scholarly journals GENERALIZATION OF EXTENDED BETA FUNCTION, HYPERGEOMETRIC AND CONFLUENT HYPERGEOMETRIC FUNCTIONS

2011 ◽  
Vol 33 (2) ◽  
pp. 187-206 ◽  
Author(s):  
Dong-Myung Lee ◽  
Arjun K. Rathie ◽  
Rakesh K. Parmar ◽  
Yong-Sup Kim
Keyword(s):  
Hypergeometric Functions ◽  
Beta Function ◽  
Confluent Hypergeometric Functions ◽  
Extended Beta Function

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

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 ◽  
Integral Representations ◽  
Cumulative Distribution ◽  
Fractional Operator ◽  
Confluent Hypergeometric Functions ◽  
Lauricella Function ◽  
Extended Beta 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.

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 Some Inequalities of Extended Hypergeometric Functions

Mathematics ◽  
2021 ◽  
Vol 9 (21) ◽  
pp. 2702
Author(s):  
Shilpi Jain ◽  
Rahul Goyal ◽  
Praveen Agarwal ◽  
Juan L. G. Guirao
Keyword(s):  
Hypergeometric Function ◽  
Integral Inequality ◽  
Hypergeometric Functions ◽  
Beta Function ◽  
Confluent Hypergeometric Function ◽  
Gauss Hypergeometric Function ◽  
Extended Type ◽  
Mathematical Sciences ◽  
Set Up ◽  
Extended Beta Function

Hypergeometric functions and their inequalities have found frequent applications in various fields of mathematical sciences. Motivated by the above, we set up certain inequalities including extended type Gauss hypergeometric function and confluent hypergeometric function, respectively, by virtue of Hölder integral inequality and Chebyshev’s integral inequality. We also studied the monotonicity, log-concavity, and log-convexity of extended hypergeometric functions, which are derived by using the inequalities on an extended beta function.

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 On the extended Appell-Lauricella hypergeometric functions and their applications

Filomat ◽  
2017 ◽  
Vol 31 (12) ◽  
pp. 3693-3713 ◽  
Author(s):  
Ravi Agarwal ◽  
Min-Jie Luo ◽  
Praveen Agarwal
Keyword(s):  
Generating Functions ◽  
Hypergeometric Functions ◽  
Fractional Integral ◽  
Laguerre Polynomials ◽  
Beta Function ◽  
Lagrange Polynomials ◽  
Integral Technique ◽  
Beta Integral ◽  
Extended Beta Function

The main object of this paper is to present a systematic introduction to the theory and applications of the extended Appell-Lauricella hypergeometric functions defined by means of the extended beta function and extended Dirichlet?s beta integral. Their connections with the Laguerre polynomials, the ordinary Lauricella functions and the Srivastava-Daoust generalized Lauricella functions are established for some specific paramters. Furthermore, by applying the various methods and known formulas (such as fractional integral technique; some results of the Lagrange polynomials), we also derive some elegant generating functions for these new functions.

Some Properties of Extended Hypergeometric Function and Its Transformations

2018 ◽  
Vol 85 (3-4) ◽  
pp. 305
Author(s):  
Aparna Chaturvedi ◽  
Prakriti Rai
Keyword(s):  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Beta Function ◽  
Gauss Hypergeometric Function ◽  
Confluent Hypergeometric Functions ◽  
Mellin Transformation ◽  
Transformation Formulas

There emerges different extended versions of Beta function and hypergeometric functions containing extra parameters. We obtain some properties of certain functions like extended Generalized Gauss hypergeometric functions, extended Confluent hypergeometric functions including transformation formulas, Mellin transformation for the generalized extended Gauss hypergeometric function in one, two and more variables.

scholarly journals A new Generalization of Extended Beta and Hypergeometric Functions

2018 ◽  
Author(s):  
Kottakkaran Sooppy Nisar ◽  
Gauhar Rahman ◽  
Shahid Mubeen
Keyword(s):  
Hypergeometric Functions ◽  
Beta Function ◽  
Integral Representations ◽  
Summation Formulas ◽  
Extended Beta Function

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.

scholarly journals New Extension of Beta Function and Its Applications

2018 ◽  
Vol 2018 ◽  
pp. 1-25 ◽  
Author(s):  
Mehar Chand ◽  
Hanaa Hachimi ◽  
Rekha Rani
Keyword(s):  
Integral Representation ◽  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Fractional Derivatives ◽  
Fractional Integral ◽  
Beta Function ◽  
Gauss Hypergeometric Function ◽  
Confluent Hypergeometric Functions ◽  
Integral Formulas ◽  
New Type

In the present paper, new type of extension of classical beta function is introduced and its convergence is proved. Further it is used to introduce the extension of Gauss hypergeometric function and confluent hypergeometric functions. Then we study their properties, integral representation, certain fractional derivatives, and fractional integral formulas and application of these functions.

scholarly journals A novel Kind of the multi-index Beta, Gauss, and confluent hypergeometric functions

2020 ◽  
Vol 23 (02) ◽  
pp. 145-154
Author(s):  
Musharraf Ali ◽  
Mohd Ghayasuddin ◽  
Waseem A. Khan ◽  
Kottakkaran Sooppy Nisar
Keyword(s):  
Hypergeometric Functions ◽  
Beta Function ◽  
Integral Representations ◽  
Multi Index ◽  
Basic Properties ◽  
Confluent Hypergeometric Functions ◽  
Summation Formulae ◽  
Research Article ◽  
New Type

This research article elaborates on a novel expansion of the beta function by using the multi-index Mittag-Leffler function. Here, we derive some basic properties of this new beta function and then present a new type of beta dispersal as an application of our proposed beta function. We also introduce a novel expansion of Gauss and confluent hypergeometric functions for our newly initiated beta function. Some important properties of our proposed hypergeometric functions (like integral representations, differential formulae, transformations formulae, summation formulae, and a generating relation) are also pointed out systematically.

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