A NEW EXTENSION OF BETA AND HYPERGEOMETRIC FUNCTIONS

The main objective of this paper is to introduce a further extension of extended (p, q)-beta function by considering two Mittag-Leffler function in the kernel. We investigate various properties of this newly defined beta function such as integral representations, summation formulas and Mellin transform. We define extended beta distribution and its mean, variance and moment generating function with the help of extension of beta function. Also, we establish an extension of extended (p, q)-hypergeometric and (p, q)-confluent hypergeometric functions by using the extension of beta function. Various properties of newly defined extended hypergeometric and confluent hypergeometric functions such as integral representations, Mellin transformations, differentiation formulas, transformation and summation formulas are investigated.


introduction
We start with the classical beta function which is defined by and its relation with well known gamma function is given by The Gauss hypergeometric and confluent hypergeometric functions which are defined (see [16]) as , respectively.The integral representation of hypergeometric and confluent hypergeometric functions are respectively defined by ) , and ) .
In the same paper, they defined the following integral representations of extended hypergeometric and confluent hypergeometric functions as and It is clear that when p = 0, then the equations (1.9)-(1.12)reduce to the well known hypergeometric and confluent hypergeometric series and their integral representation respectively.
In the same paper, they defined the following integral representations of extended hypergeometric and confluent hypergeometric functions as ) , and Φ p,q (δ 2 ; It is clear that when p = q, then the equations (1.15)-(1.18)reduce to the well known extended hypergeometric and confluent hypergeometric series and their integral representation respectively (see e.g., (1.9)-(1.12)).

Extension of beta function and its properties
In this section, we define a new extension of beta function and its properties such as Mellin transforms and integral representations.

Theorem 2.1. The extension of extended beta function have the following Mellin transform relation:
) Proof.Applying the Mellin transform on (2.1), we have Interchanging the order integrations, we have 3), we have By applying the definition of Γ λ (.) to (2.4) (see [15]), we get the following desired result.
Corollary 2.1.The following integral representation holds true Proof.By taking r = s = 1 and λ = 1 in Theorem 2.1, we get the required result.

properties of extended beta function
In this section, we investigate various properties of the extended beta function B λ p (δ 1 , δ 2 ).

Theorem 3.1. The extension of beta function satisfies the following integral representation
Proof.Consider the left hand side of (3.1), we have which proves the desired result.
Corollary 3.1.The following result holds true Proof.Setting λ = 1 in Theorem 3.1, we get the desired result.

Corollary 3.2. The following integral representation holds true
Proof.Setting λ = 1 and p = q in Theorem 3.1, we get the required result.

Corollary 3.3. The following integral representation holds true
Proof.Setting λ = 1 and p = q = 0 in Theorem 3.1, we get the required result.
Theorem 3.2.The extension of beta function satisfies the following summation formulas Proof.Consider the generalized binomial theorem Applying (3.6) to the definition (2.1) of extended beta function Now, interchanging the order of summation and integration in above equation and using (2.1) proves the desired result.
Theorem 3.3.The extension of beta function satisfies the following infinite summation formulas Proof.Replacing the following series representation in (2.1) we obtain By interchanging the order of integration and summation in above equation and using (2.1), we get the desired result.

Theorem 3.5. For extension of beta function, we have the following Mellin transformation formula:
Proof.Applying the inverse Mellin transform on both sides of (2.2), we get the desired result.

The Extended beta distribution
Like extended beta function B p (x, y) and B p,q (x, y) there will be many application of further extension of beta function B λ p,q (x, y).One application of this newly extension of beta function B λ p,q (x, y) is to define beta distribution to a variables δ 1 and δ 2 with an infinite range, which is an extension of extended beta distribution defined by Chaudhary et al. and Choi et al. (see [4,6]).We define the extension of beta distribution by It is clear that the beta distribution (4.1) is the extension of beta distribution defined by Chaudhary et al. and Choi et al. (see [4,6]).
If v is any real number, then the mean of extended beta distribution (4.1) is defined as; ) .
When v = 1, then we get the mean of the distribution .
The various of the distribution can defined by The moment generating function of the distribution is defined by The cumulative distribution of (4.1) can be defined as where ) is an extension of incomplete beta function.

Extension of hypergeometric functions
In this section, we introduce further extension of hypergeometric and confluent hypergeometric functions by using the extension of beta function (2.1).

Differentiation formulas for the extended hypergeometric functions
In this section, we derive differentiations formulas for the extended hypergeometric and confluent hypergeometric functions.
Theorem 7.1.The following formula hold true: Proof.Differentiating (5.1) with respect to z, we have Changing n to n + 1 in (7.2), we have Applying (7.4) to (7.3), we get Again differentiating (7.5) with respect to z, we obtain Continuing up to n times, we get the required result.

Transformation and summation formulas
In this section, we obtain transformation and summation formulas for the extended hypergeometric and confluent hypergeometric functions as follows: Theorem 9.1.The following transformation for extended hypergeometric function holds true for p, λ > 0: where Proof.Replacing t by (1 − t) in (6.1), we get the desired result.Proof.From (6.6) and (6.7), we can easily establish the required result.
Proof.Taking z = 1 in (6.1), we have ( δ 1 , δ 2 ; δ 3 ; 1 By applying definition (2.1) to the above equation, we get the desired result.In this paper, the authors established the extension of extended beta function.They defined several properties, integral representations and extension of beta distribution.Also, they defined further extension of extended hypergeometric and confluent hypergeometric functions with the help of newly defined beta function and established various properties, integral representations and differentiation formulas of extended hypergeometric and confluent hypergeometric functions.The authors conclude that if we letting λ = 1 throughout in the paper then all the results will be reduced to the work of Chaudhry et al. (see [6]).Also, if we letting λ = 1 and p = q throughout in the paper then all the results will be reduced to results of extended beta function, extended beta distribution, extended Gauss hypergeometric and extended confluent hypergeometric functions ( see [4,5]).In similar way, if we letting λ = 1 and p = q = 0 throughout in the paper then all the results will be reduced to the classical results of beta function, beta distribution, Gauss hypergeometric and confluent hypergeometric functions (see [8], [16]).