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 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 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.

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 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 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 Extended elliptic-type integrals with associated properties and Turán-type inequalities

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Rakesh K. Parmar ◽  
Ritu Agarwal ◽  
Naveen Kumar ◽  
S. D. Purohit
Keyword(s):  
Hypergeometric Function ◽  
Laguerre Polynomials ◽  
Beta Function ◽  
Gauss Hypergeometric Function ◽  
Elliptic Type ◽  
Integral Formulas ◽  
Special Cases ◽  
Turán Type Inequalities ◽  
Extended Beta Function ◽  
Function Of Two Variables

AbstractOur aim is to study and investigate the family of $(p, q)$ ( p , q ) -extended (incomplete and complete) elliptic-type integrals for which the usual properties and representations of various known results of the (classical) elliptic integrals are extended in a simple manner. This family of elliptic-type integrals involves a number of special cases and has a connection with $(p, q)$ ( p , q ) -extended Gauss’ hypergeometric function and $(p, q)$ ( p , q ) -extended Appell’s double hypergeometric function $F_{1}$ F 1 . Turán-type inequalities including log-convexity properties are proved for these $(p, q)$ ( p , q ) -extended complete elliptic-type integrals. Further, we establish various Mellin transform formulas and obtain certain infinite series representations containing Laguerre polynomials. We also obtain some relationship between these $(p, q)$ ( p , q ) -extended elliptic-type integrals and Meijer G-function of two variables. Moreover, we obtain several connections with $(p, q)$ ( p , q ) -extended beta function as special values and deduce numerous differential and integral formulas. In conclusion, we introduce $(p, q)$ ( p , q ) -extension of the Epstein–Hubbell (E-H) elliptic-type integral.

scholarly journals Computing sums in terms of beta, polygamma, and Gauss hypergeometric functions

2020 ◽  
Author(s):  
Feng Qi ◽  
Chuan-Jun Huang
Keyword(s):  
Hypergeometric Function ◽  
General Formula ◽  
Differentiable Function ◽  
Inversion Formula ◽  
Hypergeometric Functions ◽  
Beta Function ◽  
Higher Order ◽  
Gauss Hypergeometric Function ◽  
Partial Derivatives ◽  
Real Academia

In the paper, by virtue of the binomial inversion formula, a general formula of higher order derivatives for a ratio of two differentiable function, and other techniques, the authors compute several sums in terms of the beta function and its partial derivatives, polygamma functions, the Gauss hypergeometric function, and a determinant. These results generalize known ones in combinatorics. This preprint has been formally published as "Feng Qi and Chuan-Jun Huang, Computing sums in terms of beta, polygamma, and Gauss hypergeometric functions, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales Serie A Matematicas, Vol. 114, Paper No. 191, 9 pages (2020); available online at https://doi.org/10.1007/s13398-020-00927-y."

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.

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