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 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 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 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."

Application of the E-operator to evaluate an infinite integral

1969 ◽  
Vol 65 (3) ◽  
pp. 725-730 ◽  
Author(s):  
F. Singh
Keyword(s):  
Finite Difference ◽  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
General Nature ◽  
Difference Operators ◽  
Generalized Hypergeometric Function ◽  
Generalized Hypergeometric Functions ◽  
Confluent Hypergeometric Functions ◽  
Infinite Integral

1. The object of this paper is to evaluate an infinite integral, involving the product of H-functions, generalized hypergeometric functions and confluent hypergeometric functions by means of finite difference operators E. As the generalized hypergeometric function and H-function are of a very general nature, the integral, on specializing the parameters, leads to a generalization of many results some of which are known and others are believed to be new.

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

Integral transforms and probability distributions involving a generalized hypergeometric function

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Nabiullah Khan ◽  
Talha Usman ◽  
Mohd Aman ◽  
Shrideh Al-Omari ◽  
Junesang Choi
Keyword(s):  
Probability Distribution ◽  
Hypergeometric Function ◽  
Probability Distributions ◽  
Beta Function ◽  
Integral Transforms ◽  
Generalized Hypergeometric Function ◽  
Confluent Hypergeometric Functions ◽  
Special Cases ◽  
Legendre Transforms ◽  
Generalized Probability

Abstract Various extensions of the beta function together with their associated extended hypergeometric and confluent hypergeometric functions have been introduced and investigated. In this paper, using the very recently contrived extended beta function, we aim to introduce an extension F v p , q ; λ ; σ , τ u {{}_{u}F_{v}^{p,q;\lambda;\sigma,\tau}} of the generalized hypergeometric function F v u {{}_{u}F_{v}} and investigate certain classes of transforms and several identities of a generalized probability distribution involving this extension. In fact, we present some interesting formulas of Jacobi, Gegenbauer, pathway, Laplace, and Legendre transforms of this extension multiplied by a polynomial. We also introduce a generalized probability distribution to investigate its several related properties. Further, we consider some special cases of our main results with an argument about the derived process of a known result.

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 INTEGRALS OF CONFLUENT HYPERGEOMETRIC FUNCTIONS

1990 ◽  
Vol 21 (2) ◽  
pp. 131-135
Author(s):  
GIOVANNA PITTALUGA
Keyword(s):  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Confluent Hypergeometric Function ◽  
Weight Functions ◽  
Confluent Hypergeometric Functions

The moments of the weight functions $w(x)=e^{-x}x^\mu(\ln x)^\rho$, $\rho=0, 1, 2$, on $[0, \infty)$ with respect to the Confluent Hypergeometric function $\phi(a - n, c;x)$, $n = 0, 1, 2, \cdots$, are explicitly evaluated.

scholarly journals On Mathieu-type series for the unified Gaussian hypergeometric functions

2020 ◽  
Vol 14 (1) ◽  
pp. 138-149
Author(s):  
Rakesh Parmar ◽  
Tibor Pogány
Keyword(s):  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Integral Form ◽  
Type A ◽  
Gauss Hypergeometric Function ◽  
Type Series ◽  
Gaussian Hypergeometric Functions ◽  
Special Cases

The main purpose of this paper is to present closed integral form expressions for the Mathieu-type a-series and for the associated alternating versions whose terms contain a generalized p-extended Gauss' hypergeometric function. Related bounding inequalities for the p-generalized Mathieu-type series are also obtained. Finally, a set of various (known or new) special cases and consequences of the results earned are presented.

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