International Journal of Mathematics and Mathematical Sciences

Volume 2018, Article ID 6451592, 25 pages

https://doi.org/10.1155/2018/6451592

## New Extension of Beta Function and Its Applications

^{1}Department of Mathematics, Baba Farid College, Bathinda 151001, India^{2}BOSS Team, GS laboratory, ENSA, Ibn Tofail University, Kenitra 14000, Morocco^{3}Department of Applied Sciences, Gurukashi University, Bathinda 151302, India

Correspondence should be addressed to Mehar Chand; moc.liamg@arhdnallaj.rahem

Received 19 April 2018; Accepted 22 July 2018; Published 18 December 2018

Academic Editor: Niansheng Tang

Copyright © 2018 Mehar Chand et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction and Preliminaries

No doubt the classical beta function is one of the most fundamental special functions, because of its precious role in several field of sciences such as mathematical, physical, and statistical sciences and engineering. In many areas of applied mathematics, different types of special functions have become necessary tool for the scientists and engineers. During the recent decades or so, numerous interesting and useful extensions of the different special functions (the Gamma and beta functions, the Gauss hypergeometric function, and so on) have been introduced by different authors [1–6].

In 1997 Choudhary et al. [1] introduced the following extension of classical beta function defined as

Further Chaudhry et al. [7, p. 591, Eqs. and ] made use of the extended beta function in (1) to extend the Gauss hypergeometric function and confluent hypergeometric function as follows:

and

and present their Euler type integrals as follows:

and

If we choose , the above definitions given in (1), (2), (3), (4), and (5) reduce to the following form, respectively:

Gauss hypergeometric function and confluent hypergeometric function are special cases of the generalized hypergeometric series defined as (see [8, p.73]) and [9, pp. 71-75]:

where is the Pochhammer symbol defined (for ) by (see[9, p.2 and p.5])and denotes the set of nonpositive integers and is familiar Gamma function.

The Fox-Wright function is defined as (see, for details, Srivastava and Karisson [10])

where the coefficients , such thatMotivated from the above literature, we introduce new extension of classical beta function in (16) and its convergence is studied in Theorem 1 in Section 2. Using MATLAB(R2015a), the numerical results and graphs are presented in Section 3 and also radius of convergence of new extension of classical beta function is discussed on the basis of numerical results established by using MATLAB software. We establish the integral representations and study the properties of new extension of classical beta function.

Using the new extended beta function, extension of the beta distribution is also introduced; Gauss hypergeometric function and confluent hypergeometric function are extended by employing the new extension of classical beta function. Then we have studied the generating relations, extension of Riemann-Lioville fractional derivative operator. Fractional integrals of extended hypergeometric functions and their image formulas in the form of beta transform, Laplace transform, and Whittaker transform have been also established. The solutions of fractional kinetic equations involving extended Gauss hypergeometric function and extended confluent hypergeometric function are established. The numerical results and graphical interpretation have made it easier to study the nature of these fractional kinetic equations.

#### 2. Extension of Beta Function

In this section, we introduce new extension of classical beta function. Its convergence is proved mathematically; then numerical results are established for different values of parameters involved.

We introduce new extension of classical beta function as follows:where (where is positive number).

Theorem 1. *If (where is positive number), then the new extension of the classical beta function in equation (16) is convergent.*

*Proof. *We can write (16) as follows:and further, using the definition of classical beta function (6), (17) reduces toIn the above equation, is in series form involving (where ) and in each term of the series, is convergent, since and for and , which implies that each term of the series (18) exists.

Now we shall prove that is convergent. may be greater than or less than , so there are two cases as follows.*Case 1*. If , then we need to prove that is convergent.

Equation (18) can be written asFurther,By ratio test for positive series, is convergent for .*Case 2*. If , then we need to prove that the extension of classical beta function is convergent.

To prove this case, let (where ); then (18) becomesEquation (21) can be written asThe series (22) is an alternating series; therefore (1), (2) is decreasing(3) if ( as only if and as ) All the conditions of Leibniz’s test for alternating series have been satisfied; therefore is convergent for .

From Cases 1 and 2 it is implied that the power series in (18) is convergent.

#### 3. Numerical Results and Graphs of New Extension of the Classical Beta Function

The numerical results of new extension of classical beta function have been calculated in this section. For this purpose we choose the values of variables and parameter as and . All the numerical values of new extension of the classical beta function are presented in Tables 1 and 2, from which we can easily observe that does not exist at and it is also investigated that does not exist for and ; as and as , which implies that the behaviour of new extension of classical beta function is the same as that of classical beta function.