scholarly journals An Extension of Beta Function by Using Wiman’s Function

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 187 ◽  
Author(s):  
Rahul Goyal ◽  
Shaher Momani ◽  
Praveen Agarwal ◽  
Michael Th. Rassias
Keyword(s):  
Integral Representation ◽  
Mellin Transform ◽  
Beta Function ◽  
Functional Relations ◽  
Derivative Formulas ◽  
Extended Beta Function

The main purpose of this paper is to study extension of the extended beta function by Shadab et al. by using 2-parameter Mittag-Leffler function given by Wiman. In particular, we study some functional relations, integral representation, Mellin transform and derivative formulas for this extended beta function.

scholarly journals Some Results of Extended Beta Function and Hypergeometric Functions by Using Wiman’s Function

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2944
Author(s):  
Shilpi Jain ◽  
Rahul Goyal ◽  
Praveen Agarwal ◽  
Antonella Lupica ◽  
Clemente Cesarano
Keyword(s):  
Hypergeometric Function ◽  
Beta Function ◽  
Confluent Hypergeometric Function ◽  
Integral Representations ◽  
Gauss Hypergeometric Function ◽  
Logarithmic Convexity ◽  
Functional Relations ◽  
Derivative Formulas ◽  
Extended Beta Function ◽  
Transformation Formulas

The main aim of this research paper is to introduce a new extension of the Gauss hypergeometric function and confluent hypergeometric function by using an extended beta function. Some functional relations, summation relations, integral representations, linear transformation formulas, and derivative formulas for these extended functions are derived. We also introduce the logarithmic convexity and some important inequalities for extended beta function.

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

2020 ◽  
Vol 108 (122) ◽  
pp. 33-45
Author(s):  
S.A. Dar ◽  
R.B. Paris
Keyword(s):  
Hypergeometric Function ◽  
Mellin Transform ◽  
Beta Function ◽  
Integral Representations ◽  
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.

scholarly journals Further Generalized Beta Function with Three Parameters Mittag- Leffler Function

2018 ◽  
pp. 41-49
Author(s):  
Salem Saleh Barahmah
Keyword(s):  
Mellin Transform ◽  
Beta Function ◽  
Integral Representations ◽  
Summation Formulas ◽  
Extended Beta Function

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.

scholarly journals A New Representation of Extended Mittage-Leffler Function and Its Properties

2019 ◽  
Author(s):  
Abdul Ghaffar ◽  
Ayesha Saif ◽  
Faheem Khan
Keyword(s):  
Mellin Transform ◽  
Recursion Relation ◽  
Beta Function ◽  
Extended Beta Function

In this article, our main purpose is to establish a new extension of Mittag-Leffler function by using the known extended beta function $\mathbf{B}_{\omega}(a,b;p)$ introduced in [1]. It led to a novel extension of the applicability of Mittag-Leffler function that introduced them as distributions defined for a specific set of functions. We also, investigate some of its important properties, namely recursion relation, Mellin transform and differential formulas.

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 Note on the New Extended Beta Function with Its Application

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 ◽  
Integral Representations ◽  
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 .

scholarly journals Further Extension of Extended Fractional Derivative Operator of Riemann-Liouville

2017 ◽  
Author(s):  
Gauhar Rahman ◽  
Shahid Mubeen ◽  
Kottakkaran Sooppy Nisar
Keyword(s):  
Fractional Derivative ◽  
Generating Functions ◽  
Mellin Transform ◽  
Bessel Functions ◽  
Beta Function ◽  
Fractional Operator ◽  
Derivative Operator ◽  
Extended Beta Function

The main objective of this present paper is to establish the extension of an extended fractional derivative operator by using an extended beta function recently defined by Parmar et al. by considering the Bessel functions in its kernel. Also, we give some results related to the newly defined fractional operator such as Mellin transform and relations to extended hypergeometric and Appell's function via generating functions.

scholarly journals Generalized Wright Function and Its Properties Using Extended Beta Function

2020 ◽  
Vol 51 (4) ◽  
pp. 349-363
Author(s):  
Nabiullah Khan ◽  
Talha Usma ◽  
Mohd Aman
Keyword(s):  
Fractional Derivative ◽  
Mellin Transform ◽  
Beta Function ◽  
Linear Partial Differential Equation ◽  
Wright Function ◽  
Generalized Wright Function ◽  
Derivative Formula ◽  
Liouville Fractional Derivative ◽  
Extended Beta Function

Solving a linear partial differential equation witness a noteworthy role of Wright function. Due to its usefulness and various applications, a variety of its extentions (and generalizations) have been investigated and presented. The purpose and design of the paper is intended to study and come up with a new extention of the genralized Wright function by using generalized beta function and obtain some integral representation of the freshly defined function. Also we present the Mellin transform of this function in the form of Fox Wright function. Furthermore, we obtain the recurrence relation, derivative formula for the said function and also by using an extended Riemann-Liouville fractional derivative, we present a fractional derivative formula for the extended Wright function.

scholarly journals A (p,v)-Extension of Hurwitz-Lerch Zeta Function and its Properties

2018 ◽  
Author(s):  
Gauhar Rahman ◽  
KS Nisar ◽  
Shahid Mubeen
Keyword(s):  
Zeta Function ◽  
Beta Function ◽  
Integral Representations ◽  
Basic Properties ◽  
Mellin Transformation ◽  
Special Cases ◽  
Lerch Zeta Function ◽  
Derivative Formulas ◽  
Generating Relations

In this paper, we define a (p,v)-extension of Hurwitz-Lerch Zeta function by considering an extension of beta function defined by Parmar et al. [J. Classical Anal. 11 (2017) 81–106]. We obtain its basic properties which include integral representations, Mellin transformation, derivative formulas and certain generating relations. Also, we establish the special cases of the main results.

scholarly journals Euler Type Integrals and Integrals in Terms of Extended Beta Function

2014 ◽  
Vol 2014 ◽  
pp. 1-12
Author(s):  
Subuhi Khan ◽  
Mustafa Walid Al-Saad
Keyword(s):  
Hypergeometric Series ◽  
Beta Function ◽  
Generalized Hypergeometric Series ◽  
Special Polynomials ◽  
Extended Beta Function

We derive the evaluations of certain integrals of Euler type involving generalized hypergeometric series. Further, we establish a theorem on extended beta function, which provides evaluation of certain integrals in terms of extended beta function and certain special polynomials. The possibility of extending some of the derived results to multivariable case is also investigated.

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