A Note on the New Extended Beta Function with Its Application

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 .

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

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.

Some Inequalities of Extended Hypergeometric Functions

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.

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

Our 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.

An Extension of Beta Function by Using Wiman’s 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.

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

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.

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

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.

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

Recently, 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.

Generalized Wright Function and Its Properties Using 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.