A New Generalization of Extended Beta and Hypergeometric Functions

Abstract: A new generalization of extended beta function and its various properties, integral representations and distribution are given in this paper. In addition, we establish the generalization of extended hypergeometric and confluent hypergeometric functions using the newly extended beta function. Some properties of these extended and confluent hypergeometric functions such as integral representations, Mellin transformations, differentiation formulas, transformation and summation formulas are also investigated.


introduction
Recently many researchers introduced various extensions and generalizations of various special functions due to its applications in various fields. The latest development and properties of such extension is found in the recent work of various researchers (see e.g., [1,2,3,4,7,10,13,12,14,15]).

A new generalization of extended beta function and its properties
In this section, we establish a new generalization of extended beta function and its properties such as Mellin transforms and integral representations. Definition 1 The generalization of extended beta function is defined as: where ℜ(ς) > 0 , ℜ(β) > 0, ℜ(p) ≥ 0 and λ , m > 0 and E p λ (.) is Mittag-Leffler function. Remark 1 Note that: (i) If λ = 1, then (19) reduces the generalized extended beta function given in [17].
Interchanging the order integrations, we have By using the following formula (see [9], pp. 95), By applying the (24) to (22), we have Now, using the Euler's reflection formula on Gamma function, we get the desired result.

Corollary 1
The following integral representation holds true Proof. By taking s = 1 and λ = m = 1 in proof of Theorem 2, we get the required result.

Theorem 2 The following integral representations holds true
and Equations (26)-(29) can be easily obtained by taking the transformation z = cos 2 θ, z = u 1+u , z = 1+u 2 and z = u−a c−a in (19), respectively.

properties of generalized extended beta function
In this section, we investigate various properties of (19). Theorem 3 The extension of beta function satisfies the following integral representation Proof. Consider the left hand side of (30), we have which proves the desired result.

Corollary 2
The following result holds true Proof. Setting m = 1 in proof of Theorem 3, we get the desired result.

Corollary 3
The following result holds true Proof. Setting m = λ = 1 in proof of Theorem 3, we get the desired result.

corollary 4
The following integral representation holds true Proof. Setting m = λ = 1 and p = 0 in proof of Theorem 3, we get the required result.

Theorem 4 The extension of beta function satisfies the following summation formulas
Proof. Consider the generalized binomial theorem Applying (35) to the definition (19) of extended beta function Now, interchanging the order of summation and integration in above equation and using (19) completes the desired proof.

Theorem 5
The extension of beta function satisfies the following infinite summation formulas Proof. Replacing the following series representation in (19) ( dt. By interchanging the order of integration and summation in above equation and using (19), we get the desired result. and so on. The above series behaves like as finite binomials series does. Thus, we can finally obtain the desired relation (37). Note that, we can also prove the desired inequality by applying induction on n.

The generalized extended beta distribution
In this section, we give a new generalization of extended beta distribution. Also, we define its mean, variance and moment generating function of the newly defined beta distribution: If ν is any real number, then the mean of extended beta distribution (39) is defined by; where p > 0, m, λ > 0 and −∞ < ς < ∞. 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 (39) can be defined as where where p > 0, m, λ > 0, −∞ < ς and β < ∞ is an extension of incomplete beta function. Remark 2 Obviously, if m = 1 then the beta distribution defined in (39) reduces to the beta distribution defined by Shadab et al. [18]. Similarly m = λ = 1, then the beta distribution defined in (39) reduces to the extended beta distribution defined by Chaudhry et al. [5].

Generalization of Extended hypergeometric functions
In this section, we introduce generalization of an extension of extended hypergeometric and confluent hypergeometric functions by using the generalized extended beta function (19).

Differentiation formulas for the extended hypergeometric functions
In this section, we derive the differentiations formulas for the generalized extended hypergeometric and confluent hypergeometric functions. Theorem 11 Proof. Differentiating (43) with respect to z, we have d dz Changing n to n + 1 and using the fact that Again differentiating (55) with respect to z, we obtain Continuing up to n times, we get the required result.

Theorem 12
Proof. Applying the similar procedure used in Theorem 7, we get the desired result.
Proof. Applying Mellin transform on both sides of (45), we have dtdp.
Interchanging the order of integrations in above equation, we have Using (24) in (60), we have Now, using the Euler's reflection formula on Gamma function (25), we get the desired result.

Theorem 14
The following result holds true; Proof. Taking the inverse Mellin transform of both sides on (58), we get the required result.
In similar way, we can prove the following theorems for extended confluent hypergeometric functions.

Theorem 16
The following result holds true

Transformation and summation formulas
In this section, we obtain transformation and summation formulas for the generalized extended hypergeometric and confluent hypergeometric functions. Theorem 17 The following transformation for extended hypergeometric function holds true where p ≥ 0, λ > 0 | arg(1 − z)| < π. Proof. Replacing t by (1 − t) in (45), we get the desired result.
By applying definition (19) to the above equation, we get the desired result.

Concluding remarks
In this paper, we established the generalization of extended beta function and hypergeometric functions. In conclusion, by letting λ = 1, the results obtained in this paper will reduces to the results of [17] and by letting m = 1 throughout the paper, all the results will be reduced to the work of Shadab et al. [18]. Similarly, if we set m = λ = 1 then the results obtained in this paper will give the results of Chaudhry et al. (see [5,6]). In a similar way, if we letting m = λ = 1 and p = 0 then all the results will be reduced to the results involving the classical beta function, beta distribution, Gauss hypergeometric and confluent hypergeometric functions etc (see [8], [16]).