence Relations ◽

Beta Function ◽

Integral Operators ◽

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.

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

Mathematics ◽

10.3390/math8101715 ◽

2020 ◽

Vol 8 (10) ◽

pp. 1715

Author(s):

Ghazi S. Khammash ◽

Praveen Agarwal ◽

Junesang Choi

Keyword(s):

Gamma Function ◽

Special Functions ◽

Integral Formula ◽

Beta Function ◽

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.