New Extension of Beta Function and Its Applications

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.


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][2][3][4][5][6].
Gauss hypergeometric function and confluent hypergeometric function are special cases of the generalized hypergeometric series    (,  ∈ N) defined as (see [8, p.73]) and [9, pp. 71 where ()  is the Pochhammer symbol defined (for  ∈ C) by (see [9, p. and Z − 0 denotes the set of nonpositive integers and Γ() is familiar Gamma function.
The Fox-Wright function  Ψ  is defined as (see, for details, Srivastava and Karisson [10]) where the coefficients  1 , . . .,   ,  1 , . . .,   ∈ R + such that Motivated 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.

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.
Case 1.If  > 0, then we need to prove that    (, ) is convergent.
Case 2. If  < 0, then we need to prove that the extension of classical beta function   (, ) is convergent.
From Cases 1 and 2 it is implied that the power series in (18) is convergent.

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  1 and 2, from which we can easily observe that    (, ) does not exist at  =  = 0 and it is also investigated that    (, ) does not exist for  < −2.0335 and  > 2.0335;    (, ) → ∞ as ,  → 0 and    (, ) → 0 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  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 [−2.0335,2.0335], 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 R() lies in the interval [−2.0335, 2.0335]; i.e., −2.0335 ≤ R() ≤ 2.0335.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 2.0335.Remark 4. For R() > 0, R() > 0,  ∈ C; || <  (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).

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 [11]: (where  = m + 1 n ) and R () > 0. ( Theorem 5 (relation between new extension of the classical beta function and the classical beta function).If R(+) > 0, R( + ) > 0,  ∈ C; || <  (where  is positive number not greater than 2.0335), then we have the following relation: Proof.Multiplying both sides of ( 16) by  −1 , then integrating with respect to  from  = 0 to  = ∞, we have and interchanging the order of integration, (26) reduces to International Journal of Mathematics and Mathematical Sciences and further using the formula given in (24), after simplification, (27) and using the definition of classical beta function, we have the required result.( Theorem 8 (integral representations of the new extension of the classical beta function).If R() > 0, R() > 0,  ∈ C; || <  (where  is positive number not greater than 2.0335), then we have the following relation: Proof.The result (31) follows from the integral representation (32), since the function exp(/(1 + ) 2 ) attains its maximum value 1.6626 at  = 1 and  = 2.0335.

Properties of the New Extension of the Classical Beta Function
and after simplification (42) reduced to If we choose  = 0, we get the usual relation for the beta function from (41).
Theorem 11 (symmetry).If R() > 0, R() > 0,  ∈ C; || <  (where  is positive number), then we have the following relation: Proof.From ( 18), we have and since usual beta function is symmetric, i.e., (, ) = (, ), using this property in the right-hand side of (45), then we have where  is positive real number), then we have the following relation: Proof.The LHS of (47) can be written as and using the binomial series expansion 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 R() > 0, R() > 0,  ∈ C; || <  (where  is positive number), then we have the following relation: Proof.The LHS of (47) can be written as and using the binomial series expansion and interchanging the order of summation and integration, (52) reduces to Theorem 14 (separation).If R() > 0, R() > 0,  ∈ C; || <  (where  is positive number), then    (, ) can be separated into real and imaginary parts of  as follows: where  = √ 2 +  2 = || <  and  = tan −1 /.
Proof.Since  ∈ C, so let  =  + , where ,  ∈ R and also let  +  =  cos  +  sin  ⇒  = √ 2 +  2 and  = tan −1 /; then from ( 16), we have and after simplification (57) reduces to Equating the real and imaginary parts of  only, we have the required results.

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.

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.

International Journal of Mathematics and Mathematical Sciences
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 [12], then In particular, for ] = 1, represents the mean of the distribution and is a variance of the distribution.The moment of generating function of the distribution is The commutative distribution of (59) can be written as where is the new extended incomplete beta function.For  = 0, we must have ,  > 0 in (65) for convergence, and   0, (, ) =   (, ), where   (, ) is the incomplete beta function [11] 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.

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  Φ  (; ; ) new extension of confluent hypergeometric function.
Note 15.If we choose  = 0, the above two new extensions in (67) and (68) reduce to Gauss hypergeometric function and confluent hypergeometric function given in ( 7) and ( 8), respectively. of new extension of Gauss hypergeoemtric function and new extension of confluent hypergeoemtric function in Table 3 and Table 4 for  = −2 : 1 : 2. Further their graphs are plotted in Figure 3 and Figure 4, respectively.When  = 0 we have the values of Gauss hypergeoemtric function and confluent hypergeoemtric function.
Again if we choose  = sin 2 , we obtain the result (71).

Theorem 18.
For the new extension of confluent hypergeometric function  Φ  (; ; ), 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.

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 (, −) = (/)(+1, −) and () +1 = ( + 1)  , 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.
and replacing  →  + 1, (78) reduces to and with recursive application of this procedure in (79), we have the desired result (77).
Theorem 20.If ,  ∈ C; R() > R() > 0 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.(81)

Generating Relations Associated with Hypergeometric Functions
Proof.Let the left-hand side of (81) be denoted by S; then using the definition of new extension of Gauss hypergeometric function, we have Upon reversal of the order of summation and then using the identity ()  ( + )  = ()  ( + )  , (82) reduces to and further using the definition of binomial (1 − ) −− = ∑ ∞ =0 ( + )  (  /!), (|| < 1) in (83), we have and interpreting the above equation with the view of (67), we have the desired result (81).(85) Proof.For convenience, let the left-hand side of (85) be denoted by J. Applying the series of (67) to J, we get By changing the order of summation in (86) and using the known identity ([13, p.5]), namely, then, after little simplification, we obtain International Journal of Mathematics and Mathematical Sciences Further, upon using the generalized binomial expansion, we find that the inner sum in (88) yields Finally 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.
Now, we prove the following result, which provides the generating functions for the Gauss hypergeometric function defined above.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 to Now taking (90) into account, one can easily arrive at the desired result (92).
Remark 25.It may be noted that if we set  = 1 and replace  by  −  in (92), we are easily led to the result (85).
where the path of integration is a line from 0 to  in complex −plane.For the case  = 0, we obtain the classical Riemann-Liouville fractional operator.
We start our investigation by calculating the extended fractional derivative of some elementary functions.
Lemma 26.Let ,  ∈ C; R() > −1, R() < 0 and || <  (where  is positive real number); then we have Proof.Employing the definition given in (94) in the left-hand side of (96), we have Choosing  = , (97) reduces to Proof.Employing the definition given in (94) in the left-hand side of (99), we have Choosing  = , (100) reduces to and further employing the result in (69), after simplification, we have the required result (101).
The series ∑ ∞ =0   ()  is uniformly convergent in the disc || <  for 0 ≤  ≤ 1 and the integral ∫ | is convergent provided that R() > 0, R() < 0 and || <  (where  is positive real number), therefore we can interchange the order of integration and summation; after simplification, above equation (103) reduces to (105) Proof.Let us consider the elementary identity Expanding the left-hand side of (106) for || < |1 − |, we have Further, multiplying both sides of (107) and then applying the new extension of fractional derivative operator  −,  on both sides, we have Interchanging the order, which is valid for R() > 0 and || < |1 − |, we have Proof.To prove the theorem, we consider the following identity: Replacing  into 1 − , then using the result (113) in (69), after simplification, we have further interpreted with the view of (69), we obtain the desired result (110).

Fractional Integration of New Extension of Hypergeometric Functions
The concept of the Hadamard products (see [14]) is very useful in our investigation.
Definition 31 (Hadamard products [14]).Let () fl ∑ ∞ =0     and () fl ∑ ∞ =0     be two power series whose radii of convergence are given by   and   , respectively.Then their Hadamard product is power series defined by whose radius of convergence  satisfies   . ≤ . In The above-mentioned detailed and systematic investigation by many authors (see, for example, [4,15]) has largely motivated our present study.Therefore, the results established in this paper are of general character and hence encompass several cases of interest.
In this section, we will establish certain fractional integral formulas involving the new extension of Gauss hypergeometric function and new extension of confluent hypergeometric function.To do this, we need to recall the following pair of Saigo hypergeometric fractional integral operators.
For  > 0, , ,  ∈ C and R() > 0, we have and where the function 2  1 (.) is a special case of the generalized hypergeometric function, the Gauss hypergeometric function.
The operator  ,, 0, (.) contains the Riemann-Liouville   0, (.) fractional integral operators by means of the following relationships: and It is noted that the operator (118) unifies the Erdêlyi-Kober fractional integral operators as follows: and The following lemmas proved in Kilbas and Sebastin [16] are useful to prove our main results.
The main results are given in the following theorem.(128) Further using ()  = Γ( + )/Γ(), (128) reduces to the following form: By putting  = 0, the Saigo hypergeometric fractional integrals operators reduce to the Erdêlyi-Kober fractional integral operators; then the results in (125), ( 131), ( 132 If we choose  = 0, then new extension of Gauss hypergeometric function and new extension of confluent hypergeometric function reduce to Gauss hypergeometric function and confluent hypergeometric function; then from the formulae establisehd in (125), ( 131), ( 132) and (133), we have the following results.

Beta Transform
The Beta transform of () is defined as follows [17]: Proof.For convenience, we denote the left-hand side of the result (147) by B. Using the definition of beta transform, the LHS of (147) becomes and further using (129) and then changing the order of integration and summation, which is valid under the conditions of Theorem 1, then Applying the definition of beta transform, (149) reduced to and interpreting the above equation with the help of (67), we have  (154) Proof.The proofs of the Theorems 51, 52, and 53 are parallel to those of Theorem 50.

Laplace Transform
The Laplace transform of () is defined as follows [17]:   (162) Proof.The proofs of Theorems 55, 56, and 57 would run parallel to those of Theorem 54, so the proofs of these theorems are omitted here.(165) Now we use the following integral formula involving Whittaker function:

Whittaker Transform
and interpreting (167) with the help of (67), then, with the concept of Hadamard (116), we have the desired the result.
Proof.The proofs of Theorems 59, 60, and 61 would run parallel to those of Theorem 58, so the proofs of these theorems are omitted here.

Fractional Kinetic Equations
The importance of fractional differential equations in the field of applied science has gained more attention not only in mathematics but also in physics, dynamical systems, control systems, and engineering, to create the mathematical model of many physical phenomena.The kinetic equations especially describe the continuity of motion of substance.The extension and generalization of fractional kinetic equations involving many fractional operators were found in [18][19][20][21][22][23][24][25][26][27][28][29][30][31].
In view of the effectiveness and a great importance of the kinetic equation in certain astrophysical problems the authors develop a further generalized form of the fractional kinetic equation involving new extensions of Gauss hypergeometric function and confluent hypergeometric function.
The fractional differential equation between rate of change of the reaction, the destruction rate, and the production rate was established by Haubold and Mathai [23], given as follows: where  = () the rate of reaction,  = () the rate of destruction,  = () the rate of production, and   denotes the function defined by   ( * ) = ( −  * ),  * > 0.
In the special case of (171) for spatial fluctuations and inhomogeneities in () the quantities are neglected, that is, the equation with the initial condition that   ( = 0) =  0 is the number density of the species  at time  = 0 and   > 0. If we remove the index  and integrate the standard kinetic equation (172), we have where 0  −1  is the special case of the Riemann-Liouville integral operator 0  −]  defined as The fractional generalization of the standard kinetic equation ( 173) is given by Haubold and Mathai [23] as follows: and obtained the solution of (175) as follows: Further, (Saxena and Kalla [27]) considered the following fractional kinetic equation: where () denotes the number density of a given species at time ,  0 = (0) is the number density of that species at time  = 0,  is a constant, and  ∈ L(0, ∞).

International Journal of Mathematics and Mathematical Sciences
Equation (192) can be written as Theorem 64.If  > 0,  > 0, ] > 0; , , ,  ∈ C;R() > R() > 0 and || <  (where  is positive real number), then the solution of the equation is given by the following formula: Theorem 65.If  > 0, ] > 0; , , ,  ∈ C;R() > R() > 0 and || <  (where  is positive real number), then the solution of the equation is given by the following formula: is given by the following formula: is given by the following formula:

Figure 1 :
Figure 1: Graph of new extension of classical beta function for fixed value of ,  and  = −2 : .5 : 2.

Figure 2 :
Figure 2: Mesh-Plot of the new extension of classical beta function.

Table 1 :
Numerical values of new extension of classical beta function    (, ).

Table 2 :
Numerical values of new extension of classical beta function    (, ).
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 Figure2, Mesh-Plot is established of new extension of classical beta function, which can be easily interpreted.