International Journal of Mathematics and Mathematical Sciences
Volume 2018 (2018), Article ID 6451592, 25 pages
https://doi.org/10.1155/2018/6451592

Research Article

New Extension of Beta Function and Its Applications

Mehar Chand,¹ Hanaa Hachimi,² and Rekha Rani³

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

Correspondence should be addressed to Mehar Chand; mehar.jallandhra@gmail.com

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 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.

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 to

In 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 as

Further,

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) becomes

Equation (21) can be written as

The 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.

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

0.00
0.00
NaN
NaN
NaN
NaN

0.01
0.01
198.27177610
198.29548624
199.06411204
199.96757732

0.02
0.02
98.27460320
98.29781890
99.05065967
99.93608768

0.03
0.03
64.94418565
64.96691911
65.70435720
66.57217010

0.04
0.04
48.28052079
48.30278382
49.02519017
49.87579792

0.05
0.05
38.28360599
38.30541007
39.01314459
39.84694542

0.06
0.06
31.62010537
31.64146162
32.33487366
33.15225447

0.07
0.07
26.86144500
26.88236422
27.56179291
28.36312969

0.08
0.08
23.29333673
23.31382939
23.97960405
24.76526226

0.09
0.09
20.51895277
20.53902903
21.19146950
21.96180446

0.10
0.10
18.30019555
18.31986529
18.95928227
19.71463949

1.0
1.0
0.72102562
0.72477846
0.84887277
1.00000000

2.0
2.0
0.11166177
0.11238923
0.13665458
0.16666667

3.0
3.0
0.02163870
0.02179192
0.02692731
0.03333333

4.0
4.0
0.00455690
0.00459058
0.00572283
0.00714286

5.0
5.0
0.00100158
0.00100918
0.00126512
0.00158730

6.0
6.0
0.00022592
0.00022766
0.00028649
0.00036075

7.0
7.0
0.00005185
0.00005225
0.00006594
0.00008325

8.0
8.0
0.00001205
0.00001214
0.00001536
0.00001943

9.0
9.0
0.00000283
0.00000285
0.00000361
0.00000457

10.0
10.0
0.00000067
0.00000067
0.00000085
0.00000108

Table 2: Numerical values of new extension of classical beta function .

0.00
0.00
NaN
NaN
NaN
NaN

0.01
0.01
199.96757732
201.03554301
202.30476477
202.35127432

0.02
0.02
99.93608768
100.98330172
102.22848838
102.27412810

0.03
0.03
66.57217010
67.59910183
68.82077895
68.86556756

0.04
0.04
49.87579792
50.88290361
52.08158267
52.12553836

0.05
0.05
39.84694542
40.83466875
42.01084757
42.05398809

0.06
0.06
33.15225447
34.12102693
35.27519021
35.31753282

0.07
0.07
28.36312969
29.31337104
30.44599075
30.48755229

0.08
0.08
24.76526226
25.69738092
26.80891675
26.84971362

0.09
0.09
21.96180446
22.87619790
23.96709764
24.00714582

0.10
0.10
19.71463949
20.61169459
21.68239456
21.72170964

1.0
1.0
1.00000000
1.18459307
1.41068613
1.41909001

2.0
2.0
0.16666667
0.20385173
0.25000000
0.25172603

3.0
3.0
0.03333333
0.04133360
0.05133577
0.05171118

4.0
4.0
0.00714286
0.00892526
0.01116423
0.01124846

5.0
5.0
0.00158730
0.00199311
0.00250456
0.00252383

6.0
6.0
0.00036075
0.00045452
0.00057299
0.00057746

7.0
7.0
0.00008325
0.00010515
0.00013287
0.00013392

8.0
8.0
0.00001943
0.00002458
0.00003112
0.00003137

9.0
9.0
0.00000457
0.00000579
0.00000734
0.00000740

10.0
10.0
0.00000108
0.00000137
0.00000174
0.00000176

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.

Figure 1: Graph of new extension of classical beta function for fixed value of and .

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

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 [11]:

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 have

and interchanging the order of integration, (26) reduces to

and further using the formula given in (24), after simplification, (27) reduces to

and using the definition of classical beta function, we have the required result.

Remark 6. By setting , the result in (25) reduces to

which 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 inequality

follows 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 to

and 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 have

and 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 as

and 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 as

and 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 have

and after simplification (57) reduces to

Equating 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 [12], 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 is

The 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 [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.

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.

Table 3: Numerical values of new extension of Gauss hypergeometric function .

x

0.0000
0.7248
0.8489
1.0000
1.1846
1.4107

0.1000
0.7734
0.9080
1.0721
1.2728
1.5188

0.2000
0.8308
0.9778
1.1572
1.3768
1.6463

0.3000
0.9000
1.0618
1.2594
1.5017
1.7993

0.4000
0.9855
1.1653
1.3853
1.6552
1.9870

0.5000
1.0947
1.2972
1.5452
1.8495
2.2240

0.6000
1.2410
1.4730
1.7572
2.1061
2.5356

0.7000
1.4512
1.7236
2.0570
2.4664
2.9701

0.8000
1.7914
2.1235
2.5295
3.0270
3.6382

0.9000
2.4976
2.9335
3.4632
4.1089
4.8982

1.0000
6.7829
7.5302
8.4142
9.4648
10.7195

Table 4: Numerical values of new extension of confluent hypergeometric function .

x

0.0000
0.7248
0.8489
1.0000
1.1846
1.4107

0.1000
0.7478
0.8769
1.0342
1.2264
1.4620

0.2000
0.7721
0.9064
1.0701
1.2704
1.5159

0.3000
0.7976
0.9374
1.1080
1.3167
1.5726

0.4000
0.8245
0.9701
1.1478
1.3654
1.6324

0.5000
0.8528
1.0045
1.1898
1.4167
1.6953

0.6000
0.8826
1.0408
1.2340
1.4707
1.7616

0.7000
0.9141
1.0790
1.2806
1.5277
1.8314

0.8000
0.9473
1.1194
1.3298
1.5879
1.9051

0.9000
0.9824
1.1620
1.3817
1.6513
1.9828

1.0000
1.0195
1.2070
1.4366
1.7183
2.0649

Figure 3: Graph of the new extension of Gauss hypergeometric function for , and .

Figure 4: Graph of the new extension of confluent hypergeometric function for , and .

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 as

Setting 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 have

and replacing , (78) reduces to

and 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 have

Upon reversal of the order of summation and then using the identity , (82) reduces to

and further using the definition of binomial , in (83), we have

and 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 get

By changing the order of summation in (86) and using the known identity ([13, p.5]), namely,

then, after little simplification, we obtain

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.

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 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 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:

and for ,

where the path of integration is a line from 0 to in complex plane. For the case , 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 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

Lemma 27. Let , and (where is positive real number), then we have

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).

Theorem 28. Let be an analytic function in the disc and it has the power series ; then we have the following result:

where and (where is positive real number).

Proof.

The series is uniformly convergent in the disc for and the integral is convergent provided that and (where is positive real number), therefore we can interchange the order of integration and summation; after simplification, above equation (103) reduces to

and interpreting (104) with the view of the definition of new extension of classical beta function (16), we have the required result.

Theorem 29 (linear generating function). Let and (where is positive real number), such that ; then we have the following result:

Proof. Let us consider the elementary identity

Expanding the left-hand side of (106) for , 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 and , we have

Now applying the result established in (99) in (109), after simplification, we have the required result (105).

6.7. Transformation Formulas

Theorem 30. Let and (where is positive real number), then we have the following transformation formula:

Proof. To prove the theorem, we consider the following identity:

Replacing into , then using the result (113) in (69), after simplification, we have

further interpreted with the view of (69), we obtain the desired result (110).

7. 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 and 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 particular, let us consider the function . Its decomposition is illustrative. That is

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 and , we have

and

where the function is a special case of the generalized hypergeometric function, the Gauss hypergeometric function.

The operator contains the Riemann-Liouville fractional integral operators by means of the following relationships:

and

It is noted that the operator (118) unifies the Erdlyi-Kober fractional integral operators as follows:

and

The following lemmas proved in Kilbas and Sebastin [16] are useful to prove our main results.

Lemma 32 (Kilbas and Sebastian 2008). Letting be such that , then

Lemma 33 (Kilbas and Sebastian 2008). Letting be such that , then

The main results are given in the following theorem.

Theorem 34. Let ; and (where is positive real number), such that ; then

Proof. For convenience, we denote the left-hand side of the result (125) by . Using (67), and then interchanging the order of integration and summation, which is valid under the conditions of Theorem 34, then

Applying the result (123), (126) reduced to

After simplification, (127) reduces to

Further using , (128) reduces to the following form:

and interpreting the above equation with the help of the concept of Hadamard given in (116), we have the required result.

Theorem 35. Let ; and (where is positive real number), such that ; then

Theorem 36. Let ; and (where is positive real number), such that ; then

Theorem 37. Let ; and (where is positive real number), such that ; then

Proof. The proofs of the Theorems 35, 36, and 37 are the same as those of Theorem 34.

7.1. Some Special Cases of the above Fractional Integral Formulas

By assigning the suitable values to the parameters involved in the results established in Theorems 34–37, we have the following special cases.

By putting , the Saigo hypergeometric fractional integrals operators reduces to Riemann-Liouville fractional integral operators; then the results in (125), (131), (132) and (133) reduce to the following form.

Corollary 38. Let ; and (where is positive real number), such that ; then

Corollary 39. Let ; and (where is positive real number), such that ; then

Corollary 40. Let ; and (where is positive real number), such that ; then

Corollary 41. Let ; and (where is positive real number), such that ; then

By putting , the Saigo hypergeometric fractional integrals operators reduce to the Erdlyi-Kober fractional integral operators; then the results in (125), (131), (132) and (133) reduce to the following form.

Corollary 42. Let ; and (where is positive real number), such that ; then

Corollary 43. Let ; and (where is positive real number), such that ; then

Corollary 44. Let ; and (where is positive real number), such that ; then

Corollary 45. Let ; and (where is positive real number), such that ; then

If we choose , 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.

Corollary 46. Let ;, such that ; then

Corollary 47. Let ;, such that ; then

Corollary 48. Let ;, such that ; then

Corollary 49. Let ;, such that . Then

8. Beta Transform

The Beta transform of is defined as follows [17]:

Theorem 50. Let ; and (where is positive real number), such that ; then

Proof. For convenience, we denote the left-hand side of the result (147) by . 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

Further, interpreting (151) with the view of the concept of Hadamard (116), we have the required the result.

Theorem 51. Let ; and (where is positive real number), such that ; then

Theorem 52. Let ; and (where is positive real number), such that ; then

Theorem 53. Let ; and (where is positive real number), such that ; then

Proof. The proofs of the Theorems 51, 52, and 53 are parallel to those of Theorem 50.

9. Laplace Transform

The Laplace transform of is defined as follows [17]:

Theorem 54. Let ; and (where is positive real number), such that ; then

Proof. For convenience, we denote the left-hand side of the result (156) by . Then, applying the Laplace, we have

and further using (129) and then changing the order of integration and summation, which is valid under the conditions of Theorem 1, then

After simplification, (158) reduces to

and interpreting (159) with the view of the concept of Hadamard (116), we have the required the result.

Theorem 55. Let ; and (where is positive real number), such that ; then

Theorem 56. Let ; and (where is positive real number), such that ; then

Theorem 57. Let ; and (where is positive real number), such that ; then

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.

10. Whittaker Transform

Theorem 58. Let ; and (where is positive real number) and , such that ; then

where

Proof. For convenience, we denote the left-hand side of the result (163) by . Then using the result from (128), after changing the order of integration and summation, we get

By substituting , (164) becomes

Now we use the following integral formula involving Whittaker function:

Then we have

and interpreting (167) with the help of (67), then, with the concept of Hadamard (116), we have the desired the result.

Theorem 59. Let ; and (where is positive real number) and , such that ; then

where

Theorem 60. Let ; and (where is positive real number) and , such that ; then

where

Theorem 61. Let ; and (where is positive real number) and , such that ; then

where

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.

11. 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 [18–31].

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

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 is the number density of the species at time and . If we remove the index and integrate the standard kinetic equation (172), we have

where is the special case of the Riemann-Liouville integral operator 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 , is the number density of that species at time , is a constant, and .

By applying the Laplace transform to (177), we have

Where the Laplace transform [32] is given by

11.1. Solutions of Generalized Fractional Kinetic Equations

In this section, we investigated the solutions of the generalized fractional kinetic equations involving the new extension of Gauss hypergeometric function and confluent hypergeometric function.

Remark 62. The solutions of the fractional kinetic equations in this section are obtained in terms of the generalized Mittag-Leffler function (Mittag-Leffler[33]), which is defined as

Theorem 63. If ; ; and (where is positive real number), then the solution of the equation

is given by the following formula:

Proof. Laplace transform of Riemann-Liouville fractional integral operator is given by the following (Erdelyi et al. [34], Srivastava and Saxena[35]):

where is defined in (179). Now, applying Laplace transform on (181) gives

and interchanging the order of integration and summation in (185), we have

This leads to

Taking Laplace inverse of (188), and by using

we have

Equation (192) can be written as

Theorem 64. If ; ; and (where is positive real number), then the solution of the equation

is given by the following formula:

Theorem 65. If ; ; and (where is positive real number), then the solution of the equation

is given by the following formula:

Theorem 66. If ; ; and (where is positive real number), then the solution of the equation

is given by the following formula:

Theorem 67. If ; ; and (where is positive real number), then the solution of the equation

is given by the following formula:

Theorem 68. If ; ; and (where is positive real number), then the solution of the equation

is given by the following formula:

Proof. The proofs of Theorems 64, 65, 66, 67, and 68 are the same as those of Theorem 63, so we skip the proof of these theorems.

11.1.1. Numerical results and their graphs of solution of fractional kinetic equation

In this section, we present the numerical results in Table 5 for , , , , , , and of equation (126) and the graph of solution of fractional kinetic equation given in (126) are presented in Figures 5, 6 and 7 for , , , , , , ; the values of are chosen as , and , respectively. Figure 8 presents the Mesh-plot of the same solution fractional kinetic equation. For different values of the parameters, it can be easily interpreted and can be observed that for different values of the parameters. As the solution of fractional kinetic equations are presented in the form of summation. For the numerical results and their graphs, sum first 500 terms have been taken. If we choose more than 500 terms the results are same.

Table 5: Numerical values of solution of fractional kinetic equation given in (182) for .

t

0.00
5.6427
5.6427
5.6427
5.6427
5.6427

0.10
6.5072
5.4044
5.6143
5.6408
5.6426

0.20
7.0748
5.1461
5.5127
5.6249
5.6410

0.30
7.5810
4.9199
5.3291
5.5773
5.6330

0.40
8.0669
4.7403
5.0624
5.4788
5.6101

0.50
8.5489
4.6124
4.7172
5.3088
5.5595

0.60
9.0358
4.5376
4.3026
5.0468
5.4638

0.70
9.5338
4.5149
3.8322
4.6727
5.3011

0.80
10.0472
4.5423
3.3237
4.1680
5.0448

0.90
10.5800
4.6174
2.7980
3.5166
4.6637

1.00
11.1353
4.7378
2.2788
2.7061
4.1222

1.10
11.7165
4.9012
1.7917
1.7296
3.3808

1.20
12.3265
5.1058
1.3630
0.5864
2.3968

1.30
12.9684
5.3500
1.0193
-0.7158
1.1249

1.40
13.6456
5.6329
0.7860
-2.1593
-0.4809

1.50
14.3612
5.9541
0.6861
-3.7152
-2.4656

1.60
15.1187
6.3138
0.7398
-5.3414
-4.8712

1.70
15.9220
6.7125
0.9628
-6.9812
-7.7330

1.80
16.7748
7.1514
1.3661
-8.5628
-11.0764

1.90
17.6815
7.6323
1.9551
-9.9980
-14.9121

2.00
18.6466
8.1573
2.7289
-11.1825
-19.2305

Figure 5: Solution of fractional kinetic equation given in (182) for .

Figure 6: Solution of fractional kinetic equation given in (182) for .

Figure 7: Solution of fractional kinetic equation given in (182) for .

Figure 8: Mesh-Plot of solution of fractional kinetic equation given in (182) for .

11.2. Special Cases of Fractional Kinetic Equations

Choosing , Theorems 63, 64, 65, 66, 67, and 68 reduce to the following form.

Corollary 69. If ; ; , then the solution of the equation

is given by the following formula:

Corollary 70. If ; ; , then the solution of the equation

is given by the following formula:

Corollary 71. If ; ; , then the solution of the equation

is given by the following formula:

Corollary 72. If ; ; , then the solution of the equation

is given by the following formula:

Corollary 73. If ; ; , then the solution of the equation

is given by the following formula:

Corollary 74. If ; ; , then the solution of the equation

is given by the following formula:

12. Concluding Remarks

In the present article authors introduced the new the generalization of the classical beta function. It has been further used to study the various properties of the new extended beta function. Furthermore, on application of this new extended beta function, extension of Gauss hypergeometric function and confluent hypergeometric function are introduced. Fractional integrals of extended hypergeometric functions and their image formulas (in the form of beta transform, Laplace transform, and Whittaker transform) are established. We introduce new fractional generalizations of the standard kinetic equation and we derived the solutions for the same. Their numerical results and graphs are established to study the nature of these fractional kinetic equations involving new extended Gauss hypergeometric function and confluent hypergeometric function. From the closed relationship of the hypergeometric functions with various special functions, we can easily construct various new known results.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The author declares that they have no conflicts of interest.

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