# New Extension of Beta Function and Its Applications

Authors: Mehar Chand; Hanaa Hachimi; Rekha Rani
Journal: International Journal of Mathematics and Mathematical Sciences
 (2018)

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## 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.


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## Body

## 1. Introduction and Preliminaries

No doubt the classical beta functionB(α,β) 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(1)Bpα,β=Bα,β;p=∫01tα-11-tβ-1exp⁡-pt1-tdt,Rp≥0.Further Chaudhry et al. [7, p. 591, Eqs. (2.1) and (2.2)] made use of the extended beta function Bp(α,β) in (1) to extend the Gauss hypergeometric function and confluent hypergeometric function as follows:(2)Fpa,b;c;z=∑n=0∞anBpb+n,c-bBb,c-bznn!,z<1;Rc>Rb>0;Rp≥0and(3)Φpb;c;z=∑n=0∞Bpb+n,c-bBb,c-bznn!,z<1;Rc>Rb>0;Rp≥0.and present their Euler type integrals as follows:(4)Fpa,b;c;z=1Bb,c-b∫01tb-11-tc-b-11-zt-aexp⁡-pt1-tdt,p>0;p=0,arg⁡1-z<π;Rc>Rb>0and(5)Φpb;c;z=1Bb,c-b∫01tb-11-tc-b-1exp⁡zt-pt1-tdt,p>0;p=0;Rc>Rb>0.If we choosep=0, the above definitions given in (1), (2), (3), (4), and (5) reduce to the following form, respectively:(6)Bα,β=∫01tα-11-tβ-1dt,Rα>0,Rβ>0.(7)Fa,b;c;z=∑n=0∞anBb+n,c-bBb,c-bznn!,z<1;Rc>Rb>0.(8)Φb;c;z=∑n=0∞Bb+n,c-bBb,c-bznn!,z<1;Rc>Rb>0.(9)Fa,b;c;z=1Bb,c-b∫01tb-11-tc-b-11-zt-adt,arg⁡1-z<π;Rc>Rb>0.(10)Φb;c;z=1Bb,c-b∫01tb-11-tc-b-1exp⁡ztdt,Rc>Rb>0.Gauss hypergeometric function and confluent hypergeometric function are special cases of the generalized hypergeometric seriespFq(p,q∈N) defined as (see [8, p.73]) and [9, pp. 71-75]:(11)pFqα1,…,αp;β1,…,βq;z=∑n=0∞α1n⋯αpnβ1n⋯βqnznn!=pFqα1,…,αp;β1,…,βq;z,where(ξ)n is the Pochhammer symbol defined (for ξ∈C) by (see[9, p.2 and p.5])(12)ξn≔1n=0ξξ+1…ξ+n-1n∈N(13)=Γξ+nΓξξ∈C∖Z0-,and Z0- denotes the set of nonpositive integers and Γ(ξ) is familiar Gamma function.The Fox-Wright functionpΨq is defined as (see, for details, Srivastava and Karisson [10])(14)pΨqz=pΨqa1,α1,…,ap,αp;b1,β1,…,bq,βq;z=pΨqai,αi1,p;bj,βj1,q;z=∑n=0∞∏i=1pΓai+αin∏j=1qΓbj+βjnznn!,where the coefficientsα1,…,αp, β1,…,βq∈R+ such that(15)1+∑j=1qβj-∑i=1pαi≥0.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:(16)BMCmα,β=∫01tα-11-tβ-1emt1-tdt,where R(α)>0,R(β)>0,m∈C;|m|<M (where M is positive number).Theorem 1.
IfR(α)>0,R(β)>0,m∈C;|m|<M (where M is positive number), then the new extension of the classical beta function in equation (16) is convergent.Proof.
We can write (16) as follows:(17)BMCmα,β=∫01tα-11-tβ-1∑n=0∞mt1-tnn!dt=∑n=0∞mnn!∫01tα+n-11-tβ+n-1dt,
and further, using the definition of classical beta function (6), (17) reduces to(18)MCBmα,β=∑n=0∞mnn!Bα+n,β+n.
In the above equation,MCBm(α,β) is in series form involving B(α+n,β+n) (where n=0,1,2,…) and in each term of the series, B(α+n,β+n) is convergent, since R(α+n)>0 and R(β+n)>0 for R(α)>0 and R(β)>0, which implies that each term of the series (18) exists.
Now we shall prove thatMCBm(α,β) is convergent. m may be greater than or less than 0, so there are two cases as follows.
Case 1. If m>0, then we need to prove that MCBm(α,β) is convergent.
Equation (18) can be written as(19)BMCmα,β=∑n=0∞an,wherean=mnn!Bα+n,β+n.Further,(20)limn→∞⁡anan+1=∞>1.
By ratio test for positive series,MCBm(α,β) is convergent for m>0.
Case 2. If m<0, then we need to prove that the extension of classical beta function Bm(α,β) is convergent.
To prove this case, letm=-p (where p>0); then (18) becomes(21)BMCmα,β=∑n=0∞-pnn!Bα+n,β+n.
Equation (21) can be written as(22)BMCmα,β=-∑n=0∞-1n-1bn,wherebn=pnn!Bα+n,β+n.
The series (22) is an alternating series; therefore (1)
bn>0, ∀p>0,R(α,β)>0(2)
bn-bn+1=pn/n!B(α+n,β+n)1-(α+n)(β+n)/(α+β+2n)(α+β+2n+1)>0⇒bn is decreasing(3)
limn→∞bn=0 if p≤2 (pn/n!→0 as n→∞ only if p≤2 and B(α+n,β+n)→0 as n→∞)
All the conditions of Leibniz’s test for alternating series have been satisfied; thereforeBm(α,β) is convergent for 0<p≤2i.e.-2≤m<0.
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 variablesα,β and parameter m as α,β∈[0,10] and m∈[-2.0335,2.0335]. 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 MCBm(α,β) does not exist at α=β=0 and it is also investigated that MCBm(α,β) does not exist for m<-2.0335 and m>2.0335; MCBm(α,β)→∞ as α,β→0 and MCBm(α,β)→0 as α,β→∞, 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 functionMCBm(α,β).
α β m = - 2.0335 m = - 2.0 m=-1.0 m = 0.0 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 0.00000108Table 2
Numerical values of new extension of classical beta functionMCBm(α,β).
α β m=0.0 m = 1.0 m=2.0 m = 2.0335 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 0.00000176We also check the effect ofm on the new extension of classical beta function. For this purpose, we fix the values of α and β as shown in Figure 1, then we plot the graph which depicts that MCBm(α,β) is an increasing function as the values of m 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 α,β, and m. The value of m does not affect the nature of classical beta function; the main effect of the value of m 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 ofα,β and m=-2:.5:2.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 Tables1 and 2, we find that the interval of convergence of the series is [-2.0335,2.0335], which implies that MCBm(α,β) is convergent for |m|<M, where M is positive number not greater than 2.0335.Note 2.
From the above discussion, it is easy to conclude that the value ofR(m) lies in the interval [-2.0335,2.0335]; i.e., -2.0335≤R(m)≤2.0335.Note 3.
In the sequel of this paper,|m|<M represents the circle of convergence and M is the radius of convergence of (16), where M is not greater than 2.0335.Remark 4.
ForR(α)>0, R(β)>0, m∈C; |m|<M (where M 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:(23)BMCmα,β=∑n=0∞mnn!Γα+nΓβ+nΓα+β+2n=2Ψ1α,1,β,1α+β,2;m.
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]: (24)∫0∞xmexp⁡-βxndx=Γγnβγ,whereγ=m+1nandRβ>0.Theorem 5 (relation between new extension of the classical beta function and the classical beta function).
IfR(α+s)>0, R(β+s)>0, m∈C; |m|<M (where M is positive number not greater than 2.0335), then we have the following relation:(25)∫0∞ms-1MCBmα,βdm=-1sΓsBα+s,β+s.Proof.
Multiplying both sides of (16) by ms-1, then integrating with respect to m from m=0 to m=∞, we have(26)∫0∞ms-1MCBmα,βdm=∫0∞ms-1∫01tα-11-tβ-1exp⁡mt1-tdtdm,
and interchanging the order of integration, (26) reduces to(27)∫0∞ms-1MCBmα,βdm=∫01tα-11-tβ-1∫0∞ms-1exp⁡mt1-tdmdt,
and further using the formula given in (24), after simplification, (27) reduces to(28)∫0∞ms-1MCBmα,βdm=-1sΓs∫01tα+s-11-tβ+s-1dt,and using the definition of classical beta function, we have the required result.Remark 6.
By settings=1, the result in (25) reduces to(29)∫0∞BMCmα,βdm=-Bα+1,β+1,Rα>-1,Rβ>-1which 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 parameterm can be expressed in terms of the function as(30)∂n∂nmBMCmα,β=Bα+n,β+n,Rα+n>0,Rβ+n>0.Theorem 8 (integral representations of the new extension of the classical beta function).
IfR(α)>0, R(β)>0, m∈C; |m|<M (where M is positive number not greater than 2.0335), then we have the following relation:(31)
B
M
C
m
α
,
β
=
2
∫
0
π
/
2
c
o
s
2
α
-
1
θ
s
i
n
2
β
-
1
θ
exp
⁡
m
c
o
s
2
θ
s
i
n
2
θ
d
θ
,
(32)
M
C
B
m
α
,
β
=
∫
0
∞
u
α
-
1
1
+
u
α
+
β
exp
⁡
m
u
1
+
u
2
d
u
,
(33)
B
M
C
m
α
,
β
=
1
2
∫
0
∞
u
α
-
1
+
u
β
-
1
1
+
u
α
+
β
exp
⁡
m
u
1
+
u
2
d
u
,
(34)
B
M
C
m
α
,
β
=
c
-
a
1
-
α
-
β
∫
a
c
u
-
a
α
-
1
c
-
u
β
-
1
exp
⁡
m
u
-
a
c
-
u
c
-
a
2
d
u
,
(35)
M
C
B
m
α
,
β
=
2
1
-
α
-
β
∫
-
1
1
1
+
t
α
-
1
1
-
t
β
-
1
exp
⁡
m
1
-
t
2
4
d
u
,
(36)
B
M
C
m
α
,
β
=
2
1
-
α
-
β
∫
-
∞
∞
exp
⁡
α
-
β
x
+
m
4
c
o
s
h
2
x
d
x
cosh
⁡
x
α
+
β
,
(37)
B
M
C
m
α
,
β
=
2
2
-
α
-
β
∫
0
∞
cosh
⁡
α
-
β
x
exp
⁡
m
4
c
o
s
h
2
x
d
x
cosh
⁡
x
α
+
β
,
(38)
B
M
C
m
α
,
β
=
2
1
-
α
-
β
∫
-
∞
∞
exp
⁡
1
2
α
-
β
x
+
m
2
cosh
⁡
x
d
x
cosh
⁡
x
/
2
α
+
β
,
(39)
B
M
C
m
α
,
β
=
2
2
-
α
-
β
∫
0
∞
cosh
⁡
α
-
β
x
/
2
cosh
⁡
x
/
2
α
+
β
exp
⁡
m
2
cosh
⁡
x
d
x
.Proof.
The result (31) can be easily obtained by setting t=cos2θ in (16); to prove (32) choose t=u/(1+u); (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 t=(u-a)/(c-a), and setting a=-1,c=1 in (34) gives the result in (35) and to prove the result in (36) put u=tanh⁡x in (35). The results in (37), (38), and (39) can be easily obtained from the result (36).Remark 9 (useful inequalities).
IfR(α)>0, R(β)>0, then we have the following inequality(40)BMCmα,β≤1.6626Bα,βfollows from the integral representation (32), since the function exp(mu/(1+u)2) attains its maximum value 1.6626 at u=1 and m=2.0335.


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

Theorem 10 (functional relation).
IfR(α)>0, R(β)>0, m∈C; |m|<M (where M is positive number), then we have the following relation:(41)BMCmα,β+1+MCBmα+1,β=MCBmα,β.Proof.
Using the definition of new extension of beta function, LHS of (41) is equal to(42)∫01tα-11-tβ+tα1-tβ-1emt1-tdt,and after simplification (42) reduced to(43)∫01tα-11-tβ-1emt1-tdt=BMCmα,β.If we choosem=0, we get the usual relation for the beta function from (41).Theorem 11 (symmetry).
IfR(α)>0, R(β)>0, m∈C; |m|<M (where M is positive number), then we have the following relation:(44)MCBmα,β=MCBmβ,α.Proof.
From (18), we have(45)MCBmα,β=∑n=0∞mnn!Bα+n,β+n,
and since usual beta function is symmetric, i.e.,B(α,β)=B(β,α), using this property in the right-hand side of (45), then we have(46)MCBmα,β=∑n=0∞mnn!Bβ+n,α+n=MCBmβ,α.Theorem 12 (first summation relation).
IfR(α)>0, R(1-β)>0, m∈C; |m|<M (where M is positive real number), then we have the following relation:(47)MCBmα,1-β=∑n=0∞βnn!BMCmα+n,1.Proof.
The LHS of (47) can be written as(48)MCBmα,1-β=∫01tα-11-t-βexp⁡mt1-tdt,and using the binomial series expansion (1-t)-β=∑n=0∞(β)n/n!tn in (48) and then interchanging the order of summation and integration, the above result (48) reduced to the following form:(49)BMCmα,1-β=∑n=0∞βnn!∫01tα+n-1exp⁡mt1-tdt⟹(50)BMCmα,1-β=∑n=0∞βnn!BMCmα+n,1.Theorem 13 (second summation relation).
IfR(α)>0,R(β)>0,m∈C;|m|<M (where M is positive number), then we have the following relation:(51)MCBmα,β=∑n=0∞BMCmα+n,β+1.Proof.
The LHS of (47) can be written as(52)BMCmα,β=∫01tα-11-tβ-1exp⁡mt1-tdt,and using the binomial series expansion (1-t)β-1=(1-t)α∑n=0∞tn(|t|<1) and interchanging the order of summation and integration, (52) reduces to(53)MCBmα,β=∑n=0∞∫01tα+n-11-tβexp⁡mt1-tdt⟹(54)BMCmα,β=∑n=0∞BMCmα+n,β+1.Theorem 14 (separation).
IfR(α)>0, R(β)>0, m∈C; |m|<M (where M is positive number), then MCBm(α,β) can be separated into real and imaginary parts of m as follows:(55)BMCrcos⁡θα,β=∫01tα-11-tβ-1exp⁡rt1-tcos⁡θdt,(56)BMCrsin⁡θα,β=∫01tα-11-tβ-1exp⁡rt1-tsin⁡θdt,where r=x2+y2=|m|<M and θ=tan-1y/x.Proof.
Sincem∈C, so let m=x+iy, where x,y∈R and also let x+iy=rcos⁡θ+irsin⁡θ⇒r=x2+y2 and θ=tan-1y/x; then from (16), we have(57)MCBx+iyα,β=∫01tα-11-tβ-1exp⁡reiθt1-tdt,and after simplification (57) reduces to(58)BMCrcosθ+irsinθα,β=∫01tα-11-tβ-1exp⁡rt1-tcos⁡θdt+i∫01tα-11-tβ-1exp⁡rt1-tsin⁡θdt.Equating the real and imaginary parts of m 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(59)ft=tp-11-tq-1exp⁡mt1-tMCBmp,q,0<t<10,otherwise.A random variableX with probability density function (pdf) given in (59) will be said to have the extended beta distribution with parameters p and q, -∞<p,q<∞, and |m|<M where M is positive number. If ν is any real number [12], then(60)EXν=MCBmp+ν,qMCBmp,q.In particular, forν=1,(61)μ=EX=MCBmp+1,qMCBmp,qrepresents the mean of the distribution and(62)σ2=EX2-EX2=MCBmp,qMCBmp+2,q-MCBm2p+1,qMCBm2p,qis a variance of the distribution.The moment of generating function of the distribution is(63)Mt=∑n=0∞tnEXnn!=1MCBmp,q∑n=0∞tnn!MCBmp+n,q.The commutative distribution of (59) can be written as(64)Fx=MCBm,xp,qMCBmp,q,where(65)MCBm,xp,q=∫0xtp-11-tq-1exp⁡mt1-tdt,m<M,-∞<p,q<∞is the new extended incomplete beta function. Form=0, we must have p,q>0 in (65) for convergence, and MCB0,x(p,q)=Bx(p,q), where Bx(p,q) is the incomplete beta function [11] defined as(66)Bxp,q=BMC0,xp,q=xppF21p,1-q;p=1;x.It is to be noted that the problem of expressingMCBm,x(p,q) 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:(67)MCFmα,β;γ;z≔∑n=0∞αnBMCmβ+n,γ-βBβ,γ-βznn!,z<1;Rγ>Rβ>0;m<MwhereMisapositiverealnumber.(68)MCΦmβ;γ;z≔∑n=0∞BMCmβ+n,γ-βBβ,γ-βznn!,z<1;Rγ>Rβ>0;m<MwhereMisapositiverealnumber.We callMCFm(a,b;c;z) new extension of Gauss hypergeometric function and MCΦm(b;c;z) new extension of confluent hypergeometric function.Note 15.
If we choosem=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.


### 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 Table3 and Table 4 for m=-2:1:2. Further their graphs are plotted in Figure 3 and Figure 4, respectively. When m=0 we have the values of Gauss hypergeoemtric function and confluent hypergeoemtric function.Table 3
Numerical values of new extension of Gauss hypergeometric functionMCFm(2,1;3;x).
x m = - 2 m = - 1 m = 0 m = 1 m = 2 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 10.7195Table 4
Numerical values of new extension of confluent hypergeometric functionMC1F1,m(1;3;x).
x m = - 2 m = - 1 m = 0 m = 1 m = 2 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 2.0649Figure 3
Graph of the new extension of Gauss hypergeometric function forα=2,β=1,γ=3, and z=0:0.1:1.Figure 4
Graph of the new extension of confluent hypergeometric function forβ=1,γ=3, and z=0:0.1:1.


### 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 functionMCFm(α,β;γ;z), we have the following integral representations:(69)MCFmα,β;γ;z≔1Bβ,γ-β∫01tβ-11-tγ-β-11-zt-αemt1-tdt,(70)MCFmα,β;γ;z≔1Bβ,γ-β∫01uβ-11+uα-γ-11+1-zu-αemu/1+u2du,(71)MCFmα,β;γ;z≔2Bβ,γ-β∫0π/2sin2β-1θcos2γ-2β-1θ1-zsin2θ-αemsin2θcos2θdθ.Proof.
Equation (67) can be written as(72)MCFmα,β;γ;z≔∑n=0∞αnBMCmβ+n,γ-βBβ,γ-βznn!=1Bβ,γ-β∑n=0∞αn∫01tβ+n-11-tγ-β-1emt1-tznn!dt=1Bβ,γ-β∫01tβ-11-tγ-β-1emt1-t∑n=0∞αntznn!dt=1Bβ,γ-β∫01tβ-11-tγ-β-11-tz-αemt1-t.
Settingu=t/1-t in (69), we have the required result (70).
Again if we chooset=sin2θ, we obtain the result (71).Remark 17.
Choosingz=1 in (69), we have the following relation between new extensions of Gauss hypergeometric function:(73)MCFmα,β;γ;1≔1Bβ,γ-β∫01tβ-11-tγ-α-β-1emt1-tdt=BMCmβ,γ-α-βBβ,γ-β.Theorem 18.
For the new extension of confluent hypergeometric functionMCΦm(α;β;x), we have the following integral representations:(74)MCΦmβ;γ;z≔1Bβ,γ-β∫01tβ-11-tγ-β-1eztemt1-tdt,(75)MCΦmβ;γ;z≔1Bβ,γ-β∫01uβ-11+u-γezu/1+uemu/1+u2du,(76)MCΦmβ;γ;z≔2Bβ,γ-β∫0π/2sin2β-1θcos2γ-2β-1θezsin2θemsin2θcos2θdθ.Proof.
The proof of this theorem would run parallel to those of Theorem16, 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 formulasB(β,γ-β)=γ/βB(β+1,γ-β) and (α)n+1=α(α+1)n, 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 z; we have the following.Theorem 19.
Ifα,β,γ∈C; R(γ)>R(β)>0 and |m|<M (where M is positive real number), then we have the following result:(77)dndznFMCmα,β;γ;z=βnαnγnFMCmα+n,β+n;γ+n;z.Proof.
Taking the derivative ofMCFm(α,β;γ;z) with respect to z, we have(78)ddzFMCmα,β;γ;z=∑n=1∞αnBMCmβ+n,γ-βBβ,γ-βzn-1n-1!,and replacing n→n+1, (78) reduces to(79)ddzFMCmα,β;γ;z=βαγ∑n=0∞α+1nBMCmβ+n+1,γ-βBβ+1,γ-βzn-1n-1!=βαγFMCmα+1,β+1;γ+1;z,and with recursive application of this procedure in (79), we have the desired result (77).Theorem 20.
Ifβ,γ∈C; R(γ)>R(β)>0 and |m|<M (where M is positive real number), then we have the following result:(80)dndznΦMCmβ;γ;z=βnαnγnΦMCmβ+n;γ+n;z.Proof.
The proof of Theorem20 is as that of Theorem 19, so it can be omitted here.


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

Theorem 21.
Ifα,β,γ∈C;k∈R; R(γ)>R(β)>0 and |m|<M (where M is positive real number), then the following generating functions hold:(81)∑n=0∞αnFMCmα+n,β;γ;ztnn!=1-t-αFMCmα,β;γ;;z1-t,t<1.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(82)S=∑n=0∞αn∑k=0∞α+nkBMCmβ+k,γ-βBβ,γ-βzkk!tnn!.
Upon reversal of the order of summation and then using the identity(α)n(α+n)k=(α)k(α+k)n, (82) reduces to(83)S=∑k=0∞αkBMCmβ+k,γ-βBβ,γ-β∑n=0∞a+kntnn!zkk!,
and further using the definition of binomial(1-t)-α-k=∑n=0∞(α+k)ntn/n!, (|t|<1) in (83), we have(84)S=∑k=0∞αkBMCmβ+k,γ-βBβ,γ-β1-t-α-kzkk!,
and interpreting the above equation with the view of (67), we have the desired result (81).Theorem 22.
Ifα,β,γ∈C;k∈R; R(γ)>R(β)>0 and |m|<M (where M is positive real number), then the following generating functions hold:(85)∑k=0∞α+k-1kFMCm,kα+k,β;γ;ztk=1-t-αFMCm,kα,β;γ;;z1-t,t<1.Proof.
For convenience, let the left-hand side of (85) be denoted by J. Applying the series of (67) to J, we get(86)J=∑k=0∞α+k-1k∑n=0∞α+knBMCmβ+n,γ-βBβ,γ-βznn!tk.
By changing the order of summation in (86) and using the known identity ([13, p.5]), namely,(87)αn=Γα+1n!Γα-n+1,n∈No;α∈C,then, after little simplification, we obtain(88)J=∑n=0∞αnBMCmβ+n,γ-βBβ,γ-β∑k=0∞α+n+k-1ktkznn!.Further, upon using the generalized binomial expansion, we find that the inner sum in (88) yields(89)∑k=0∞α+n+k-1ktk=1-t-α+n,t<0.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 sequenceζk(α,ρ) defined by(90)ζkα,ρz=ζkα,ρα,β;γ;z=FMCm,kΔρ;α+k,β;γ;zwhereα,β,γ∈C;k∈R;ρ∈N; R(γ)>R(β)>0 and |m|<M (where M is positive real number); for convenience, Δ(ρ;α) abbreviates the array of ρ parameters(91)αρ,α+1ρ,α+2ρ,…,α+ρ-1ρ,ρ∈N.Now, we prove the following result, which provides the generating functions for the Gauss hypergeometric function defined above.Theorem 24.
For eachρ∈N, the following generating functions hold true:(92)∑k=0∞α+l+k-1kζkα,ρztk=1-t-α-lζlα,ρz1-tρ,t<1where α,β,γ∈C;k∈R;ρ∈N;l∈N0R(γ)>R(β)>0 and |m|<M (where M 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 I) leads to(93)I=∑n=0∞α+lρn⋯α+l+ρ-1ρnBMCmβ+n,γ-βBβ,γ-β∑k=0∞α+l+nρ+k-1ktkznn!.Now taking (90) into account, one can easily arrive at the desired result (92).Remark 25.
It may be noted that if we setρ=1 and replace α by α-l 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:(94)MCDzμ,mfz≔1Γ-μ∫0zftz-t-μ-1emtz-t/z2dtand forn-1<R(μ)<n, n=1,2,..(95)MCDzμ,mfz≔dndznDMCzμ-nfz=dndzn1Γ-μ+n∫0zftz-t-μ+n-1emtz-t/z2dt,where the path of integration is a line from 0 toz in complex t-plane. For the case m=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.
Letξ,μ∈C;R(ξ)>-1,R(μ)<0 and |m|<M (where M is positive real number); then we have(96)MCDzμ,mzξ=BMCmξ+1,-μΓ-μzξ-μ.Proof.
Employing the definition given in (94) in the left-hand side of (96), we have(97)MCDzμ,mzξ≔1Γ-μ∫0ztξz-t-μ-1emtz-t/z2dt.Choosing t=zu, (97) reduces to(98)MCDzμ,mzξ=1Γ-μ∫01zuξz-μ-11-u-μ-1emu1-uzdu=zξ-μΓ-μ∫01uξ1-u-μ-1emu1-udu=BMCmξ+1,-μΓ-μzξ-μ.Lemma 27.
Letξ,α,μ∈C;R(ξ)>0,R(α)>0,R(μ)<0, |z|<1 and |m|<M (where M is positive real number), then we have(99)MCDzξ-μ,mzξ-11-z-α≔=ΓξΓμzμ-1FMCmα,ξ;μ;z.Proof.
Employing the definition given in (94) in the left-hand side of (99), we have(100)MCDzξ-μ,mzξ-11-z-α≔1Γμ-ξ∫0ztξ-11-t-αemtz-t/z2z-tμ-ξ-1dt.
Choosingt=zu, (100) reduces to(101)MCDzξ-μ,mzξ-11-z-α≔zμ-1Γμ-ξ∫01tξ-11-uz-α1-uμ-ξ-1emu1-udu=zμ-1Γμ-ξ∫01tξ-11-uz-α1-uμ-ξ-1emu1-udu,and further employing the result in (69), after simplification, we have the required result (101).Theorem 28.
Letf(z) be an analytic function in the disc |z|<ρ and it has the power series f(z)=∑n=0∞anzn; then we have the following result:(102)MCDzμ,mzξ-1fz=DMCzμ,mzξ-1∑n=0∞anzn=zξ-μ-1Γ-μ∑n=0∞anBMCmξ+n,-μzn,where ξ,μ∈C;R(ξ)>0,R(μ)<0 and |m|<M (where M is positive real number).Proof.
(103) M C D z μ , m z ξ - 1 f z = D   M C z μ , m z ξ - 1 ∑ n = 0 ∞ a n z n = 1 Γ - μ ∫ 0 z t ξ - 1 ∑ n = 0 ∞ a n t n z - t - μ - 1 e m t z - t / z 2 d t = 1 Γ - μ ∫ 0 1 z ξ ξ - 1 z - μ - 1 1 - ξ - μ - 1 e m ξ 1 - ξ ∑ n = 0 ∞ a n z ξ n z  d ξ .The series ∑n=0∞an(zξ)n is uniformly convergent in the disc |z|<ρ for 0≤ξ≤1 and the integral ∫01|ξξ+n-1(1-ξ)-μ-1emξ(1-ξ)|dξ is convergent provided that R(ξ)>0,R(μ)<0 and |m|<M (where M is positive real number), therefore we can interchange the order of integration and summation; after simplification, above equation (103) reduces to (104)MCDzμ,mzξ-1fz=zξ-μ-1Γ-μ∑n=0∞anzn∫01ξξ+n-11-ξ-μ-1emξ1-ξdξ,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).
Letα,β,ξ,∈C;R(ξ)>0,R(β)>R(α)>0 and |m|<M (where M is positive real number), such that |x|<min{1,|1-t|}; then we have the following result:(105)∑n=0∞ξnn!FMCmξ+n,α;β;xtn=1-t-ξFMCmξ,α;β;x1-t.Proof.
Let us consider the elementary identity(106)1-x-t-ξ=1-t-ξ1-x1-t-ξ.Expanding the left-hand side of (106) for |t|<|1-x|, we have(107)∑n=0∞ξnn!1-x-ξt1-xn=1-t-ξ1-x1-t-ξ.Further, multiplying both sides of (107) and then applying the new extension of fractional derivative operator Dxα-β,m on both sides, we have(108)MCDxα-β,m∑n=0∞ξnn!1-x-ξt1-xnxα-1=1-t-ξDMCxα-β,mxα-11-x1-t-ξ.
Interchanging the order, which is valid forR(α)>0 and |t|<|1-x|, we have(109)∑n=0∞ξnn!DMCxα-β,mxα-11-x-ξ-ntn=1-t-ξDMCxα-β,mxα-11-x1-t-ξ.Now applying the result established in (99) in (109), after simplification, we have the required result (105).


### 6.7. Transformation Formulas

Theorem 30.
Letα,β,ξ,∈C;R(ξ)>0,R(β)>R(α)>0 and |m|<M (where M is positive real number), then we have the following transformation formula:(110)MCFmα,β;γ;z=1-z-αFMCmα,γ-β;β;zz-1,arg⁡1-z<π,(111)MCFmα,β;γ;1-1z=zαFMCmα,γ-β;β;1-z,arg⁡z<π,(112)MCFmα,β;γ;z1+z=1+zαFMCmα,γ-β;β;z,arg⁡1+z<π.Proof.
To prove the theorem, we consider the following identity:(113)1-z1-t-α=1-z-α1+z1-zt-α.
Replacingt into 1-t, then using the result (113) in (69), after simplification, we have(114)MCFmα,β;γ;z≔1-z-αBβ,γ-β∫01tγ-β-11-tβ-11+z1-zt-αemt1-tdt,
further interpreted with the view of (69), we obtain the desired result (110).


## 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(59)ft=tp-11-tq-1exp⁡mt1-tMCBmp,q,0<t<10,otherwise.A random variableX with probability density function (pdf) given in (59) will be said to have the extended beta distribution with parameters p and q, -∞<p,q<∞, and |m|<M where M is positive number. If ν is any real number [12], then(60)EXν=MCBmp+ν,qMCBmp,q.In particular, forν=1,(61)μ=EX=MCBmp+1,qMCBmp,qrepresents the mean of the distribution and(62)σ2=EX2-EX2=MCBmp,qMCBmp+2,q-MCBm2p+1,qMCBm2p,qis a variance of the distribution.The moment of generating function of the distribution is(63)Mt=∑n=0∞tnEXnn!=1MCBmp,q∑n=0∞tnn!MCBmp+n,q.The commutative distribution of (59) can be written as(64)Fx=MCBm,xp,qMCBmp,q,where(65)MCBm,xp,q=∫0xtp-11-tq-1exp⁡mt1-tdt,m<M,-∞<p,q<∞is the new extended incomplete beta function. Form=0, we must have p,q>0 in (65) for convergence, and MCB0,x(p,q)=Bx(p,q), where Bx(p,q) is the incomplete beta function [11] defined as(66)Bxp,q=BMC0,xp,q=xppF21p,1-q;p=1;x.It is to be noted that the problem of expressingMCBm,x(p,q) 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:(67)MCFmα,β;γ;z≔∑n=0∞αnBMCmβ+n,γ-βBβ,γ-βznn!,z<1;Rγ>Rβ>0;m<MwhereMisapositiverealnumber.(68)MCΦmβ;γ;z≔∑n=0∞BMCmβ+n,γ-βBβ,γ-βznn!,z<1;Rγ>Rβ>0;m<MwhereMisapositiverealnumber.We callMCFm(a,b;c;z) new extension of Gauss hypergeometric function and MCΦm(b;c;z) new extension of confluent hypergeometric function.Note 15.
If we choosem=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.


## 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 Table3 and Table 4 for m=-2:1:2. Further their graphs are plotted in Figure 3 and Figure 4, respectively. When m=0 we have the values of Gauss hypergeoemtric function and confluent hypergeoemtric function.Table 3
Numerical values of new extension of Gauss hypergeometric functionMCFm(2,1;3;x).
x m = - 2 m = - 1 m = 0 m = 1 m = 2 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 10.7195Table 4
Numerical values of new extension of confluent hypergeometric functionMC1F1,m(1;3;x).
x m = - 2 m = - 1 m = 0 m = 1 m = 2 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 2.0649Figure 3
Graph of the new extension of Gauss hypergeometric function forα=2,β=1,γ=3, and z=0:0.1:1.Figure 4
Graph of the new extension of confluent hypergeometric function forβ=1,γ=3, and z=0:0.1:1.


## 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 functionMCFm(α,β;γ;z), we have the following integral representations:(69)MCFmα,β;γ;z≔1Bβ,γ-β∫01tβ-11-tγ-β-11-zt-αemt1-tdt,(70)MCFmα,β;γ;z≔1Bβ,γ-β∫01uβ-11+uα-γ-11+1-zu-αemu/1+u2du,(71)MCFmα,β;γ;z≔2Bβ,γ-β∫0π/2sin2β-1θcos2γ-2β-1θ1-zsin2θ-αemsin2θcos2θdθ.Proof.
Equation (67) can be written as(72)MCFmα,β;γ;z≔∑n=0∞αnBMCmβ+n,γ-βBβ,γ-βznn!=1Bβ,γ-β∑n=0∞αn∫01tβ+n-11-tγ-β-1emt1-tznn!dt=1Bβ,γ-β∫01tβ-11-tγ-β-1emt1-t∑n=0∞αntznn!dt=1Bβ,γ-β∫01tβ-11-tγ-β-11-tz-αemt1-t.
Settingu=t/1-t in (69), we have the required result (70).
Again if we chooset=sin2θ, we obtain the result (71).Remark 17.
Choosingz=1 in (69), we have the following relation between new extensions of Gauss hypergeometric function:(73)MCFmα,β;γ;1≔1Bβ,γ-β∫01tβ-11-tγ-α-β-1emt1-tdt=BMCmβ,γ-α-βBβ,γ-β.Theorem 18.
For the new extension of confluent hypergeometric functionMCΦm(α;β;x), we have the following integral representations:(74)MCΦmβ;γ;z≔1Bβ,γ-β∫01tβ-11-tγ-β-1eztemt1-tdt,(75)MCΦmβ;γ;z≔1Bβ,γ-β∫01uβ-11+u-γezu/1+uemu/1+u2du,(76)MCΦmβ;γ;z≔2Bβ,γ-β∫0π/2sin2β-1θcos2γ-2β-1θezsin2θemsin2θcos2θdθ.Proof.
The proof of this theorem would run parallel to those of Theorem16, 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 formulasB(β,γ-β)=γ/βB(β+1,γ-β) and (α)n+1=α(α+1)n, 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 z; we have the following.Theorem 19.
Ifα,β,γ∈C; R(γ)>R(β)>0 and |m|<M (where M is positive real number), then we have the following result:(77)dndznFMCmα,β;γ;z=βnαnγnFMCmα+n,β+n;γ+n;z.Proof.
Taking the derivative ofMCFm(α,β;γ;z) with respect to z, we have(78)ddzFMCmα,β;γ;z=∑n=1∞αnBMCmβ+n,γ-βBβ,γ-βzn-1n-1!,and replacing n→n+1, (78) reduces to(79)ddzFMCmα,β;γ;z=βαγ∑n=0∞α+1nBMCmβ+n+1,γ-βBβ+1,γ-βzn-1n-1!=βαγFMCmα+1,β+1;γ+1;z,and with recursive application of this procedure in (79), we have the desired result (77).Theorem 20.
Ifβ,γ∈C; R(γ)>R(β)>0 and |m|<M (where M is positive real number), then we have the following result:(80)dndznΦMCmβ;γ;z=βnαnγnΦMCmβ+n;γ+n;z.Proof.
The proof of Theorem20 is as that of Theorem 19, so it can be omitted here.


### 6.5.1. Generating Relations Associated with Hypergeometric Functions

Theorem 21.
Ifα,β,γ∈C;k∈R; R(γ)>R(β)>0 and |m|<M (where M is positive real number), then the following generating functions hold:(81)∑n=0∞αnFMCmα+n,β;γ;ztnn!=1-t-αFMCmα,β;γ;;z1-t,t<1.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(82)S=∑n=0∞αn∑k=0∞α+nkBMCmβ+k,γ-βBβ,γ-βzkk!tnn!.
Upon reversal of the order of summation and then using the identity(α)n(α+n)k=(α)k(α+k)n, (82) reduces to(83)S=∑k=0∞αkBMCmβ+k,γ-βBβ,γ-β∑n=0∞a+kntnn!zkk!,
and further using the definition of binomial(1-t)-α-k=∑n=0∞(α+k)ntn/n!, (|t|<1) in (83), we have(84)S=∑k=0∞αkBMCmβ+k,γ-βBβ,γ-β1-t-α-kzkk!,
and interpreting the above equation with the view of (67), we have the desired result (81).Theorem 22.
Ifα,β,γ∈C;k∈R; R(γ)>R(β)>0 and |m|<M (where M is positive real number), then the following generating functions hold:(85)∑k=0∞α+k-1kFMCm,kα+k,β;γ;ztk=1-t-αFMCm,kα,β;γ;;z1-t,t<1.Proof.
For convenience, let the left-hand side of (85) be denoted by J. Applying the series of (67) to J, we get(86)J=∑k=0∞α+k-1k∑n=0∞α+knBMCmβ+n,γ-βBβ,γ-βznn!tk.
By changing the order of summation in (86) and using the known identity ([13, p.5]), namely,(87)αn=Γα+1n!Γα-n+1,n∈No;α∈C,then, after little simplification, we obtain(88)J=∑n=0∞αnBMCmβ+n,γ-βBβ,γ-β∑k=0∞α+n+k-1ktkznn!.Further, upon using the generalized binomial expansion, we find that the inner sum in (88) yields(89)∑k=0∞α+n+k-1ktk=1-t-α+n,t<0.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 sequenceζk(α,ρ) defined by(90)ζkα,ρz=ζkα,ρα,β;γ;z=FMCm,kΔρ;α+k,β;γ;zwhereα,β,γ∈C;k∈R;ρ∈N; R(γ)>R(β)>0 and |m|<M (where M is positive real number); for convenience, Δ(ρ;α) abbreviates the array of ρ parameters(91)αρ,α+1ρ,α+2ρ,…,α+ρ-1ρ,ρ∈N.Now, we prove the following result, which provides the generating functions for the Gauss hypergeometric function defined above.Theorem 24.
For eachρ∈N, the following generating functions hold true:(92)∑k=0∞α+l+k-1kζkα,ρztk=1-t-α-lζlα,ρz1-tρ,t<1where α,β,γ∈C;k∈R;ρ∈N;l∈N0R(γ)>R(β)>0 and |m|<M (where M 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 I) leads to(93)I=∑n=0∞α+lρn⋯α+l+ρ-1ρnBMCmβ+n,γ-βBβ,γ-β∑k=0∞α+l+nρ+k-1ktkznn!.Now taking (90) into account, one can easily arrive at the desired result (92).Remark 25.
It may be noted that if we setρ=1 and replace α by α-l in (92), we are easily led to the result (85).


## 6.5.1. Generating Relations Associated with Hypergeometric Functions

Theorem 21.
Ifα,β,γ∈C;k∈R; R(γ)>R(β)>0 and |m|<M (where M is positive real number), then the following generating functions hold:(81)∑n=0∞αnFMCmα+n,β;γ;ztnn!=1-t-αFMCmα,β;γ;;z1-t,t<1.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(82)S=∑n=0∞αn∑k=0∞α+nkBMCmβ+k,γ-βBβ,γ-βzkk!tnn!.
Upon reversal of the order of summation and then using the identity(α)n(α+n)k=(α)k(α+k)n, (82) reduces to(83)S=∑k=0∞αkBMCmβ+k,γ-βBβ,γ-β∑n=0∞a+kntnn!zkk!,
and further using the definition of binomial(1-t)-α-k=∑n=0∞(α+k)ntn/n!, (|t|<1) in (83), we have(84)S=∑k=0∞αkBMCmβ+k,γ-βBβ,γ-β1-t-α-kzkk!,
and interpreting the above equation with the view of (67), we have the desired result (81).Theorem 22.
Ifα,β,γ∈C;k∈R; R(γ)>R(β)>0 and |m|<M (where M is positive real number), then the following generating functions hold:(85)∑k=0∞α+k-1kFMCm,kα+k,β;γ;ztk=1-t-αFMCm,kα,β;γ;;z1-t,t<1.Proof.
For convenience, let the left-hand side of (85) be denoted by J. Applying the series of (67) to J, we get(86)J=∑k=0∞α+k-1k∑n=0∞α+knBMCmβ+n,γ-βBβ,γ-βznn!tk.
By changing the order of summation in (86) and using the known identity ([13, p.5]), namely,(87)αn=Γα+1n!Γα-n+1,n∈No;α∈C,then, after little simplification, we obtain(88)J=∑n=0∞αnBMCmβ+n,γ-βBβ,γ-β∑k=0∞α+n+k-1ktkznn!.Further, upon using the generalized binomial expansion, we find that the inner sum in (88) yields(89)∑k=0∞α+n+k-1ktk=1-t-α+n,t<0.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 sequenceζk(α,ρ) defined by(90)ζkα,ρz=ζkα,ρα,β;γ;z=FMCm,kΔρ;α+k,β;γ;zwhereα,β,γ∈C;k∈R;ρ∈N; R(γ)>R(β)>0 and |m|<M (where M is positive real number); for convenience, Δ(ρ;α) abbreviates the array of ρ parameters(91)αρ,α+1ρ,α+2ρ,…,α+ρ-1ρ,ρ∈N.Now, we prove the following result, which provides the generating functions for the Gauss hypergeometric function defined above.Theorem 24.
For eachρ∈N, the following generating functions hold true:(92)∑k=0∞α+l+k-1kζkα,ρztk=1-t-α-lζlα,ρz1-tρ,t<1where α,β,γ∈C;k∈R;ρ∈N;l∈N0R(γ)>R(β)>0 and |m|<M (where M 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 I) leads to(93)I=∑n=0∞α+lρn⋯α+l+ρ-1ρnBMCmβ+n,γ-βBβ,γ-β∑k=0∞α+l+nρ+k-1ktkznn!.Now taking (90) into account, one can easily arrive at the desired result (92).Remark 25.
It may be noted that if we setρ=1 and replace α by α-l 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:(94)MCDzμ,mfz≔1Γ-μ∫0zftz-t-μ-1emtz-t/z2dtand forn-1<R(μ)<n, n=1,2,..(95)MCDzμ,mfz≔dndznDMCzμ-nfz=dndzn1Γ-μ+n∫0zftz-t-μ+n-1emtz-t/z2dt,where the path of integration is a line from 0 toz in complex t-plane. For the case m=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.
Letξ,μ∈C;R(ξ)>-1,R(μ)<0 and |m|<M (where M is positive real number); then we have(96)MCDzμ,mzξ=BMCmξ+1,-μΓ-μzξ-μ.Proof.
Employing the definition given in (94) in the left-hand side of (96), we have(97)MCDzμ,mzξ≔1Γ-μ∫0ztξz-t-μ-1emtz-t/z2dt.Choosing t=zu, (97) reduces to(98)MCDzμ,mzξ=1Γ-μ∫01zuξz-μ-11-u-μ-1emu1-uzdu=zξ-μΓ-μ∫01uξ1-u-μ-1emu1-udu=BMCmξ+1,-μΓ-μzξ-μ.Lemma 27.
Letξ,α,μ∈C;R(ξ)>0,R(α)>0,R(μ)<0, |z|<1 and |m|<M (where M is positive real number), then we have(99)MCDzξ-μ,mzξ-11-z-α≔=ΓξΓμzμ-1FMCmα,ξ;μ;z.Proof.
Employing the definition given in (94) in the left-hand side of (99), we have(100)MCDzξ-μ,mzξ-11-z-α≔1Γμ-ξ∫0ztξ-11-t-αemtz-t/z2z-tμ-ξ-1dt.
Choosingt=zu, (100) reduces to(101)MCDzξ-μ,mzξ-11-z-α≔zμ-1Γμ-ξ∫01tξ-11-uz-α1-uμ-ξ-1emu1-udu=zμ-1Γμ-ξ∫01tξ-11-uz-α1-uμ-ξ-1emu1-udu,and further employing the result in (69), after simplification, we have the required result (101).Theorem 28.
Letf(z) be an analytic function in the disc |z|<ρ and it has the power series f(z)=∑n=0∞anzn; then we have the following result:(102)MCDzμ,mzξ-1fz=DMCzμ,mzξ-1∑n=0∞anzn=zξ-μ-1Γ-μ∑n=0∞anBMCmξ+n,-μzn,where ξ,μ∈C;R(ξ)>0,R(μ)<0 and |m|<M (where M is positive real number).Proof.
(103) M C D z μ , m z ξ - 1 f z = D   M C z μ , m z ξ - 1 ∑ n = 0 ∞ a n z n = 1 Γ - μ ∫ 0 z t ξ - 1 ∑ n = 0 ∞ a n t n z - t - μ - 1 e m t z - t / z 2 d t = 1 Γ - μ ∫ 0 1 z ξ ξ - 1 z - μ - 1 1 - ξ - μ - 1 e m ξ 1 - ξ ∑ n = 0 ∞ a n z ξ n z  d ξ .The series ∑n=0∞an(zξ)n is uniformly convergent in the disc |z|<ρ for 0≤ξ≤1 and the integral ∫01|ξξ+n-1(1-ξ)-μ-1emξ(1-ξ)|dξ is convergent provided that R(ξ)>0,R(μ)<0 and |m|<M (where M is positive real number), therefore we can interchange the order of integration and summation; after simplification, above equation (103) reduces to (104)MCDzμ,mzξ-1fz=zξ-μ-1Γ-μ∑n=0∞anzn∫01ξξ+n-11-ξ-μ-1emξ1-ξdξ,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).
Letα,β,ξ,∈C;R(ξ)>0,R(β)>R(α)>0 and |m|<M (where M is positive real number), such that |x|<min{1,|1-t|}; then we have the following result:(105)∑n=0∞ξnn!FMCmξ+n,α;β;xtn=1-t-ξFMCmξ,α;β;x1-t.Proof.
Let us consider the elementary identity(106)1-x-t-ξ=1-t-ξ1-x1-t-ξ.Expanding the left-hand side of (106) for |t|<|1-x|, we have(107)∑n=0∞ξnn!1-x-ξt1-xn=1-t-ξ1-x1-t-ξ.Further, multiplying both sides of (107) and then applying the new extension of fractional derivative operator Dxα-β,m on both sides, we have(108)MCDxα-β,m∑n=0∞ξnn!1-x-ξt1-xnxα-1=1-t-ξDMCxα-β,mxα-11-x1-t-ξ.
Interchanging the order, which is valid forR(α)>0 and |t|<|1-x|, we have(109)∑n=0∞ξnn!DMCxα-β,mxα-11-x-ξ-ntn=1-t-ξDMCxα-β,mxα-11-x1-t-ξ.Now applying the result established in (99) in (109), after simplification, we have the required result (105).


## 6.7. Transformation Formulas

Theorem 30.
Letα,β,ξ,∈C;R(ξ)>0,R(β)>R(α)>0 and |m|<M (where M is positive real number), then we have the following transformation formula:(110)MCFmα,β;γ;z=1-z-αFMCmα,γ-β;β;zz-1,arg⁡1-z<π,(111)MCFmα,β;γ;1-1z=zαFMCmα,γ-β;β;1-z,arg⁡z<π,(112)MCFmα,β;γ;z1+z=1+zαFMCmα,γ-β;β;z,arg⁡1+z<π.Proof.
To prove the theorem, we consider the following identity:(113)1-z1-t-α=1-z-α1+z1-zt-α.
Replacingt into 1-t, then using the result (113) in (69), after simplification, we have(114)MCFmα,β;γ;z≔1-z-αBβ,γ-β∫01tγ-β-11-tβ-11+z1-zt-αemt1-tdt,
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]).
Letf(z)≔∑n=0∞anzn and g(z)≔∑n=0∞bnzn be two power series whose radii of convergence are given by Rf and Rg, respectively. Then their Hadamard product is power series defined by(115)f∗gz≔∑n=0∞anbnzn,whose radius of convergence R satisfies Rf.Rg≤R.In particular, let us consider the functionpFp+r(α,β;ρ,ξ)[z;b]. Its decomposition is illustrative. That is(116)pFp+rα,β;ρ,ξx1,…,xpy1,…,yp+r;z;b=1Fr1y1,…,yr;z;b∗pFpα,β;ρ,ξx1,…,xpy1+r,…,yp+r;z;b.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.Forx>0,ξ,ζ,η∈C and R(ξ)>0, we have(117)I0,xξ,ζ,ηftx=x-ξ-ζΓξ∫0xx-tξ-12F1ξ+ζ,-η;ξ;1-txftdtand(118)Jx,∞ξ,ζ,ηftx=1Γξ∫x∞t-xξ-1t-ξ-ζ2F1ξ+ζ,-η;ξ;1-xtftdt,where the function2F1(.) is a special case of the generalized hypergeometric function, the Gauss hypergeometric function.The operatorI0,xξ,ζ,η(.) contains the Riemann-Liouville R0,xξ(.) fractional integral operators by means of the following relationships:(119)R0,xξftx=I0,xξ,-ξ,ηftx=1Γξ∫0xx-tξ-1ftdtand(120)Wx,∞ξftx=Jx,∞ξ,-ξ,ηftx=1Γξ∫x∞t-xξ-1ftdt.It is noted that the operator (118) unifies the Erde^lyi-Kober fractional integral operators as follows:(121)E0,xξ,ηftx=I0,xξ,0,ηftx=x-ξ-ηΓξ∫0xx-tξ-1tηftdtand(122)Kx,∞ξ,ηftx=Jx,∞ξ,0,ηftx=xηΓξ∫x∞t-xξ-1t-ξ-ηftdt.The following lemmas proved in Kilbas and Sebastin [16] are useful to prove our main results.Lemma 32 (Kilbas and Sebastian 2008).
Lettingξ,ζ,η∈C be such that R(ξ)>0,R(ρ)>max⁡[0,R(ζ-η)], then(123)I0,xξ,ζ,ηtρ-1x=ΓρΓρ+η-ζΓρ-ζΓρ+ξ+ηxρ-ζ-1.Lemma 33 (Kilbas and Sebastian 2008).
Lettingξ,ζ,η∈C be such that R(ξ)>0,R(ρ)<1+min⁡[R(ζ),R(η)], then(124)Jx,∞ξ,ζ,ηtρ-1x=Γζ-ρ+1Γη-ρ+1Γ1-ρΓξ+ζ+η-ρ+1xρ-ζ-1.The main results are given in the following theorem.Theorem 34.
Letξ,ζ,η,ρ,α,β,γ,m∈C;R(ξ)>0;R(γ)>R(β)>0 and |m|<M (where M is positive real number), such that R(ρ)>max[0,R(ζ-η)]; then(125)I0,xξ,ζ,ηtρ-1FMCmα,β;γ;tx=xρ-ζ-1ΓρΓρ+η-ζΓρ-ζΓρ+ξ+η×FMCmα,β;γ;x∗2F2ρ,ρ+η-ζ;ρ-ζ,ρ+ξ+η;x.Proof.
For convenience, we denote the left-hand side of the result (125) by I. Using (67), and then interchanging the order of integration and summation, which is valid under the conditions of Theorem 34, then(126)I=∑n=0∞αnBMCmβ+n,γ-βBβ,γ-β1n!I0,xξ,ζ,ηtn+ρ-1x.
Applying the result (123), (126) reduced to(127)I=∑n=0∞αnBMCmβ+n,γ-βBβ,γ-βΓρ+nΓρ+η-ζ+nΓρ-ζ+nΓρ+ξ+η+nxρ+n-ζ-1n!.
After simplification, (127) reduces to(128)I=xρ-ζ-1∑n=0∞anBMCmβ+n,γ-βBβ,γ-βΓρ+nΓρ+η-ζ+nΓρ-ζ+nΓρ+ξ+η+nxnn!.
Further using(a)n=Γ(a+n)/Γ(a), (128) reduces to the following form:(129)I=xρ-ζ-1ΓρΓρ+η-ζΓρ-ζΓρ+ξ+η∑n=0∞anBMCmβ+n,γ-βBβ,γ-β×ρnρ+η-ζnρ-ζnρ+ξ+ηnxnn!,(130)I=xρ-ζ-1ΓρΓρ+η-ζΓρ-ζΓρ+ξξ+ηMC2Fm,2α,β,ρ,ρ+η-ζ;γ,ρ-ζ,ρ+ξ+η;x,
and interpreting the above equation with the help of the concept of Hadamard given in (116), we have the required result.Theorem 35.
Letξ,ζ,η,ρ,α,β,γ,m∈C;R(ξ)>0;R(γ)>R(β)>0 and |m|<M (where M is positive real number), such that R(ρ)<1+min⁡[R(ζ),R(η)]; then(131)Jx,∞ξ,ζ,ηtρ-1FMCmα,β;γ;1tx=xρ-ζ-1Γζ-ρ+1Γη-ρ+1Γ1-ρΓξ+ζ+η-ρ+1×FMCmα,β;γ;x∗2F2ζ-ρ+1,η-ρ+1;1-ρ,ξ+ζ+η-ρ+1;x.Theorem 36.
Letξ,ζ,η,ρ,β,γ,m∈C;R(ξ)>0;R(γ)>R(β)>0 and |m|<M (where M is positive real number), such that R(ρ)>max⁡[0,R(ζ-η)]; then(132)I0,xξ,ζ,ηtρ-1ΦMCmβ;γ;tx=xρ-ζ-1ΓρΓρ+η-ζΓρ-ζΓρ+ξ+η×ΦMCmβ;γ;x∗2F2ρ,ρ+η-ζ;ρ-ζ,ρ+ξ+η;x.Theorem 37.
Letξ,ζ,η,ρ,β,γ,m∈C;R(ξ)>0;R(γ)>R(β)>0 and m<M (where M is positive real number), such that R(ρ)<1+min⁡[R(ζ),R(η)]; then(133)Jx,∞ξ,ζ,ηtρ-1ΦMCmβ;γ;1tx=xρ-ζ-1Γζ-ρ+1Γη-ρ+1Γ1-ρΓξ+ζ+η-ρ+1×ΦMCmβ;γ;x∗2F2ζ-ρ+1,η-ρ+1;1-ρ,ξ+ζ+η-ρ+1;x.Proof.
The proofs of the Theorems35, 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 Theorems34–37, we have the following special cases.By puttingζ=-ξ, 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.
Letξ,ρ,α,β,γ,m∈C;R(ξ)>0;R(γ)>R(β)>0 and |m|<M (where M is positive real number), such that R(ρ)>max⁡[0,R(-ξ)]; then(134)R0,xξtρ-1FMCmα,β;γ;tx=xρ+ξ-1ΓρΓρ+ξFMCmα,β;γ;x∗1F1ρ;ρ+ξ;x.Corollary 39.
Letξ,ρ,α,β,γ,m∈C;R(ξ)>0;R(γ)>R(β)>0 and |m|<M (where M is positive real number), such that R(ρ)<1+R(ξ); then(135)Wx,∞ξtρ-1FMCmα,β;γ;1tx=xρ+ξ-1Γ1-ξ-ρΓ1-ρFMCmα,β;γ;x∗1F11-ξ-ρ;1-ρ;x.Corollary 40.
Letξ,ρ,β,γ,m∈C;R(ξ)>0;R(γ)>R(β)>0 and |m|<M (where M is positive real number), such that R(ρ)>max⁡[0,R(-ξ)]; then(136)R0,xξtρ-1ΦMCmβ;γ;tx=xρ+ξ-1ΓρΓρ+ξΦMCmβ;γ;x∗1F1ρ;ρ+ξ;x.Corollary 41.
Letξ,ρ,β,γ,m∈C;R(ξ)>0;R(γ)>R(β)>0 and |m|<M (where M is positive real number), such that R(ρ)<1+R(ξ); then(137)Wx,∞ξtρ-1ΦMCmβ;γ;1tx=xρ+ξ-1Γ1-ξ-ρΓ1-ρΦMCmβ;γ;x∗1F11-ξ-ρ;1-ρ;x.By puttingζ=0, the Saigo hypergeometric fractional integrals operators reduce to the Erde^lyi-Kober fractional integral operators; then the results in (125), (131), (132) and (133) reduce to the following form.Corollary 42.
Letξ,η,ρ,α,β,γ,m∈C;R(ξ)>0;R(γ)>R(β)>0 and |m|<M (where M is positive real number), such that R(ρ)>max⁡[0,R(-η)]; then(138)E0,xξ,ηtρ-1FMCmα,β;γ;tx=xρ-1Γρ+ηΓρ+ξ+ηFMCmα,β;γ;x∗1F1ρ+η;ρ+ξ+η;x.Corollary 43.
Letξ,η,ρ,α,β,γ,m∈C;R(ξ)>0;R(γ)>R(β)>0 and |m|<M (where M is positive real number), such that R(ρ)<1+min⁡[R(ζ),R(η)]; then(139)Kx,∞ξ,ηtρ-1FMCmα,β;γ;1tx=xρ-1Γη-ρ+1Γξ+η-ρ+1FMCmα,β;γ;x∗1F1η-ρ+1;ξ+η-ρ+1;x.Corollary 44.
Letξ,η,ρ,β,γ,m∈C;R(ξ)>0;R(γ)>R(β)>0 and |m|<M (where M is positive real number), such that R(ρ)>max⁡[0,R(-η)]; then(140)E0,xξ,ηtρ-1ΦMCmβ;γ;tx=xρ-1Γρ+ηΓρ+ξ+ηΦMCmβ;γ;x∗1F1ρ+η;ρ+ξ+η;x.Corollary 45.
Letξ,η,ρ,β,γ,m∈C;R(ξ)>0;R(γ)>R(β)>0 and |m|<M (where M is positive real number), such that R(ρ)<1+min⁡[R(ζ),R(η)]; then(141)Kx,∞ξ,ηtρ-1ΦMCmβ;γ;1tx=xρ-1Γη-ρ+1Γξ+η-ρ+1ΦMCmβ;γ;x∗1F1η-ρ+1;ξ+η-ρ+1;x.If we choosem=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.Corollary 46.
Letξ,ζ,η,ρ,α,β,γ∈C;R(ξ)>0;R(γ)>R(β)>0, such that R(ρ)>max⁡[0,R(ζ-η)]; then(142)I0,xξ,ζ,ηtρ-12F1α,β;γ;tx=xρ-ζ-1ΓρΓρ+η-ζΓρ-ζΓρ+ξ+η×2F1α,β;γ;x∗2F2ρ,ρ+η-ζ;ρ-ζ,ρ+ξ+η;x.Corollary 47.
Letξ,ζ,η,ρ,α,β,γ∈C;R(ξ)>0;R(γ)>R(β)>0, such that R(ρ)<1+min⁡[R(ζ),R(η)]; then(143)Jx,∞ξ,ζ,ηtρ-12F1α,β;γ;1tx=xρ-ζ-1Γζ-ρ+1Γη-ρ+1Γ1-ρΓξ+ζ+η-ρ+1×2F1α,β;γ;x∗2F2ζ-ρ+1,η-ρ+1;1-ρ,ξ+ζ+η-ρ+1;x.Corollary 48.
Letξ,ζ,η,ρ,β,γ∈C;R(ξ)>0;R(γ)>R(β)>0, such that R(ρ)>max⁡[0,R(ζ-η)]; then(144)I0,xξ,ζ,ηtρ-1Φβ;γ;tx=xρ-ζ-1ΓρΓρ+η-ζΓρ-ζΓρ+ξ+η×Φβ;γ;x∗2F2ρ,ρ+η-ζ;ρ-ζ,ρ+ξ+η;x.Corollary 49.
Letξ,ζ,η,ρ,β,γ∈C;R(ξ)>0;R(γ)>R(β)>0, such that R(ρ)<1+min⁡[R(ζ),R(η)]. Then(145)Jx,∞ξ,ζ,ηtρ-1Φβ;γ;1tx=xρ-ζ-1Γζ-ρ+1Γη-ρ+1Γ1-ρΓξ+ζ+η-ρ+1×Φβ;γ;x∗2F2ζ-ρ+1,η-ρ+1;1-ρ,ξ+ζ+η-ρ+1;x.


## 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 Theorems34–37, we have the following special cases.By puttingζ=-ξ, 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.
Letξ,ρ,α,β,γ,m∈C;R(ξ)>0;R(γ)>R(β)>0 and |m|<M (where M is positive real number), such that R(ρ)>max⁡[0,R(-ξ)]; then(134)R0,xξtρ-1FMCmα,β;γ;tx=xρ+ξ-1ΓρΓρ+ξFMCmα,β;γ;x∗1F1ρ;ρ+ξ;x.Corollary 39.
Letξ,ρ,α,β,γ,m∈C;R(ξ)>0;R(γ)>R(β)>0 and |m|<M (where M is positive real number), such that R(ρ)<1+R(ξ); then(135)Wx,∞ξtρ-1FMCmα,β;γ;1tx=xρ+ξ-1Γ1-ξ-ρΓ1-ρFMCmα,β;γ;x∗1F11-ξ-ρ;1-ρ;x.Corollary 40.
Letξ,ρ,β,γ,m∈C;R(ξ)>0;R(γ)>R(β)>0 and |m|<M (where M is positive real number), such that R(ρ)>max⁡[0,R(-ξ)]; then(136)R0,xξtρ-1ΦMCmβ;γ;tx=xρ+ξ-1ΓρΓρ+ξΦMCmβ;γ;x∗1F1ρ;ρ+ξ;x.Corollary 41.
Letξ,ρ,β,γ,m∈C;R(ξ)>0;R(γ)>R(β)>0 and |m|<M (where M is positive real number), such that R(ρ)<1+R(ξ); then(137)Wx,∞ξtρ-1ΦMCmβ;γ;1tx=xρ+ξ-1Γ1-ξ-ρΓ1-ρΦMCmβ;γ;x∗1F11-ξ-ρ;1-ρ;x.By puttingζ=0, the Saigo hypergeometric fractional integrals operators reduce to the Erde^lyi-Kober fractional integral operators; then the results in (125), (131), (132) and (133) reduce to the following form.Corollary 42.
Letξ,η,ρ,α,β,γ,m∈C;R(ξ)>0;R(γ)>R(β)>0 and |m|<M (where M is positive real number), such that R(ρ)>max⁡[0,R(-η)]; then(138)E0,xξ,ηtρ-1FMCmα,β;γ;tx=xρ-1Γρ+ηΓρ+ξ+ηFMCmα,β;γ;x∗1F1ρ+η;ρ+ξ+η;x.Corollary 43.
Letξ,η,ρ,α,β,γ,m∈C;R(ξ)>0;R(γ)>R(β)>0 and |m|<M (where M is positive real number), such that R(ρ)<1+min⁡[R(ζ),R(η)]; then(139)Kx,∞ξ,ηtρ-1FMCmα,β;γ;1tx=xρ-1Γη-ρ+1Γξ+η-ρ+1FMCmα,β;γ;x∗1F1η-ρ+1;ξ+η-ρ+1;x.Corollary 44.
Letξ,η,ρ,β,γ,m∈C;R(ξ)>0;R(γ)>R(β)>0 and |m|<M (where M is positive real number), such that R(ρ)>max⁡[0,R(-η)]; then(140)E0,xξ,ηtρ-1ΦMCmβ;γ;tx=xρ-1Γρ+ηΓρ+ξ+ηΦMCmβ;γ;x∗1F1ρ+η;ρ+ξ+η;x.Corollary 45.
Letξ,η,ρ,β,γ,m∈C;R(ξ)>0;R(γ)>R(β)>0 and |m|<M (where M is positive real number), such that R(ρ)<1+min⁡[R(ζ),R(η)]; then(141)Kx,∞ξ,ηtρ-1ΦMCmβ;γ;1tx=xρ-1Γη-ρ+1Γξ+η-ρ+1ΦMCmβ;γ;x∗1F1η-ρ+1;ξ+η-ρ+1;x.If we choosem=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.Corollary 46.
Letξ,ζ,η,ρ,α,β,γ∈C;R(ξ)>0;R(γ)>R(β)>0, such that R(ρ)>max⁡[0,R(ζ-η)]; then(142)I0,xξ,ζ,ηtρ-12F1α,β;γ;tx=xρ-ζ-1ΓρΓρ+η-ζΓρ-ζΓρ+ξ+η×2F1α,β;γ;x∗2F2ρ,ρ+η-ζ;ρ-ζ,ρ+ξ+η;x.Corollary 47.
Letξ,ζ,η,ρ,α,β,γ∈C;R(ξ)>0;R(γ)>R(β)>0, such that R(ρ)<1+min⁡[R(ζ),R(η)]; then(143)Jx,∞ξ,ζ,ηtρ-12F1α,β;γ;1tx=xρ-ζ-1Γζ-ρ+1Γη-ρ+1Γ1-ρΓξ+ζ+η-ρ+1×2F1α,β;γ;x∗2F2ζ-ρ+1,η-ρ+1;1-ρ,ξ+ζ+η-ρ+1;x.Corollary 48.
Letξ,ζ,η,ρ,β,γ∈C;R(ξ)>0;R(γ)>R(β)>0, such that R(ρ)>max⁡[0,R(ζ-η)]; then(144)I0,xξ,ζ,ηtρ-1Φβ;γ;tx=xρ-ζ-1ΓρΓρ+η-ζΓρ-ζΓρ+ξ+η×Φβ;γ;x∗2F2ρ,ρ+η-ζ;ρ-ζ,ρ+ξ+η;x.Corollary 49.
Letξ,ζ,η,ρ,β,γ∈C;R(ξ)>0;R(γ)>R(β)>0, such that R(ρ)<1+min⁡[R(ζ),R(η)]. Then(145)Jx,∞ξ,ζ,ηtρ-1Φβ;γ;1tx=xρ-ζ-1Γζ-ρ+1Γη-ρ+1Γ1-ρΓξ+ζ+η-ρ+1×Φβ;γ;x∗2F2ζ-ρ+1,η-ρ+1;1-ρ,ξ+ζ+η-ρ+1;x.


## 8. Beta Transform

The Beta transform off(z) is defined as follows [17]:(146)Bfz:a,b=∫01za-11-zb-1fzdz.Theorem 50.
Letξ,ζ,η,ρ,β,γ,m∈C;R(ξ)>0;R(γ)>R(β)>0 and |m|<M (where M is positive real number), such that R(ρ)>max⁡[0,R(ζ-η)]; then(147)BI0,xξ,ζ,ηtρ-1FMCmα,β;γ;tx:l,m=Γmxρ-ζ-1ΓρΓρ+η-ζΓρ-ζΓρ+ξ+η×FMCmα,β;γ;x∗3F3ρ,ρ+η-ζ,l;ρ-ζ,ρ+ξ+η,l+m;x.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(148)B=∫01zl-11-zm-1I0,xξ,ζ,ηtρ-1FMCmα,β;γ;tzxdz,
and further using (129) and then changing the order of integration and summation, which is valid under the conditions of Theorem 1, then(149)B=xρ-ζ-1ΓρΓρ+η-ζΓρ-ζΓρ+ξ+η∑n=0∞anBMCmβ+n,γ-βBβ,γ-β×ρnρ+η-ζnρ-ζnρ+ξ+ηnxnn!∫01zl+n-11-zm-1dz.
Applying the definition of beta transform, (149) reduced to(150)B=xρ-ζ-1ΓρΓρ+η-ζΓρ-ζΓρ+ξ+η∑n=0∞anBMCmβ+n,γ-βBβ,γ-βρnρ+η-ζnρ-ζnρ+ξ+ηn×Γl+nΓmΓl+m+nxnn!,
and interpreting the above equation with the help of (67), we have(151)B=Γmxρ-ζ-1ΓρΓρ+η-ζΓρ-ζΓρ+ξ+ηMC5Fm,3α,β,ρ,ρ+η-ζ,l;γ,ρ-ζ,ρ+ξ+η,l+m;x.Further, interpreting (151) with the view of the concept of Hadamard (116), we have the required the result.Theorem 51.
Letξ,ζ,η,ρ,β,γ,m∈C;R(ξ)>0;R(γ)>R(β)>0 and |m|<M (where M is positive real number), such that R(ρ)<1+min⁡[R(ζ),R(η)]; then(152)BJx,∞ξ,ζ,ηtρ-1FMCmα,β;γ;ztx:l,m=Γmxρ-ζ-1Γζ-ρ+1Γη-ρ+1Γ1-ρΓξ+ζ+η-ρ+1FMCmα,β;γ;x∗3F3ζ-ρ+1,η-ρ+1,l;1-ρ,ξ+ζ+η-ρ+1,l+m;x.Theorem 52.
Letξ,ζ,η,ρ,β,γ,m∈C;R(ξ)>0;R(γ)>R(β)>0 and |m|<M (where M is positive real number), such that R(ρ)>max⁡[0,R(ζ-η)]; then(153)BI0,xξ,ζ,ηtρ-1ΦMCmβ;γ;tx:l,m=Γmxρ-ζ-1ΓρΓρ+η-ζΓρ-ζΓρ+ξ+η×ΦMCmβ;γ;x∗3F3ρ,ρ+η-ζ,l;ρ-ζ,ρ+ξ+η,l+m;x.Theorem 53.
Letξ,ζ,η,ρ,β,γ,m∈C;R(ξ)>0;R(γ)>R(β)>0 and |m|<M (where M is positive real number), such that R(ρ)<1+min[R(ζ),R(η)]; then(154)BJx,∞ξ,ζ,ηtρ-1ΦMCmβ;γ;ztx:l,m=Γmxρ-ζ-1Γζ-ρ+1Γη-ρ+1Γ1-ρΓξ+ζ+η-ρ+1ΦMCmβ;γ;x∗3F3ζ-ρ+1,η-ρ+1,l;1-ρ,ξ+ζ+η-ρ+1,l+m;x.Proof.
The proofs of the Theorems51, 52, and 53 are parallel to those of Theorem 50.


## 9. Laplace Transform

The Laplace transform off(z) is defined as follows [17]:(155)Lfz=∫0∞e-szfzdz.Theorem 54.
Letξ,ζ,η,ρ,β,γ,m∈C;R(ξ)>0;R(γ)>R(β)>0 and |m|<M (where M is positive real number), such that R(ρ)>max[0,R(ζ-η)]; then(156)Lzl-1I0,xξ,ζ,ηtρ-1FMCmα,β;γ;tzx=xρ-ζ-1slΓρΓρ+η-ζΓρ-ζΓρ+ξ+η×FMCmα,β;γ;x∗3F2ρ,ρ+η-ζ,l;ρ-ζ,ρ+ξ+η;x.Proof.
For convenience, we denote the left-hand side of the result (156) by L. Then, applying the Laplace, we have(157)L=∫0∞e-szzl-1I0,xξ,ζ,ηtρ-1FMCmα,β;γ;tzxdz,
and further using (129) and then changing the order of integration and summation, which is valid under the conditions of Theorem 1, then(158)L=xρ-ζ-1ΓρΓρ+η-ζΓρ-ζΓρ+ξ+η∑n=0∞anBMCmβ+n,γ-βBβ,γ-β×ρnρ+η-ζnρ-ζnρ+ξ+ηnxnn!∫0∞e-szzn+l-1dz.
After simplification, (158) reduces to(159)L=xρ-ζ-1ΓρΓρ+η-ζΓρ-ζΓρ+ξ+η∑n=0∞anBMCmβ+n,γ-βBβ,γ-βρnρ+η-ζnρ-ζnρ+ξ+ηnxnn!Γn+lsn+l=xρ-ζ-1ΓρΓρ+η-ζΓlΓρ-ζΓρ+ξ+ηMC3Fm,2α,β,ρ,ρ+η-ζ,l;ρ-ζ,ρ+ξ+η,γ;x,
and interpreting (159) with the view of the concept of Hadamard (116), we have the required the result.Theorem 55.
Letξ,ζ,η,ρ,β,γ,m∈C;R(ξ)>0;R(γ)>R(β)>0 and |m|<M (where M is positive real number), such that R(ρ)<1+min[R(ζ),R(η)]; then(160)Lzl-1Jx,∞ξ,ζ,ηtρ-1FMCmα,β;γ;ztx=xρ-ζ-1slΓζ-ρ+1Γη-ρ+1Γ1-ρΓξ+ζ+η-ρ+1FMCmα,β;γ;x∗3F2ζ-ρ+1,η-ρ+1,l;1-ρ,ξ+ζ+η-ρ+1;x.Theorem 56.
Letξ,ζ,η,ρ,β,γ,m∈C;R(ξ)>0;R(γ)>R(β)>0 and |m|<M (where M is positive real number), such that R(ρ)>max[0,R(ζ-η)]; then(161)Lzl-1I0,xξ,ζ,ηtρ-1ΦMCmβ;γ;tzx=xρ-ζ-1slΓρΓρ+η-ζΓρ-ζΓρ+ξ+η×ΦMCmβ;γ;x∗3F2ρ,ρ+η-ζ,l;ρ-ζ,ρ+ξ+η;x.Theorem 57.
Letξ,ζ,η,ρ,β,γ,m∈C;R(ξ)>0;R(γ)>R(β)>0 and |m|<M (where M is positive real number), such that R(ρ)<1+min[R(ζ),R(η)]; then(162)Lzl-1Jx,∞ξ,ζ,ηtρ-1ΦMCmβ;γ;ztx=xρ-ζ-1slΓζ-ρ+1Γη-ρ+1Γ1-ρΓξ+ζ+η-ρ+1ΦMCmβ;γ;x∗3F2ζ-ρ+1,η-ρ+1,l;1-ρ,ξ+ζ+η-ρ+1;x.Proof.
The proofs of Theorems55, 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.
Letξ,ζ,η,ρ,β,γ,m∈C;R(ξ)>0;R(γ)>R(β)>0 and |m|<M (where M is positive real number) and R(σ±ω)>-1/2, such that R(ρ)>max[0,R(ζ-η)]; then(163)∫0∞zσ-1e-δz/2Wτ,ωκzW1zdz=xρ-ζ-1κσ-1ΓρΓρ+η-ζΓρ-ζΓρ+ξ+ηFMCmα,β;γ;xκ∗4F3ρ,ρ+η-ζ,12+ω+σ,12-ω+σ;ρ-ζ,ρ+ξ+η,12-τ+σ;xκ,
whereW1(z)=I0,xξ,ζ,ηtρ-1MCFm(α,β;γ;tz)(x).Proof.
For convenience, we denote the left-hand side of the result (163) by W. Then using the result from (128), after changing the order of integration and summation, we get(164)W=xρ-ζ-1ΓρΓρ+η-ζΓρ-ζΓρ+ξ+η∑n=0∞anBMCmβ+n,γ-βBβ,γ-β×ρnρ+η-ζnρ-ζnρ+ξ+ηnxnn!∫0∞zn+σ-1e-κz/2Wτ,ωκzdz.
By substitutingκz=ς, (164) becomes(165)W=xρ-ζ-1ΓρΓρ+η-ζΓρ-ζΓρ+ξ+η∑n=0∞anBMCmβ+n,γ-βBβ,γ-β×ρnρ+η-ζnρ-ζnρ+ξ+ηnxnn!1κn+σ-1∫0∞ςn+σ-1e-ς/2Wτ,ωςdς.
Now we use the following integral formula involving Whittaker function:(166)∫0∞tν-1e-t/2Wτ,ωtdt=Γ1/2+ω+νΓ1/2-ω+νΓ1/2-τ+ν,Rν±ω>-12.Then we have(167)W=xρ-ζ-1κσ-1ΓρΓρ+η-ζΓρ-ζΓρ+ξ+η∑n=0∞anBMCmβ+n,γ-βBβ,γ-βρnρ+η-ζnρ-ζnρ+ξ+ηn×Γ1/2+ω+σ+nΓ1/2-ω+σ+nΓ1/2-τ+σ+n1n!xκn,
and interpreting (167) with the help of (67), then, with the concept of Hadamard (116), we have the desired the result.Theorem 59.
Letξ,ζ,η,ρ,β,γ,m∈C;R(ξ)>0;R(γ)>R(β)>0 and |m|<M (where M is positive real number) and R(σ±ω)>-1/2, such that R(ρ)<1+min[R(ζ),R(η)]; then(168)∫0∞zσ-1e-δz/2Wτ,ωκzW2zdz=xρ-ζ-1κσ-1Γζ-ρ+1Γη-ρ+1Γ1-ρΓξ+ζ+η-ρ+1FMCmα,β;γ;xκ∗4F3ζ-ρ+1,η-ρ+1,12+ω+σ,12-ω+σ;1-ρ,ξ+ζ+η-ρ+1,12-τ+σ;xκ,
whereW2(z)=Jx,∞ξ,ζ,ηtρ-1MCFm(α,β;γ;z/t)(x).Theorem 60.
Letξ,ζ,η,ρ,β,γ,m∈C;R(ξ)>0;R(γ)>R(β)>0 and |m|<M (where M is positive real number) and R(σ±ω)>-1/2, such that R(ρ)>max[0,R(ζ-η)]; then(169)∫0∞zσ-1e-δz/2Wτ,ωκzW3zdz=xρ-ζ-1κσ-1ΓρΓρ+η-ζΓρ-ζΓρ+ξ+ηΦMCmβ;γ;xκ∗4F3ρ,ρ+η-ζ,12+ω+σ,12-ω+σ;ρ-ζ,ρ+ξ+η,12-τ+σ;xκ,
whereW3(z)=I0,xξ,ζ,ηtρ-1MCΦm(β;γ;tz)(x).Theorem 61.
Letξ,ζ,η,ρ,β,γ,m∈C;R(ξ)>0;R(γ)>R(β)>0 and |m|<M (where M is positive real number) and R(σ±ω)>-1/2, such that R(ρ)<1+min[R(ζ),R(η)]; then(170)∫0∞zσ-1e-δz/2Wτ,ωκzW4zdz=xρ-ζ-1κσ-1Γζ-ρ+1Γη-ρ+1Γ1-ρΓξ+ζ+η-ρ+1ΦMCmβ;γ;xκ∗4F3ζ-ρ+1,η-ρ+1,12+ω+σ,12-ω+σ;1-ρ,ξ+ζ+η-ρ+1,12-τ+σ;xκ,
whereW4(z)=Jx,∞ξ,ζ,ηtρ-1MCΦm(β;γ;z/t)(x).Proof.
The proofs of Theorems59, 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:(171)dNdt=-dNt+pNt,whereN=N(t) the rate of reaction, d=d(N) the rate of destruction, p=p(N) the rate of production, and Nt denotes the function defined by Nt(t∗)=N(t-t∗),t∗>0.In the special case of (171) for spatial fluctuations and inhomogeneities in N(t) the quantities are neglected, that is, the equation(172)dNdt=-ciNit,with the initial condition thatNi(t=0)=N0 is the number density of the species i at time t=0 and ci>0. If we remove the index i and integrate the standard kinetic equation (172), we have(173)Nt-N0=-c0Dt-1Nt,where0Dt-1 is the special case of the Riemann-Liouville integral operator 0Dt-ν defined as(174)0Dt-νft=1Γν∫0tt-sν-1fsds,t>0,Rν>0.The fractional generalization of the standard kinetic equation (173) is given by Haubold and Mathai[23] as follows:(175)Nt-N0=-cν0Dt-1Ntand obtained the solution of (175) as follows:(176)Nt=N0∑k=0∞-1kΓνk+1ctνk.Further, (Saxena and Kalla [27]) considered the following fractional kinetic equation:(177)Nt-N0ft=-cν0Dt-νNt,Rv>0,whereN(t) denotes the number density of a given species at time t, N0=N(0) is the number density of that species at time t=0, c is a constant, and f∈L(0,∞).By applying the Laplace transform to (177), we have(178)LNt;p=N0Fp1+cνp-ν=N0∑n=0∞-cνnp-νnFp,n∈N0,cp<1.Where the Laplace transform [32] is given by(179)Fp=LNt;p=∫0∞e-ptftdt,Rp>0.


### 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 functionEα,β(x) (Mittag-Leffler[33]), which is defined as(180)Eα,βz=∑n=0∞znΓαn+β,Rα>0,Rβ>0.Theorem 63.
Ifa>0,d>0,ν>0; α,β,γ,m∈C; R(γ)>R(β)>0 and |m|<M (where M is positive real number), then the solution of the equation(181)Nt-N0FMCmα,β;γ;dνtν=-aν0Dt-νNtis given by the following formula:(182)Nt=N0∑n=0∞αnBMCmβ+n,γ-βBβ,γ-βdνtνnn!Γνn+1Eν,νn+1-aνtν.Proof.
Laplace transform of Riemann-Liouville fractional integral operator is given by the following (Erdelyi et al. [34], Srivastava and Saxena[35]):(183)L0Dt-νft;p=p-νFp,
whereF(p) is defined in (179). Now, applying Laplace transform on (181) gives(184)LNt;p=N0LFMCma,b;c;dνtν;p-aνL0Dt-νNt;p(185)i.e.Np=N0∫0∞e-pt∑n=0∞αnBMCmβ+n,γ-βBβ,γ-βdνtνnn!dt-aνp-νNpand interchanging the order of integration and summation in (185), we have(186)Np+aνp-νNp=N0∑n=0∞αnBMCmβ+n,γ-βBβ,γ-βdνnn!∫0∞e-pttνndt(187)=N0∑n=0∞αnBMCmβ+n,γ-βBβ,γ-βdνnn!Γνn+1pνn+1.This leads to(188)Np=N0∑n=0∞αnBMCmβ+n,γ-βBβ,γ-βdνnn!Γνn+1p-νn+1∑l=0∞-pa-νl.
Taking Laplace inverse of (188), and by using(189)L-1p-ν;t=tν-1Γν,Rν>0,
we have(190)L-1Np=N0∑n=0∞αnBMCmβ+n,γ-βBβ,γ-βdνnn!×Γνr+1L-1∑l=0∞-1laνlp-νn+l+1(191)i.e.Nt=N0∑n=0∞αnBMCmβ+n,γ-βBβ,γ-βdνnn!×Γνn+1∑l=0∞-1laνltνn+lΓνn+l+1(192)=N0∑n=0∞αnBMCmβ+n,γ-βBβ,γ-βdνtνnn!×Γνn+1∑l=0∞-1laνtνlΓνn+l+1.
Equation (192) can be written as(193)Nt=N0∑n=0∞αnBMCmβ+n,γ-βBβ,γ-βdνtνnn!Γνn+1Eν,νn+1-aνtν.Theorem 64.
Ifa>0,d>0,ν>0; α,β,γ,m∈C;R(γ)>R(β)>0 and m<M (where M is positive real number), then the solution of the equation(194)Nt-N0ΦMCmβ;γ;dνtν=-aνD0t-νNtis given by the following formula:(195)Nt=N0∑n=0∞BMCmβ+n,γ-βBβ,γ-βdνtνnn!Γνn+1Eν,νn+1-aνtν.Theorem 65.
Ifd>0,ν>0; α,β,γ,m∈C;R(γ)>R(β)>0 and m<M (where M is positive real number), then the solution of the equation (196)Nt-N0FMCmα,β;γ;dνtν=-dν0Dt-νNtis given by the following formula:(197)Nt=N0∑n=0∞αnBMCmβ+n,γ-βBβ,γ-βdνtνnn!Γνn+1Eν,νn+1-dνtν.Theorem 66.
Ifd>0,ν>0; α,β,γ,m∈C; R(γ)>R(β)>0 and |m|<M (where M is positive real number), then the solution of the equation(198)Nt-N0ΦMCmα,β;γ;dνtν=-dν0Dt-νNtis given by the following formula:(199)Nt=N0∑n=0∞BMCmβ+n,γ-βBβ,γ-βdνtνnn!Γνn+1Eν,νn+1-dνtν.Theorem 67.
Ifd>0,ν>0; α,β,γ,m∈C; R(γ)>R(β)>0 and |m|<M (where M is positive real number), then the solution of the equation(200)Nt-N0FMCmα,β;γ;t=-dν0Dt-νNtis given by the following formula:(201)Nt=N0∑n=0∞αnBMCmβ+n,γ-βBβ,γ-βtnEν,n+1-dνtν.Theorem 68.
Ifd>0,ν>0; α,β,γ,m∈C; R(γ)>R(β)>0 and |m|<M (where M is positive real number), then the solution of the equation(202)Nt-N0ΦMCmα,β;γ;t=-dν0Dt-νNtis given by the following formula:(203)Nt=N0∑n=0∞BMCmβ+n,γ-βBβ,γ-βtnEν,n+1-dνtν.Proof.
The proofs of Theorems64, 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 Table5 for N0=1, α=2, β=1, γ=3, m=2, d=1, a=2 and ν=0.2:1:4.2 of equation (126) and the graph of solution of fractional kinetic equation given in (126) are presented in Figures 5, 6 and 7 for N0=1, α=2, β=1, γ=3, m=2, d=1, a=2; the values of νare chosen as ν=0.2:.2:0.8, ν=1.2:.2:1.8 and ν=2.2:.2:2.8, 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 N(t)>0 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 N0=1,α=2,β=1,γ=3,m=2,d=1,a=2.
t ν = 0.2 ν = 1.2 ν = 2.2 ν = 3.2 ν = 4.2 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 -19.2305Figure 5
Solution of fractional kinetic equation given in (182) for N0=1,α=2,β=1,γ=3,m=2,d=1,a=2,ν=0.2:.2:0.8.Figure 6
Solution of fractional kinetic equation given in (182) for N0=1,α=2,β=1,γ=3,m=2,d=1,a=2,ν=1.2:.2:1.8.Figure 7
Solution of fractional kinetic equation given in (182) for N0=1,α=2,β=1,γ=3,m=2,d=1,a=2,ν=2.2:.2:2.8.Figure 8
Mesh-Plot of solution of fractional kinetic equation given in (182) for N0=3,α=2,β=1,γ=3,m=2,d=1,a=2.


### 11.2. Special Cases of Fractional Kinetic Equations

Choosingm=0, Theorems 63, 64, 65, 66, 67, and 68 reduce to the following form.Corollary 69.
Ifa>0,d>0,ν>0; α,β,γ∈C; R(γ)>R(β)>0, then the solution of the equation(204)Nt-N02F1α,β;γ;dνtν=-aν0Dt-νNtis given by the following formula:(205)Nt=N0∑n=0∞αnBβ+n,γ-βBβ,γ-βdνtνnn!Γνn+1Eν,νn+1-aνtν.Corollary 70.
Ifa>0,d>0,ν>0; α,β,γ∈C; R(γ)>R(β)>0, then the solution of the equation(206)Nt-N0Φβ;γ;dνtν=-aν0Dt-νNtis given by the following formula:(207)Nt=N0∑n=0∞Bβ+n,γ-βBβ,γ-βdνtνnn!Γνn+1Eν,νn+1-aνtν.Corollary 71.
Ifd>0,ν>0; α,β,γ∈C; R(γ)>R(β)>0, then the solution of the equation(208)Nt-N02F1α,β;γ;dνtν=-dν0Dt-νNt is given by the following formula:(209)Nt=N0∑n=0∞αnBβ+n,γ-βBβ,γ-βdνtνnn!Γνn+1Eν,νn+1-dνtν.Corollary 72.
Ifd>0,ν>0; α,β,γ∈C; R(γ)>R(β)>0, then the solution of the equation(210)Nt-N0Φβ;γ;dνtν=-dν0Dt-νNtis given by the following formula:(211)Nt=N0∑n=0∞Bβ+n,γ-βBβ,γ-βdνtνnn!Γνn+1Eν,νn+1-dνtν.Corollary 73.
Ifd>0,ν>0; α,β,γ∈C; R(γ)>R(β)>0, then the solution of the equation(212)Nt-N02F1α,β;γ;t=-dν0Dt-νNtis given by the following formula:(213)Nt=N0∑n=0∞αnBβ+n,γ-βBβ,γ-βtnEν,n+1-dνtν.Corollary 74.
Ifd>0,ν>0; α,β,γ∈C; R(γ)>R(β)>0, then the solution of the equation(214)Nt-N0Φβ;γ;t=-dνD0t-νNtis given by the following formula:(215)Nt=N0∑n=0∞Bβ+n,γ-βBβ,γ-βtnEν,n+1-dνtν.


## 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 functionEα,β(x) (Mittag-Leffler[33]), which is defined as(180)Eα,βz=∑n=0∞znΓαn+β,Rα>0,Rβ>0.Theorem 63.
Ifa>0,d>0,ν>0; α,β,γ,m∈C; R(γ)>R(β)>0 and |m|<M (where M is positive real number), then the solution of the equation(181)Nt-N0FMCmα,β;γ;dνtν=-aν0Dt-νNtis given by the following formula:(182)Nt=N0∑n=0∞αnBMCmβ+n,γ-βBβ,γ-βdνtνnn!Γνn+1Eν,νn+1-aνtν.Proof.
Laplace transform of Riemann-Liouville fractional integral operator is given by the following (Erdelyi et al. [34], Srivastava and Saxena[35]):(183)L0Dt-νft;p=p-νFp,
whereF(p) is defined in (179). Now, applying Laplace transform on (181) gives(184)LNt;p=N0LFMCma,b;c;dνtν;p-aνL0Dt-νNt;p(185)i.e.Np=N0∫0∞e-pt∑n=0∞αnBMCmβ+n,γ-βBβ,γ-βdνtνnn!dt-aνp-νNpand interchanging the order of integration and summation in (185), we have(186)Np+aνp-νNp=N0∑n=0∞αnBMCmβ+n,γ-βBβ,γ-βdνnn!∫0∞e-pttνndt(187)=N0∑n=0∞αnBMCmβ+n,γ-βBβ,γ-βdνnn!Γνn+1pνn+1.This leads to(188)Np=N0∑n=0∞αnBMCmβ+n,γ-βBβ,γ-βdνnn!Γνn+1p-νn+1∑l=0∞-pa-νl.
Taking Laplace inverse of (188), and by using(189)L-1p-ν;t=tν-1Γν,Rν>0,
we have(190)L-1Np=N0∑n=0∞αnBMCmβ+n,γ-βBβ,γ-βdνnn!×Γνr+1L-1∑l=0∞-1laνlp-νn+l+1(191)i.e.Nt=N0∑n=0∞αnBMCmβ+n,γ-βBβ,γ-βdνnn!×Γνn+1∑l=0∞-1laνltνn+lΓνn+l+1(192)=N0∑n=0∞αnBMCmβ+n,γ-βBβ,γ-βdνtνnn!×Γνn+1∑l=0∞-1laνtνlΓνn+l+1.
Equation (192) can be written as(193)Nt=N0∑n=0∞αnBMCmβ+n,γ-βBβ,γ-βdνtνnn!Γνn+1Eν,νn+1-aνtν.Theorem 64.
Ifa>0,d>0,ν>0; α,β,γ,m∈C;R(γ)>R(β)>0 and m<M (where M is positive real number), then the solution of the equation(194)Nt-N0ΦMCmβ;γ;dνtν=-aνD0t-νNtis given by the following formula:(195)Nt=N0∑n=0∞BMCmβ+n,γ-βBβ,γ-βdνtνnn!Γνn+1Eν,νn+1-aνtν.Theorem 65.
Ifd>0,ν>0; α,β,γ,m∈C;R(γ)>R(β)>0 and m<M (where M is positive real number), then the solution of the equation (196)Nt-N0FMCmα,β;γ;dνtν=-dν0Dt-νNtis given by the following formula:(197)Nt=N0∑n=0∞αnBMCmβ+n,γ-βBβ,γ-βdνtνnn!Γνn+1Eν,νn+1-dνtν.Theorem 66.
Ifd>0,ν>0; α,β,γ,m∈C; R(γ)>R(β)>0 and |m|<M (where M is positive real number), then the solution of the equation(198)Nt-N0ΦMCmα,β;γ;dνtν=-dν0Dt-νNtis given by the following formula:(199)Nt=N0∑n=0∞BMCmβ+n,γ-βBβ,γ-βdνtνnn!Γνn+1Eν,νn+1-dνtν.Theorem 67.
Ifd>0,ν>0; α,β,γ,m∈C; R(γ)>R(β)>0 and |m|<M (where M is positive real number), then the solution of the equation(200)Nt-N0FMCmα,β;γ;t=-dν0Dt-νNtis given by the following formula:(201)Nt=N0∑n=0∞αnBMCmβ+n,γ-βBβ,γ-βtnEν,n+1-dνtν.Theorem 68.
Ifd>0,ν>0; α,β,γ,m∈C; R(γ)>R(β)>0 and |m|<M (where M is positive real number), then the solution of the equation(202)Nt-N0ΦMCmα,β;γ;t=-dν0Dt-νNtis given by the following formula:(203)Nt=N0∑n=0∞BMCmβ+n,γ-βBβ,γ-βtnEν,n+1-dνtν.Proof.
The proofs of Theorems64, 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 Table5 for N0=1, α=2, β=1, γ=3, m=2, d=1, a=2 and ν=0.2:1:4.2 of equation (126) and the graph of solution of fractional kinetic equation given in (126) are presented in Figures 5, 6 and 7 for N0=1, α=2, β=1, γ=3, m=2, d=1, a=2; the values of νare chosen as ν=0.2:.2:0.8, ν=1.2:.2:1.8 and ν=2.2:.2:2.8, 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 N(t)>0 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 N0=1,α=2,β=1,γ=3,m=2,d=1,a=2.
t ν = 0.2 ν = 1.2 ν = 2.2 ν = 3.2 ν = 4.2 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 -19.2305Figure 5
Solution of fractional kinetic equation given in (182) for N0=1,α=2,β=1,γ=3,m=2,d=1,a=2,ν=0.2:.2:0.8.Figure 6
Solution of fractional kinetic equation given in (182) for N0=1,α=2,β=1,γ=3,m=2,d=1,a=2,ν=1.2:.2:1.8.Figure 7
Solution of fractional kinetic equation given in (182) for N0=1,α=2,β=1,γ=3,m=2,d=1,a=2,ν=2.2:.2:2.8.Figure 8
Mesh-Plot of solution of fractional kinetic equation given in (182) for N0=3,α=2,β=1,γ=3,m=2,d=1,a=2.


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

In this section, we present the numerical results in Table5 for N0=1, α=2, β=1, γ=3, m=2, d=1, a=2 and ν=0.2:1:4.2 of equation (126) and the graph of solution of fractional kinetic equation given in (126) are presented in Figures 5, 6 and 7 for N0=1, α=2, β=1, γ=3, m=2, d=1, a=2; the values of νare chosen as ν=0.2:.2:0.8, ν=1.2:.2:1.8 and ν=2.2:.2:2.8, 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 N(t)>0 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 N0=1,α=2,β=1,γ=3,m=2,d=1,a=2.
t ν = 0.2 ν = 1.2 ν = 2.2 ν = 3.2 ν = 4.2 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 -19.2305Figure 5
Solution of fractional kinetic equation given in (182) for N0=1,α=2,β=1,γ=3,m=2,d=1,a=2,ν=0.2:.2:0.8.Figure 6
Solution of fractional kinetic equation given in (182) for N0=1,α=2,β=1,γ=3,m=2,d=1,a=2,ν=1.2:.2:1.8.Figure 7
Solution of fractional kinetic equation given in (182) for N0=1,α=2,β=1,γ=3,m=2,d=1,a=2,ν=2.2:.2:2.8.Figure 8
Mesh-Plot of solution of fractional kinetic equation given in (182) for N0=3,α=2,β=1,γ=3,m=2,d=1,a=2.


## 11.2. Special Cases of Fractional Kinetic Equations

Choosingm=0, Theorems 63, 64, 65, 66, 67, and 68 reduce to the following form.Corollary 69.
Ifa>0,d>0,ν>0; α,β,γ∈C; R(γ)>R(β)>0, then the solution of the equation(204)Nt-N02F1α,β;γ;dνtν=-aν0Dt-νNtis given by the following formula:(205)Nt=N0∑n=0∞αnBβ+n,γ-βBβ,γ-βdνtνnn!Γνn+1Eν,νn+1-aνtν.Corollary 70.
Ifa>0,d>0,ν>0; α,β,γ∈C; R(γ)>R(β)>0, then the solution of the equation(206)Nt-N0Φβ;γ;dνtν=-aν0Dt-νNtis given by the following formula:(207)Nt=N0∑n=0∞Bβ+n,γ-βBβ,γ-βdνtνnn!Γνn+1Eν,νn+1-aνtν.Corollary 71.
Ifd>0,ν>0; α,β,γ∈C; R(γ)>R(β)>0, then the solution of the equation(208)Nt-N02F1α,β;γ;dνtν=-dν0Dt-νNt is given by the following formula:(209)Nt=N0∑n=0∞αnBβ+n,γ-βBβ,γ-βdνtνnn!Γνn+1Eν,νn+1-dνtν.Corollary 72.
Ifd>0,ν>0; α,β,γ∈C; R(γ)>R(β)>0, then the solution of the equation(210)Nt-N0Φβ;γ;dνtν=-dν0Dt-νNtis given by the following formula:(211)Nt=N0∑n=0∞Bβ+n,γ-βBβ,γ-βdνtνnn!Γνn+1Eν,νn+1-dνtν.Corollary 73.
Ifd>0,ν>0; α,β,γ∈C; R(γ)>R(β)>0, then the solution of the equation(212)Nt-N02F1α,β;γ;t=-dν0Dt-νNtis given by the following formula:(213)Nt=N0∑n=0∞αnBβ+n,γ-βBβ,γ-βtnEν,n+1-dνtν.Corollary 74.
Ifd>0,ν>0; α,β,γ∈C; R(γ)>R(β)>0, then the solution of the equation(214)Nt-N0Φβ;γ;t=-dνD0t-νNtis given by the following formula:(215)Nt=N0∑n=0∞Bβ+n,γ-βBβ,γ-βtnEν,n+1-dνtν.


## 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.



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