#### 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 .

In 1997 Choudhary et al.  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 )

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.

We also check the effect of on the new extension of classical beta function. For this purpose, we fix the values of and as shown in Figure 1, then we plot the graph which depicts that is an increasing function as the values of increase. It is very clear from Figure 1 that for the graph of classical beta function, new extension of classical beta function remains concave upward (or convex downward) for different values of , and . The value of does not affect the nature of classical beta function; the main effect of the value of is that it just pushes the curve up or drags down the curve from the curve of the classical beta function. In Figure 2, Mesh-Plot is established of new extension of classical beta function, which can be easily interpreted.

From the above proof of radius of convergence of series and further numerical investigation of the power series in Tables 1 and 2, we find that the interval of convergence of the series is , which implies that is convergent for , where is positive number not greater than 2.0335.

Note 2. From the above discussion, it is easy to conclude that the value of lies in the interval ; i.e., .

Note 3. In the sequel of this paper, represents the circle of convergence and is the radius of convergence of (16), where is not greater than

Remark 4. For , , ; (where is positive number not greater than 2.0335), the new extension of classical beta function can be presented in the relation Fox-Wright function (see (14)) as follows:The above result is obtained from (18).

#### 4. Integral Representation of the New Extension of Classical Beta Function

The integral representation of the new extended beta function is important both to check whether the extension is natural and simple and for later use. It is also important to investigate the relationship between the classical beta function and the new extension of the classical beta function. In this connection, we first provide a relationship between them. The following integral formula is useful for further investigation :

Theorem 5 (relation between new extension of the classical beta function and the classical beta function). If , , ; (where is positive number not greater than 2.0335), then we have the following relation:

Proof. Multiplying both sides of (16) by , then integrating with respect to from to , we haveand interchanging the order of integration, (26) reduces toand further using the formula given in (24), after simplification, (27) reduces toand using the definition of classical beta function, we have the required result.

Remark 6. By setting , the result in (25) reduces towhich gives the interesting relation between classical beta function and new extended beta function.

Remark 7. All the derivatives of the new extension of classical beta function with respect to the parameter can be expressed in terms of the function as

Theorem 8 (integral representations of the new extension of the classical beta function). If , , ; (where is positive number not greater than 2.0335), then we have the following relation:

Proof. The result (31) can be easily obtained by setting in (16); to prove (32) choose ; (33) can be easily obtained by applying the symmetric property in (32) then adding new one and (32); the result in (34) is obtained by taking , and setting in (34) gives the result in (35) and to prove the result in (36) put in (35). The results in (37), (38), and (39) can be easily obtained from the result (36).

Remark 9 (useful inequalities). If , , then we have the following inequalityfollows from the integral representation (32), since the function attains its maximum value at and .

#### 5. Properties of the New Extension of the Classical Beta Function

Theorem 10 (functional relation). If , , ; (where is positive number), then we have the following relation:

Proof. Using the definition of new extension of beta function, LHS of (41) is equal toand after simplification (42) reduced to

If we choose , we get the usual relation for the beta function from (41).

Theorem 11 (symmetry). If , , ; (where is positive number), then we have the following relation:

Proof. From (18), we haveand since usual beta function is symmetric, i.e., , using this property in the right-hand side of (45), then we have

Theorem 12 (first summation relation). If , , ; (where is positive real number), then we have the following relation:

Proof. The LHS of (47) can be written asand using the binomial series expansion in (48) and then interchanging the order of summation and integration, the above result (48) reduced to the following form:

Theorem 13 (second summation relation). If (where is positive number), then we have the following relation:

Proof. The LHS of (47) can be written asand using the binomial series expansion and interchanging the order of summation and integration, (52) reduces to

Theorem 14 (separation). If , , ; (where is positive number), then can be separated into real and imaginary parts of as follows:where and

Proof. Since , so let , where and also let and ; then from (16), we haveand after simplification (57) reduces toEquating the real and imaginary parts of only, we have the required results.

#### 6. Applications of New Extension of the Classical Beta Function

It is expected that there will be many applications of the new extension of the classical beta function, e.g., new extension of the beta distribution, new extensions of Gauss hypergeometric functions and confluent hypergeometric function, generating relations, and extension of Riemann-Liouville derivatives. All these have been introduced in the following subsections.

##### 6.1. The New Extension of the Beta Distribution

One application that springs to mind is to statistics. For example, the conventional beta distribution can be extended, by using our new extension of the classical beta function, to variables p and q with an infinite range. It appears that such an extension may be desirable for the project evaluation and review technique used in some special cases.

We define the extension of the beta distribution by

A random variable with probability density function (pdf) given in (59) will be said to have the extended beta distribution with parameters and , , and where is positive number. If is any real number , then

In particular, for ,

represents the mean of the distribution and

is a variance of the distribution.

The moment of generating function of the distribution isThe commutative distribution of (59) can be written as

where

is the new extended incomplete beta function. For , we must have in (65) for convergence, and , where is the incomplete beta function  defined as

It is to be noted that the problem of expressing in terms of other special functions remains open. Presumably, this distribution should be useful in extending the statistical results for strictly positive variables to deal with variables that can take arbitrarily large negative values as well.

##### 6.2. Extensions of Gauss and Confluent Hypergeometric Function Using the New Extension of Beta Function

In this section, we extended the Gauss hypergeometric function and confluent hypergeometric function via new extension of classical beta function, which is defined as follows:

We call new extension of Gauss hypergeometric function and new extension of confluent hypergeometric function.

Note 15. If we choose , the above two new extensions in (67) and (68) reduce to Gauss hypergeometric function and confluent hypergeometric function given in (7) and (8), respectively.

##### 6.3. Numerical Results of New Extension of Gauss Hypergeometric Function and New Extension of Confluent Hypergeometric Function

intoHere, we present the numerical values of new extension of Gauss hypergeoemtric function and new extension of confluent hypergeoemtric function in Table 3 and Table 4 for . Further their graphs are plotted in Figure 3 and Figure 4, respectively. When we have the values of Gauss hypergeoemtric function and confluent hypergeoemtric function.

##### 6.4. Integral Representation of New Extension of Gauss Hypergeometric Function and New Extension of Confluent Hypergeometric Function

The new extension of Gauss hypergeometric function and new extension of confluent hypergeometric function can be provided with an integral representation by using the definition of the new extension of classical beta function (16); we have the following.

Theorem 16. For the new extension of Gauss hypergeometric function , we have the following integral representations:

Proof. Equation (67) can be written asSetting in (69), we have the required result (70).
Again if we choose , we obtain the result (71).

Remark 17. Choosing in (69), we have the following relation between new extensions of Gauss hypergeometric function:

Theorem 18. For the new extension of confluent hypergeometric function , we have the following integral representations:

Proof. The proof of this theorem would run parallel to those of Theorem 16, so we skip the proof of this theorem.

##### 6.5. Differentiation Formulas for the Representation of the New Extension of Gauss Hypergeometric Function and New Extension of Confluent Hypergeometric Function

In the present section, by using the formulas and , we obtain new formulas including derivatives of the new extension of Gauss hypergeometric function and new extension of confluent hypergeometric function with respect to the variable ; we have the following.

Theorem 19. If ; and (where is positive real number), then we have the following result:

Proof. Taking the derivative of with respect to , we haveand replacing , (78) reduces toand with recursive application of this procedure in (79), we have the desired result (77).

Theorem 20. If ; and (where is positive real number), then we have the following result:

Proof. The proof of Theorem 20 is as that of Theorem 19, so it can be omitted here.

###### 6.5.1. Generating Relations Associated with Hypergeometric Functions

Theorem 21. If ; and (where is positive real number), then the following generating functions hold:

Proof. Let the left-hand side of (81) be denoted by ; then using the definition of new extension of Gauss hypergeometric function, we haveUpon reversal of the order of summation and then using the identity , (82) reduces toand further using the definition of binomial , in (83), we haveand interpreting the above equation with the view of (67), we have the desired result (81).

Theorem 22. If ; and (where is positive real number), then the following generating functions hold:

Proof. For convenience, let the left-hand side of (85) be denoted by . Applying the series of (67) to , we getBy changing the order of summation in (86) and using the known identity ([13, p.5]), namely,then, after little simplification, we obtainFurther, upon using the generalized binomial expansion, we find that the inner sum in (88) yieldsFinally in view of (88) and (89), we get the desired assertion (85) of Theorem 1.

A further generalized Gauss hypergeometric function (67) is given in the following definition.

Definition 23. Let us introduce a sequence defined by

where ; and (where is positive real number); for convenience, abbreviates the array of parameters

Now, we prove the following result, which provides the generating functions for the Gauss hypergeometric function defined above.

Theorem 24. For each , the following generating functions hold true:where and (where is positive real number).

Proof. Using the definition introduced in (90) and the new extended Gauss hypergeometric function introduced in (67); then changing the order of summations, the left hand side of (92) (say ) leads toNow taking (90) into account, one can easily arrive at the desired result (92).

Remark 25. It may be noted that if we set and replace by in (92), we are easily led to the result (85).

##### 6.6. Extension of Riemann-Liouville Fractional Derivative

In this section, we introduce new extension of Riemann-Liouville fractional derivative operator: