# A Note on the New Extended Beta Function with Its Application

Authors: Vandana Palsaniya; Ekta Mittal; Sunil Joshi; D. L. Suthar
Journal: Journal of Mathematics
 (2021)

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## Abstract

The purpose of this research is to provide a systematic review of a new type of extended beta function and hypergeometric function using a confluent hypergeometric function, as well as to examine various belongings and formulas of the new type of extended beta function, such as integral representations, derivative formulas, transformation formulas, and summation formulas. In addition, we also investigate extended Riemann–Liouville (R-L) fractional integral operator with associated properties. Furthermore, we develop new beta distribution and present graphically the relation between moment generating function andℓ.


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## Body

## 1. Introduction

Special functions, often denoted by series, emerge as a result of solving various problems in classical physics. Typically, these problems demand the flow of electromagnetic, acoustic, or thermal energy. It has developed into a vital resource for the particular and specialized roles played by scientists and engineers these days. These characteristics are critical in a variety of domains, including physical science, mathematics, and engineering. The Gaussian hypergeometric functionF21 is among most important special functions. The hypergeometric function F21 and its numerous generalizations have been investigated by many researchers who have provided a notably large number of their formula properties (see, e.g., [1–7]). Over the last decades, many researchers have established the generalizations of the classical gamma and beta functions and provided a number of intriguing and useful properties for the extended functions (see, e.g., [8–16]).We begin with recalling some definitions of extended beta functions.Chaudhary and Zubair [2] extended the classical gamma function as follows:(1)Γϱx=∫0∞ℓx−1e−ℓ−ϱ/ℓdℓ,where ℜeϱ>0,ℜex>0.Subsequently, in 1997, Chaudhary et al. [17] developed the beta function extension(2)Bϱx,y=∫01ℓx−11−ℓy−1e−ϱ/ℓ1−ℓdℓ,where ℜeϱ>0,ℜex>0,ℜey>0.Further, Chaudhary et al. [3] used the extended beta function (2) to present the following extension of the hypergeometric function:(3)Fϱa´,b´;c´;ϑ=∑n=0∞a´nBϱb´+n,c´−b´Bb´,c´−b´ϑnn!,where ℜeϱ≥0;ϑ<1;minℜea′,ℜec′>ℜeb′>0 and(4)φϱb´;c´;ϑ=∑n=0∞Bϱb´+n,c´−b´Bb´,c´−b´ϑnn!,where ℜeϱ≥0;ϑ<1;ℜec′>ℜeb′>0.The Pochhammer symbolσn is defined (for σ,n∈C), in terms of the familiar gamma function Γ, by (see, e.g., [17], p. 2 and p. 5)(5)σn≔Γσ+nΓσ,σ+n∈C\Z0−=1n=0σσ+1…σ+n−1n∈N.Here, letC,Z0−, and N denote the sets of complex numbers, nonpositive integers, and positive integers, respectively.Ozergin et al. [11] described the generalization of the beta function such as(6)Bϱσ,δx,y=∫01ℓx−11−ℓy−1F11σ;δ;−ϱℓ1−ℓdℓ.Chand et al. [18] recently described the latest extension of the beta function as follows:(7)BκMCx,y=∫01ℓx−11−ℓy−1eκℓ1−ℓdℓ,where ℜex>0,ℜey>0,κ∈C;κ<K and {K∈R+ and not greater than 2.033 5}.In this paper, we aim to introduce and investigate extended beta function, extended hypergeometric function, and extended confluent hypergeometric function. To do this, we have divided the paper into three portions. The first part discusses the introductory segment that has been documented by many researchers. In the second part, we gain Mellin transformations, summation relationships, beta distributions, integral representations of hypergeometric and confluent hypergeometric functions, differentiation formulas, and transformation formulas incorporating this new extended beta function. In the last part, we develop a novel property-rich extension of the fractional derivative Riemann–Liouville operator.


## 2. New Extended Beta Function and Its Properties

In this segment, we propose new generalization of classical beta function as follows:(8)Bσ,δ;κMCx,y=∫01ℓx−11−ℓy−1F11σ;δ;κℓ1−ℓdℓ,where {ℜeσ>0,ℜeδ>0,κ∈C,κ<K;K∈R+ and less than 2.0335}.Ifκ=0 and σ=δ, then equation (8) reduces to classical beta function.


### 2.1. Main Result

#### 2.1.1. Melline Transform Representation of the New Generalisation of Beta Function

Theorem 1.
Ifℜex−τ>0,ℜey−τ>0,ℜeσ>ℜeδ>ℜeτ>0,κ∈C and κ<K;K∈R+ and less than 2.0335, then(9)∫0∞κτ−1Bσ,δ;κMCx,ydκ=−1τΓτΓδΓσ−δΓσΓδ−τBx−τ,y−τ.Proof.
Taking the left hand side and using equation (8), we have(10)∫0∞κτ−1Bσ,δ;κMCx,ydκ=∫0∞κτ−1∫01ℓx−11−ℓy−1F11σ;δ;κℓ1−ℓdℓdκ.
Now, interchanging the order of integration of equation (10) and forming uniform convergence, we have(11)=∫01ℓx−11−ℓy−1∫0∞κτ−1F11σ;δ;κℓ1−ℓdκdℓ.
Substitutingκℓ1−ℓ=−ς in the above expression, we get(12)=∫01ℓx−11−ℓy−1∫0∞−ςℓ1−ℓτ−1F11σ;δ;−ς−dςℓ1−ℓdℓ=∫01ℓx−τ−11−ℓy−τ−1−1τ∫0∞ςτ−1F11σ;δ;−ςdςdℓ.
Using result (page no. 821, [5]), we obtain(13)=−1τΓτΓδΓσ−δΓσΓδ−τBx−τ,y−τ.Theorem 2.
For the new generalized beta function, we have the subsequent integral representation of new classical beta function. Ifℜex>0,ℜey>0,ℜeσ>ℜeδ>0,κ∈C, and κ<K;K∈R+ and less than 2.0335, then(14)Bσ,δ;κMCx,y=2∫0π/2cos2x−1θsin2y−1θF11σ;δ;κcos2θsin2θdθ,(15)Bσ,δ;κMCx,y=∫0∞ςx−11+ςx+yF11σ;δ;κς1+ς2dς,(16)Bσ,δ;κMCx,y=∫0∞ςy−11+ςx+yF11σ;δ;κς1+ς2dς,(17)Bσ,δ;κMCx,y=12∫0∞ςx−1+ςy−11+ςx+yF11σ;δ;κς1+ς2dς,(18)Bσ,δ;κMCx,y=c′−a′1−x−y∫a′c′ς−a′x−1c′−ςy−1F11σ;δ;κς−a′c′−ςc′−a′2dς,(19)Bσ,δ;κMCx,y=21−x−y∫−111+ςx−11−ςy−1F11σ;δ;κ1−ς24dς,(20)Bσ,δ;κMCx,y=21−x−y∫−∞∞ex−yℓF11σ;δ;κ4cosh2ℓ1coshℓx+ydℓ.Proof: .
Equations (14) and (15) can be easily obtained by substituting ℓ=cos2θ and ℓ=ς/1+ς in equation (8). Equation (16) can be obtained by interchanging x and y; equation (17) can be obtained by adding equations (15) and (16) with some simplifications. Equation (18) can be achieved by putting ℓ=ς−a′/c′−a′ in equation (8), and replacing a′=−1, c′=1 in equation (18), we get equation (19) and finally for the result given in equation (20), it can be obtained by putting ς=tanhℓ and using tanhℓ=eℓ−e−ℓ/eℓ+e−ℓ in equation (19).
Throughout the paper, we consider the conditionℜeσ>ℜeδ>0,κ∈C, and κ<K;K∈R+ and less than 2.033 5.


#### 2.1.2. Summation Relation

Theorem 3.
The new generalized beta function: ifℜex>0,ℜey>0, then we have the subsequent functional relation:(21)Bσ,δ;κMCx,y+1+Bσ,δ;κMCx+1,y=Bσ,δ;κMCx,y.Proof: .
Taking the left hand side of equation (21), we get(22)Bσ,δ;κMCx,y+1+Bσ,δ;κMCx+1,y=∫01ℓx−11−ℓy+1−1F11σ;δ;κℓ1−ℓdℓ+∫01ℓx+1−11−ℓy−1F11σ;δ;κℓ1−ℓdℓ=∫01ℓx−11−ℓy+ℓx1−ℓy−1F11σ;δ;κℓ1−ℓdℓ=∫01ℓx−11−ℓy−1F11σ;δ;κℓ1−ℓdℓ,which is a required result.Remark 1.
If we chooseκ=0 and σ=δ, in the above equation (21), then it will convert into the classical beta function.Theorem 4.
Ifℜex>0,ℜey>0, then we have the subsequent summation relation:(23)Bσ,δ;κMCx,1−y=∑n=0∞ynn!Bσ,δ;κMCx+n,1.Proof.
We start with the left hand side of equation (23) with the help of equation (8),(24)Bσ,δ;κMCx,1−y=∫01ℓx−11−ℓ1−y−1F11σ;δ;κℓ1−ℓdℓ=∫01ℓx−11−ℓ−yF11σ;δ;κℓ1−ℓdℓ.
Using the binomial expansion1−ℓ−y=∑n=0∞ynℓn/n!,ℓ<1 in the above expression, and switching the order of summation and integration, we attain(25)Bσ,δ;κMCx,1−y=∑n=0∞ynn!∫01ℓx+n−1F11σ;δ;κℓ1−ℓdℓ=∑n=0∞ynn!Bσ,δ;κMCx+n,1.Theorem 5.
Ifℜex>0,ℜey>0, we have the subsequent summation relation:(26)Bσ,δ;κMCx,y=∑n=0∞Bσ,δ;κMCx+n,1+y.Proof: .
Considering the binomial series expansion,(27)1−ℓy−1=1−ℓy1−ℓ−1=1−ℓy∑n=0∞ℓn,ℓ<1,using the above result in equation (8) and interchanging the summation and integration, then we have(28)Bσ,δ;κMCx,y=∫01ℓx+n−11−ℓy+1−1F11σ;δ;κℓ1−ℓdℓ=∑n=0∞Bσ,δ;κMCx+n,y+1.


#### 2.1.3. The Beta Distribution of Extended Beta FunctionBσ,δ;κMCx,y

The specification of the new generalized beta function is the beta distribution in statistics. We are now describing new generalized beta distributions(29)fℓ=1Bσ,δ;κMCx,yℓx−11−ℓy−1F11σ;δ;κℓ1−ℓ0<ℓ<10otherwise.The generalized beta distribution is with parametersx and y such that −∞<x<∞,−∞<y<∞ and ℜeσ>0,ℜeδ>0,κ<K;K∈ℜ+ and less than 2.0335 will be applied to a random variable X with a probability density function (pdf) assumed by fℓ.Ifυ is any real number, then(30)EXυ=Bσ,δ;κMCx+υ,yBσ,δ;κMCx,y,and for υ=1, mean of the distribution is(31)μ=EX=Bσ,δ;κMCx+1,yBσ,δ;κMCx,y,and variance of the distribution is(32)σ2=EX2−EX2=Bσ,δ;κMCx+2,yBσ,δ:κMCx,y−Bσ,δ:κMCx+1,yBσ,δ;κMCx,y2=Bσ,δ;κMCx,yBσ,δ;κMCx+2,y−Bσ,δ;κMCx+1,y2Bσ,δ;κMCx,y2.The moment generating function of the distribution is(33)Mxℓ=∑n=0∞ℓnn!EXn=1Bσ,δ:κMCx,y∑n=0∞Bσ,δ:κMCx+n,yℓnn!,and cumulative frequency distribution of fℓ can be expressed as(34)Fx=Bσ,δ:κ,ϑMCx,yBσ,δ:κMCx,y,where(35)Bσ,δ;κ,ϑMCx,y=∫0ϑℓx−11−ℓy−1F11σ;δ;κℓ1−ℓdℓ,ℜeσ>0,ℜeδ>0,κ∈C,κ<K;K∈ℜ+,−∞<x<∞,−∞<y<∞,which is the new extended incomplete beta function, for κ=0,x,y>0, equation (35) convergence, and Bσ,δ;0,ϑMCx,y=Bϑx,y, where Bϑx,y is the incomplete beta function (page no. 960, [5]) demarcated by(36)∫0ϑℓx−11−ℓy−1dℓ=Bϑx,y=Bσ,δ;0,ϑMCx,y=ϑxxF21x;1−y;x+1;ϑ.It is also given by the series(37)=ϑx∑n=0∞1−ynϑnn!x+n,where 1−yn is the Pochhammer symbol. It is to be noted that the problem of stating Bσ,δ;κ,ϑMCx,y in the form of other special functions leftovers open. Apparently, this distribution should be convenient in extending the statistical results for variables that are strictly positive to deal with those variables that can take large negative values arbitrarily.


### 2.2. Graphical Representation betweenMx and ℓ

Figure1 describes the graphical form of equation (33) by using different values of κ within the range (−2, 2) and some fixed values of x,y,σ,δ. We can see that the graph is decreasing in nature. The different values between Mxℓ and ℓ,κ are given in Table 1.Figure 1
Curve betweenℓ and M.G.F with different value of κ with different values with some suitable assumption x=4,y=5,σ=2,δ=3;κ<2.0335.Table 1
Tabular representation ofMxℓ and ℓ,κ.
ℓ↓Mxℓκ⟶−2−1.4−0.8−0.20.41.01.6−1000190.189210.91236.212261.006313.452372.309460.187−700144.256159.945179.103197.876237.586282.15348.687−40089.435 799.120 1110.946122.534147.046174.555215.627−10024.855 427.467 630.657 333.78340.394 847.814 758.893 1200−50.420 4−56.050 7−62.925 7−69.662 7−83.913 6−99.906 3−123.784500−137.396−152.548−171.061−189.182−227.536−270.677−334.84800−237.147−263.219−295.057−326.254−326.254−466.309−576.884


#### 2.2.1. Generalised Gauss Hypergeometric and Confluent Hypergeometric Function

Here, we define the generalized hypergeometric function and confluent hypergeometric function with the help of new generalization of classical beta function given in equation (8).(38)Fσ,δ;κMCa′,b′;c′;ϑ=∑n=0∞a′nBσ,δ;κMCb′+n,c′−b′Bb′,c′−b′ϑnn!,(39)φσ,δ;κMCb′;c′;ϑ=∑n=0∞Bσ,δ;κMCb′+n,c′−b′Bb′,c′−b′ϑnn!,where ϑ<1,ℜea′>0,ℜec′>ℜeb′>0, then(40)Fσ,σ;κMCa′,b′;c′;ϑ=FκMCa′,b′;c′;ϑ,Fσ,σ;0MCa′,b′;c′;ϑ=F21a′,b′;c′;ϑ,φσ,σ;κMCb′;c′;ϑ=φκMCb′;c′;ϑ,φσ,σ;0MCb′;c′;ϑ=φ11b′;c′;ϑ.


#### 2.2.2. Integral Representation of Gauss and Confluent Hypergeometric Function by Using New Generalization of Beta Function

Theorem 6.
Ifarg1−ϑ<π, ℜea′>0,ℜec′>ℜeb′>0, then for the new generalized Gauss hypergeometric function Fσ,δ;κMCa′,b′;c′;ϑ, we have the subsequent integral representation demarcated as(41)Fσ,δ;κMCa′,b′;c′;ϑ=1Bb′,c′−b′∫01ℓb′−11−ℓc′−b′−1F11σ;δ;κℓ1−ℓ1−ϑℓ−a′dℓ,(42)Fσ,δ;κMCa′,b′;c′;ϑ=1Bb′,c′−b′∫0∞ςb′−11+ςa′−c′F11σ;δ;κς1+ς21+ς1−ϑ−a′dς,(43)Fσ,δ;κMCa′,b′;c′;ϑ=2Bb′,c′−b′∫0π/2sin2b′−1θcos2c′−2b′−1θ1−ϑsin2θ−a′F11σ;δ;κsin2θcos2θdθ.Proof: .
Considering the left hand side of equation (41) by means of equations (38) and (8), we yield(44)Fσ,δ;κMCa′,b′;c′;ϑ=1Bb′,c′−b′∑n=0∞a′n∫01ℓb′+n−11−ℓc′−b′−1F11σ;δ;κℓ1−ℓϑnn!dℓ.
Changing the order of integration and summation with some simplifications, we attain the right hand side of equation (41).
Further, puttingς=ℓ/1−ℓ in (41), we get(45)=1Bb′,c′−b′∫0∞ςb′−11+ςa′−c′F11σ;δ;κς1+ς21+ς1−ϑ−a′dς.
Again, if we chooseℓ=sin2θ in equation (41), we have(46)Fσ,δ;κMCa′,b′;c′;ϑ=2Bb′,c′−b′∫0π/2sin2b′−1θcos2c′−2b′−1θ1−ϑsin2θ−a′F11σ;δ;κsin2θcos2θdθ.Remark 2.
By choosingϑ=1 in equation (41), we have the subsequent relation between new generalization of Gauss hypergeometric function(47)Fσ,δ;κMCa′,b′;c′;1=1Bb′,c′−b′∫01ℓb′−11−ℓc′−a′−b′−1F11σ;δ;κℓ1−ℓdℓ=1Bb′,c′−b′Bσ,δ;κMCb′,c′−a′−b′.Theorem 7.
Ifℜec′>ℜeb′>0, then new generalization of confluent hypergeometric function φσ,δ;κMCb′;c′;ϑ is defined as follows:(48)φσ,δ;κMCb′;c′;ϑ=1Bb′,c′−b′∫01ℓb′−11−ℓc′−b′−1expϑℓF11σ;δ;κℓ1−ℓdℓ,(49)φσ,δ;κMCb′;c′;ϑ=1Bb′,c′−b′∫0∞ςb′−11+ς−c′expκς1+ςF11σ;δ;κς1+ς2dς,(50)φσ,δ;κMCb′;c′;ϑ=2Bb′,c′−b′∫0π/2sin2b′−1θcos2c′−2b′−1θexpϑsin2θF11σ;δ;κsin2θcos2θdθ.Proof: .
Considering the left hand side of (48), by using equation (39) and equation (8), we yield(51)φσ,δ;κMCb′;c′;ϑ=1Bb′,c′−b′∑n=0∞∫01ℓb′+n−11−ℓc′−b′−1F11σ;δ;κℓ1−ℓϑnn!dℓ.
Changing the order of integration and summation, we obtain the required result (48). By replacing ℓ=ς/1+ς and ℓ=sin2θ separately in equation (48) and simplifying it, we obtain equations (49) and (50).


#### 2.2.3. Differentiation Formulas on New Extension of Beta Function

To represent the new generalization of Gauss hypergeometric function and confluent hypergeometric function, we use the following conditions:(52)Bb′,c′−b′=c′b′Bb′+1,c′−b′,a′n+1=a′a′+1n.Theorem 8.
Ifℜea′>0,ℜec′>ℜeb′>0, then we have the following differential formula:(53)dndϑnFσ,δ;κMCa′,b′;c′;ϑ=a′nb′nc′nFσ,δ;κMCa′+n,b′+n;c′+n;ϑ.Proof: .
Taking the derivative ofFσ,δ;κMCa′,b′;c′;ϑ with respect to ϑ, we have(54)ddϑFσ,δ;κMCa′,b′;c′;ϑ=∑n=0∞a′nBσ,δ;κMCb′+n,c′−b′Bb′,c′−b′ddϑϑnn!=∑n=1∞a′nBσ,δ;κMCb′+n,c′−b′Bb′,c′−b′ϑn−1n−1!.
Replacingn⟶n+1 in the above equation (54), then it will reduce to(55)ddϑFσ,δ;κMCa′,b′;c′;ϑ=a′b′c′∑n=0∞a′+1nBσ,δ;κMCb′+n+1,c′−b′Bb′+1,c′−b′ϑnn!=a′b′c′Fσ,δ;κMCa+1′,b+1′;c+1′;ϑ.
Recursive application of this technique provides us the general form of equation (53).Theorem 9.
For confluent hypergeometric function, we have the following differential formula:(56)dndϑnφσ,δ;κMCb′;c′;ϑ=b′nc′nφσ,δ;κMCb′+n;c′+n;ϑ.Proof: .
Proof is same as in the above theorem.


#### 2.2.4. Transformation Formula on New Extension of Beta Function

Theorem 10.
For the new generalization of Gauss hypergeometric function with the conditionϑ<π;ℜec′>ℜeb′>0, we have the subsequent transformation formula:(57)Fσ,δ;κMCa′,b′;c′;ϑ=1−ϑ−a′Fσ,δ;κMCa′,c′−b′;b′;ϑϑ−1,(58)Fσ,δ;κMCa′,b′;c′;1−1ϑ=ϑa′Fσ,δ;κMCa′,c′−b′;b′;1−ϑ,(59)Fσ,δ;κMCa′,b′;c′;ϑ1+ϑ=1+ϑa′Fσ,δ;κMCa′,c′−b′;b′;ϑ.Proof: .
Using the expression,(60)1−ϑ1−ℓ−a′=1−ϑ−a′1+ϑ1−ϑℓ−a′,and replacing ℓ⟶1−ℓ in equation (41), we obtain(61)Fσ,δ;κMCa′,b′;c′;ϑ=1Bb′,c′−b′∫01ℓb′−11−ℓc′−b′−11−ϑ1−ℓ−a′F11σ;δ;κℓ1−ℓdℓ=1−ϑ−a′Bb′,c′−b′∫01tb′−11−ℓc′−b′−11−ϑϑ−1ℓ−a′F11σ;δ;κℓ1−ℓdℓ.
Hence,(62)Fσ,δ;κMCa′,b′;c′;ϑ=1−ϑ−a′Fσ,δ;κMCa′,c′−b′;b′;ϑϑ−1.
Replacingϑ⟶1−1/ϑ in equation (57), we obtain the result given in equation (58), and again replacing ϑ by ϑ/1+ϑ in equation (57), we get equation (59).


## 2.1. Main Result

### 2.1.1. Melline Transform Representation of the New Generalisation of Beta Function

Theorem 1.
Ifℜex−τ>0,ℜey−τ>0,ℜeσ>ℜeδ>ℜeτ>0,κ∈C and κ<K;K∈R+ and less than 2.0335, then(9)∫0∞κτ−1Bσ,δ;κMCx,ydκ=−1τΓτΓδΓσ−δΓσΓδ−τBx−τ,y−τ.Proof.
Taking the left hand side and using equation (8), we have(10)∫0∞κτ−1Bσ,δ;κMCx,ydκ=∫0∞κτ−1∫01ℓx−11−ℓy−1F11σ;δ;κℓ1−ℓdℓdκ.
Now, interchanging the order of integration of equation (10) and forming uniform convergence, we have(11)=∫01ℓx−11−ℓy−1∫0∞κτ−1F11σ;δ;κℓ1−ℓdκdℓ.
Substitutingκℓ1−ℓ=−ς in the above expression, we get(12)=∫01ℓx−11−ℓy−1∫0∞−ςℓ1−ℓτ−1F11σ;δ;−ς−dςℓ1−ℓdℓ=∫01ℓx−τ−11−ℓy−τ−1−1τ∫0∞ςτ−1F11σ;δ;−ςdςdℓ.
Using result (page no. 821, [5]), we obtain(13)=−1τΓτΓδΓσ−δΓσΓδ−τBx−τ,y−τ.Theorem 2.
For the new generalized beta function, we have the subsequent integral representation of new classical beta function. Ifℜex>0,ℜey>0,ℜeσ>ℜeδ>0,κ∈C, and κ<K;K∈R+ and less than 2.0335, then(14)Bσ,δ;κMCx,y=2∫0π/2cos2x−1θsin2y−1θF11σ;δ;κcos2θsin2θdθ,(15)Bσ,δ;κMCx,y=∫0∞ςx−11+ςx+yF11σ;δ;κς1+ς2dς,(16)Bσ,δ;κMCx,y=∫0∞ςy−11+ςx+yF11σ;δ;κς1+ς2dς,(17)Bσ,δ;κMCx,y=12∫0∞ςx−1+ςy−11+ςx+yF11σ;δ;κς1+ς2dς,(18)Bσ,δ;κMCx,y=c′−a′1−x−y∫a′c′ς−a′x−1c′−ςy−1F11σ;δ;κς−a′c′−ςc′−a′2dς,(19)Bσ,δ;κMCx,y=21−x−y∫−111+ςx−11−ςy−1F11σ;δ;κ1−ς24dς,(20)Bσ,δ;κMCx,y=21−x−y∫−∞∞ex−yℓF11σ;δ;κ4cosh2ℓ1coshℓx+ydℓ.Proof: .
Equations (14) and (15) can be easily obtained by substituting ℓ=cos2θ and ℓ=ς/1+ς in equation (8). Equation (16) can be obtained by interchanging x and y; equation (17) can be obtained by adding equations (15) and (16) with some simplifications. Equation (18) can be achieved by putting ℓ=ς−a′/c′−a′ in equation (8), and replacing a′=−1, c′=1 in equation (18), we get equation (19) and finally for the result given in equation (20), it can be obtained by putting ς=tanhℓ and using tanhℓ=eℓ−e−ℓ/eℓ+e−ℓ in equation (19).
Throughout the paper, we consider the conditionℜeσ>ℜeδ>0,κ∈C, and κ<K;K∈R+ and less than 2.033 5.


### 2.1.2. Summation Relation

Theorem 3.
The new generalized beta function: ifℜex>0,ℜey>0, then we have the subsequent functional relation:(21)Bσ,δ;κMCx,y+1+Bσ,δ;κMCx+1,y=Bσ,δ;κMCx,y.Proof: .
Taking the left hand side of equation (21), we get(22)Bσ,δ;κMCx,y+1+Bσ,δ;κMCx+1,y=∫01ℓx−11−ℓy+1−1F11σ;δ;κℓ1−ℓdℓ+∫01ℓx+1−11−ℓy−1F11σ;δ;κℓ1−ℓdℓ=∫01ℓx−11−ℓy+ℓx1−ℓy−1F11σ;δ;κℓ1−ℓdℓ=∫01ℓx−11−ℓy−1F11σ;δ;κℓ1−ℓdℓ,which is a required result.Remark 1.
If we chooseκ=0 and σ=δ, in the above equation (21), then it will convert into the classical beta function.Theorem 4.
Ifℜex>0,ℜey>0, then we have the subsequent summation relation:(23)Bσ,δ;κMCx,1−y=∑n=0∞ynn!Bσ,δ;κMCx+n,1.Proof.
We start with the left hand side of equation (23) with the help of equation (8),(24)Bσ,δ;κMCx,1−y=∫01ℓx−11−ℓ1−y−1F11σ;δ;κℓ1−ℓdℓ=∫01ℓx−11−ℓ−yF11σ;δ;κℓ1−ℓdℓ.
Using the binomial expansion1−ℓ−y=∑n=0∞ynℓn/n!,ℓ<1 in the above expression, and switching the order of summation and integration, we attain(25)Bσ,δ;κMCx,1−y=∑n=0∞ynn!∫01ℓx+n−1F11σ;δ;κℓ1−ℓdℓ=∑n=0∞ynn!Bσ,δ;κMCx+n,1.Theorem 5.
Ifℜex>0,ℜey>0, we have the subsequent summation relation:(26)Bσ,δ;κMCx,y=∑n=0∞Bσ,δ;κMCx+n,1+y.Proof: .
Considering the binomial series expansion,(27)1−ℓy−1=1−ℓy1−ℓ−1=1−ℓy∑n=0∞ℓn,ℓ<1,using the above result in equation (8) and interchanging the summation and integration, then we have(28)Bσ,δ;κMCx,y=∫01ℓx+n−11−ℓy+1−1F11σ;δ;κℓ1−ℓdℓ=∑n=0∞Bσ,δ;κMCx+n,y+1.


### 2.1.3. The Beta Distribution of Extended Beta FunctionBσ,δ;κMCx,y

The specification of the new generalized beta function is the beta distribution in statistics. We are now describing new generalized beta distributions(29)fℓ=1Bσ,δ;κMCx,yℓx−11−ℓy−1F11σ;δ;κℓ1−ℓ0<ℓ<10otherwise.The generalized beta distribution is with parametersx and y such that −∞<x<∞,−∞<y<∞ and ℜeσ>0,ℜeδ>0,κ<K;K∈ℜ+ and less than 2.0335 will be applied to a random variable X with a probability density function (pdf) assumed by fℓ.Ifυ is any real number, then(30)EXυ=Bσ,δ;κMCx+υ,yBσ,δ;κMCx,y,and for υ=1, mean of the distribution is(31)μ=EX=Bσ,δ;κMCx+1,yBσ,δ;κMCx,y,and variance of the distribution is(32)σ2=EX2−EX2=Bσ,δ;κMCx+2,yBσ,δ:κMCx,y−Bσ,δ:κMCx+1,yBσ,δ;κMCx,y2=Bσ,δ;κMCx,yBσ,δ;κMCx+2,y−Bσ,δ;κMCx+1,y2Bσ,δ;κMCx,y2.The moment generating function of the distribution is(33)Mxℓ=∑n=0∞ℓnn!EXn=1Bσ,δ:κMCx,y∑n=0∞Bσ,δ:κMCx+n,yℓnn!,and cumulative frequency distribution of fℓ can be expressed as(34)Fx=Bσ,δ:κ,ϑMCx,yBσ,δ:κMCx,y,where(35)Bσ,δ;κ,ϑMCx,y=∫0ϑℓx−11−ℓy−1F11σ;δ;κℓ1−ℓdℓ,ℜeσ>0,ℜeδ>0,κ∈C,κ<K;K∈ℜ+,−∞<x<∞,−∞<y<∞,which is the new extended incomplete beta function, for κ=0,x,y>0, equation (35) convergence, and Bσ,δ;0,ϑMCx,y=Bϑx,y, where Bϑx,y is the incomplete beta function (page no. 960, [5]) demarcated by(36)∫0ϑℓx−11−ℓy−1dℓ=Bϑx,y=Bσ,δ;0,ϑMCx,y=ϑxxF21x;1−y;x+1;ϑ.It is also given by the series(37)=ϑx∑n=0∞1−ynϑnn!x+n,where 1−yn is the Pochhammer symbol. It is to be noted that the problem of stating Bσ,δ;κ,ϑMCx,y in the form of other special functions leftovers open. Apparently, this distribution should be convenient in extending the statistical results for variables that are strictly positive to deal with those variables that can take large negative values arbitrarily.


## 2.1.1. Melline Transform Representation of the New Generalisation of Beta Function

Theorem 1.
Ifℜex−τ>0,ℜey−τ>0,ℜeσ>ℜeδ>ℜeτ>0,κ∈C and κ<K;K∈R+ and less than 2.0335, then(9)∫0∞κτ−1Bσ,δ;κMCx,ydκ=−1τΓτΓδΓσ−δΓσΓδ−τBx−τ,y−τ.Proof.
Taking the left hand side and using equation (8), we have(10)∫0∞κτ−1Bσ,δ;κMCx,ydκ=∫0∞κτ−1∫01ℓx−11−ℓy−1F11σ;δ;κℓ1−ℓdℓdκ.
Now, interchanging the order of integration of equation (10) and forming uniform convergence, we have(11)=∫01ℓx−11−ℓy−1∫0∞κτ−1F11σ;δ;κℓ1−ℓdκdℓ.
Substitutingκℓ1−ℓ=−ς in the above expression, we get(12)=∫01ℓx−11−ℓy−1∫0∞−ςℓ1−ℓτ−1F11σ;δ;−ς−dςℓ1−ℓdℓ=∫01ℓx−τ−11−ℓy−τ−1−1τ∫0∞ςτ−1F11σ;δ;−ςdςdℓ.
Using result (page no. 821, [5]), we obtain(13)=−1τΓτΓδΓσ−δΓσΓδ−τBx−τ,y−τ.Theorem 2.
For the new generalized beta function, we have the subsequent integral representation of new classical beta function. Ifℜex>0,ℜey>0,ℜeσ>ℜeδ>0,κ∈C, and κ<K;K∈R+ and less than 2.0335, then(14)Bσ,δ;κMCx,y=2∫0π/2cos2x−1θsin2y−1θF11σ;δ;κcos2θsin2θdθ,(15)Bσ,δ;κMCx,y=∫0∞ςx−11+ςx+yF11σ;δ;κς1+ς2dς,(16)Bσ,δ;κMCx,y=∫0∞ςy−11+ςx+yF11σ;δ;κς1+ς2dς,(17)Bσ,δ;κMCx,y=12∫0∞ςx−1+ςy−11+ςx+yF11σ;δ;κς1+ς2dς,(18)Bσ,δ;κMCx,y=c′−a′1−x−y∫a′c′ς−a′x−1c′−ςy−1F11σ;δ;κς−a′c′−ςc′−a′2dς,(19)Bσ,δ;κMCx,y=21−x−y∫−111+ςx−11−ςy−1F11σ;δ;κ1−ς24dς,(20)Bσ,δ;κMCx,y=21−x−y∫−∞∞ex−yℓF11σ;δ;κ4cosh2ℓ1coshℓx+ydℓ.Proof: .
Equations (14) and (15) can be easily obtained by substituting ℓ=cos2θ and ℓ=ς/1+ς in equation (8). Equation (16) can be obtained by interchanging x and y; equation (17) can be obtained by adding equations (15) and (16) with some simplifications. Equation (18) can be achieved by putting ℓ=ς−a′/c′−a′ in equation (8), and replacing a′=−1, c′=1 in equation (18), we get equation (19) and finally for the result given in equation (20), it can be obtained by putting ς=tanhℓ and using tanhℓ=eℓ−e−ℓ/eℓ+e−ℓ in equation (19).
Throughout the paper, we consider the conditionℜeσ>ℜeδ>0,κ∈C, and κ<K;K∈R+ and less than 2.033 5.


## 2.1.2. Summation Relation

Theorem 3.
The new generalized beta function: ifℜex>0,ℜey>0, then we have the subsequent functional relation:(21)Bσ,δ;κMCx,y+1+Bσ,δ;κMCx+1,y=Bσ,δ;κMCx,y.Proof: .
Taking the left hand side of equation (21), we get(22)Bσ,δ;κMCx,y+1+Bσ,δ;κMCx+1,y=∫01ℓx−11−ℓy+1−1F11σ;δ;κℓ1−ℓdℓ+∫01ℓx+1−11−ℓy−1F11σ;δ;κℓ1−ℓdℓ=∫01ℓx−11−ℓy+ℓx1−ℓy−1F11σ;δ;κℓ1−ℓdℓ=∫01ℓx−11−ℓy−1F11σ;δ;κℓ1−ℓdℓ,which is a required result.Remark 1.
If we chooseκ=0 and σ=δ, in the above equation (21), then it will convert into the classical beta function.Theorem 4.
Ifℜex>0,ℜey>0, then we have the subsequent summation relation:(23)Bσ,δ;κMCx,1−y=∑n=0∞ynn!Bσ,δ;κMCx+n,1.Proof.
We start with the left hand side of equation (23) with the help of equation (8),(24)Bσ,δ;κMCx,1−y=∫01ℓx−11−ℓ1−y−1F11σ;δ;κℓ1−ℓdℓ=∫01ℓx−11−ℓ−yF11σ;δ;κℓ1−ℓdℓ.
Using the binomial expansion1−ℓ−y=∑n=0∞ynℓn/n!,ℓ<1 in the above expression, and switching the order of summation and integration, we attain(25)Bσ,δ;κMCx,1−y=∑n=0∞ynn!∫01ℓx+n−1F11σ;δ;κℓ1−ℓdℓ=∑n=0∞ynn!Bσ,δ;κMCx+n,1.Theorem 5.
Ifℜex>0,ℜey>0, we have the subsequent summation relation:(26)Bσ,δ;κMCx,y=∑n=0∞Bσ,δ;κMCx+n,1+y.Proof: .
Considering the binomial series expansion,(27)1−ℓy−1=1−ℓy1−ℓ−1=1−ℓy∑n=0∞ℓn,ℓ<1,using the above result in equation (8) and interchanging the summation and integration, then we have(28)Bσ,δ;κMCx,y=∫01ℓx+n−11−ℓy+1−1F11σ;δ;κℓ1−ℓdℓ=∑n=0∞Bσ,δ;κMCx+n,y+1.


## 2.1.3. The Beta Distribution of Extended Beta FunctionBσ,δ;κMCx,y

The specification of the new generalized beta function is the beta distribution in statistics. We are now describing new generalized beta distributions(29)fℓ=1Bσ,δ;κMCx,yℓx−11−ℓy−1F11σ;δ;κℓ1−ℓ0<ℓ<10otherwise.The generalized beta distribution is with parametersx and y such that −∞<x<∞,−∞<y<∞ and ℜeσ>0,ℜeδ>0,κ<K;K∈ℜ+ and less than 2.0335 will be applied to a random variable X with a probability density function (pdf) assumed by fℓ.Ifυ is any real number, then(30)EXυ=Bσ,δ;κMCx+υ,yBσ,δ;κMCx,y,and for υ=1, mean of the distribution is(31)μ=EX=Bσ,δ;κMCx+1,yBσ,δ;κMCx,y,and variance of the distribution is(32)σ2=EX2−EX2=Bσ,δ;κMCx+2,yBσ,δ:κMCx,y−Bσ,δ:κMCx+1,yBσ,δ;κMCx,y2=Bσ,δ;κMCx,yBσ,δ;κMCx+2,y−Bσ,δ;κMCx+1,y2Bσ,δ;κMCx,y2.The moment generating function of the distribution is(33)Mxℓ=∑n=0∞ℓnn!EXn=1Bσ,δ:κMCx,y∑n=0∞Bσ,δ:κMCx+n,yℓnn!,and cumulative frequency distribution of fℓ can be expressed as(34)Fx=Bσ,δ:κ,ϑMCx,yBσ,δ:κMCx,y,where(35)Bσ,δ;κ,ϑMCx,y=∫0ϑℓx−11−ℓy−1F11σ;δ;κℓ1−ℓdℓ,ℜeσ>0,ℜeδ>0,κ∈C,κ<K;K∈ℜ+,−∞<x<∞,−∞<y<∞,which is the new extended incomplete beta function, for κ=0,x,y>0, equation (35) convergence, and Bσ,δ;0,ϑMCx,y=Bϑx,y, where Bϑx,y is the incomplete beta function (page no. 960, [5]) demarcated by(36)∫0ϑℓx−11−ℓy−1dℓ=Bϑx,y=Bσ,δ;0,ϑMCx,y=ϑxxF21x;1−y;x+1;ϑ.It is also given by the series(37)=ϑx∑n=0∞1−ynϑnn!x+n,where 1−yn is the Pochhammer symbol. It is to be noted that the problem of stating Bσ,δ;κ,ϑMCx,y in the form of other special functions leftovers open. Apparently, this distribution should be convenient in extending the statistical results for variables that are strictly positive to deal with those variables that can take large negative values arbitrarily.


## 2.2. Graphical Representation betweenMx and ℓ

Figure1 describes the graphical form of equation (33) by using different values of κ within the range (−2, 2) and some fixed values of x,y,σ,δ. We can see that the graph is decreasing in nature. The different values between Mxℓ and ℓ,κ are given in Table 1.Figure 1
Curve betweenℓ and M.G.F with different value of κ with different values with some suitable assumption x=4,y=5,σ=2,δ=3;κ<2.0335.Table 1
Tabular representation ofMxℓ and ℓ,κ.
ℓ↓Mxℓκ⟶−2−1.4−0.8−0.20.41.01.6−1000190.189210.91236.212261.006313.452372.309460.187−700144.256159.945179.103197.876237.586282.15348.687−40089.435 799.120 1110.946122.534147.046174.555215.627−10024.855 427.467 630.657 333.78340.394 847.814 758.893 1200−50.420 4−56.050 7−62.925 7−69.662 7−83.913 6−99.906 3−123.784500−137.396−152.548−171.061−189.182−227.536−270.677−334.84800−237.147−263.219−295.057−326.254−326.254−466.309−576.884


### 2.2.1. Generalised Gauss Hypergeometric and Confluent Hypergeometric Function

Here, we define the generalized hypergeometric function and confluent hypergeometric function with the help of new generalization of classical beta function given in equation (8).(38)Fσ,δ;κMCa′,b′;c′;ϑ=∑n=0∞a′nBσ,δ;κMCb′+n,c′−b′Bb′,c′−b′ϑnn!,(39)φσ,δ;κMCb′;c′;ϑ=∑n=0∞Bσ,δ;κMCb′+n,c′−b′Bb′,c′−b′ϑnn!,where ϑ<1,ℜea′>0,ℜec′>ℜeb′>0, then(40)Fσ,σ;κMCa′,b′;c′;ϑ=FκMCa′,b′;c′;ϑ,Fσ,σ;0MCa′,b′;c′;ϑ=F21a′,b′;c′;ϑ,φσ,σ;κMCb′;c′;ϑ=φκMCb′;c′;ϑ,φσ,σ;0MCb′;c′;ϑ=φ11b′;c′;ϑ.


### 2.2.2. Integral Representation of Gauss and Confluent Hypergeometric Function by Using New Generalization of Beta Function

Theorem 6.
Ifarg1−ϑ<π, ℜea′>0,ℜec′>ℜeb′>0, then for the new generalized Gauss hypergeometric function Fσ,δ;κMCa′,b′;c′;ϑ, we have the subsequent integral representation demarcated as(41)Fσ,δ;κMCa′,b′;c′;ϑ=1Bb′,c′−b′∫01ℓb′−11−ℓc′−b′−1F11σ;δ;κℓ1−ℓ1−ϑℓ−a′dℓ,(42)Fσ,δ;κMCa′,b′;c′;ϑ=1Bb′,c′−b′∫0∞ςb′−11+ςa′−c′F11σ;δ;κς1+ς21+ς1−ϑ−a′dς,(43)Fσ,δ;κMCa′,b′;c′;ϑ=2Bb′,c′−b′∫0π/2sin2b′−1θcos2c′−2b′−1θ1−ϑsin2θ−a′F11σ;δ;κsin2θcos2θdθ.Proof: .
Considering the left hand side of equation (41) by means of equations (38) and (8), we yield(44)Fσ,δ;κMCa′,b′;c′;ϑ=1Bb′,c′−b′∑n=0∞a′n∫01ℓb′+n−11−ℓc′−b′−1F11σ;δ;κℓ1−ℓϑnn!dℓ.
Changing the order of integration and summation with some simplifications, we attain the right hand side of equation (41).
Further, puttingς=ℓ/1−ℓ in (41), we get(45)=1Bb′,c′−b′∫0∞ςb′−11+ςa′−c′F11σ;δ;κς1+ς21+ς1−ϑ−a′dς.
Again, if we chooseℓ=sin2θ in equation (41), we have(46)Fσ,δ;κMCa′,b′;c′;ϑ=2Bb′,c′−b′∫0π/2sin2b′−1θcos2c′−2b′−1θ1−ϑsin2θ−a′F11σ;δ;κsin2θcos2θdθ.Remark 2.
By choosingϑ=1 in equation (41), we have the subsequent relation between new generalization of Gauss hypergeometric function(47)Fσ,δ;κMCa′,b′;c′;1=1Bb′,c′−b′∫01ℓb′−11−ℓc′−a′−b′−1F11σ;δ;κℓ1−ℓdℓ=1Bb′,c′−b′Bσ,δ;κMCb′,c′−a′−b′.Theorem 7.
Ifℜec′>ℜeb′>0, then new generalization of confluent hypergeometric function φσ,δ;κMCb′;c′;ϑ is defined as follows:(48)φσ,δ;κMCb′;c′;ϑ=1Bb′,c′−b′∫01ℓb′−11−ℓc′−b′−1expϑℓF11σ;δ;κℓ1−ℓdℓ,(49)φσ,δ;κMCb′;c′;ϑ=1Bb′,c′−b′∫0∞ςb′−11+ς−c′expκς1+ςF11σ;δ;κς1+ς2dς,(50)φσ,δ;κMCb′;c′;ϑ=2Bb′,c′−b′∫0π/2sin2b′−1θcos2c′−2b′−1θexpϑsin2θF11σ;δ;κsin2θcos2θdθ.Proof: .
Considering the left hand side of (48), by using equation (39) and equation (8), we yield(51)φσ,δ;κMCb′;c′;ϑ=1Bb′,c′−b′∑n=0∞∫01ℓb′+n−11−ℓc′−b′−1F11σ;δ;κℓ1−ℓϑnn!dℓ.
Changing the order of integration and summation, we obtain the required result (48). By replacing ℓ=ς/1+ς and ℓ=sin2θ separately in equation (48) and simplifying it, we obtain equations (49) and (50).


### 2.2.3. Differentiation Formulas on New Extension of Beta Function

To represent the new generalization of Gauss hypergeometric function and confluent hypergeometric function, we use the following conditions:(52)Bb′,c′−b′=c′b′Bb′+1,c′−b′,a′n+1=a′a′+1n.Theorem 8.
Ifℜea′>0,ℜec′>ℜeb′>0, then we have the following differential formula:(53)dndϑnFσ,δ;κMCa′,b′;c′;ϑ=a′nb′nc′nFσ,δ;κMCa′+n,b′+n;c′+n;ϑ.Proof: .
Taking the derivative ofFσ,δ;κMCa′,b′;c′;ϑ with respect to ϑ, we have(54)ddϑFσ,δ;κMCa′,b′;c′;ϑ=∑n=0∞a′nBσ,δ;κMCb′+n,c′−b′Bb′,c′−b′ddϑϑnn!=∑n=1∞a′nBσ,δ;κMCb′+n,c′−b′Bb′,c′−b′ϑn−1n−1!.
Replacingn⟶n+1 in the above equation (54), then it will reduce to(55)ddϑFσ,δ;κMCa′,b′;c′;ϑ=a′b′c′∑n=0∞a′+1nBσ,δ;κMCb′+n+1,c′−b′Bb′+1,c′−b′ϑnn!=a′b′c′Fσ,δ;κMCa+1′,b+1′;c+1′;ϑ.
Recursive application of this technique provides us the general form of equation (53).Theorem 9.
For confluent hypergeometric function, we have the following differential formula:(56)dndϑnφσ,δ;κMCb′;c′;ϑ=b′nc′nφσ,δ;κMCb′+n;c′+n;ϑ.Proof: .
Proof is same as in the above theorem.


### 2.2.4. Transformation Formula on New Extension of Beta Function

Theorem 10.
For the new generalization of Gauss hypergeometric function with the conditionϑ<π;ℜec′>ℜeb′>0, we have the subsequent transformation formula:(57)Fσ,δ;κMCa′,b′;c′;ϑ=1−ϑ−a′Fσ,δ;κMCa′,c′−b′;b′;ϑϑ−1,(58)Fσ,δ;κMCa′,b′;c′;1−1ϑ=ϑa′Fσ,δ;κMCa′,c′−b′;b′;1−ϑ,(59)Fσ,δ;κMCa′,b′;c′;ϑ1+ϑ=1+ϑa′Fσ,δ;κMCa′,c′−b′;b′;ϑ.Proof: .
Using the expression,(60)1−ϑ1−ℓ−a′=1−ϑ−a′1+ϑ1−ϑℓ−a′,and replacing ℓ⟶1−ℓ in equation (41), we obtain(61)Fσ,δ;κMCa′,b′;c′;ϑ=1Bb′,c′−b′∫01ℓb′−11−ℓc′−b′−11−ϑ1−ℓ−a′F11σ;δ;κℓ1−ℓdℓ=1−ϑ−a′Bb′,c′−b′∫01tb′−11−ℓc′−b′−11−ϑϑ−1ℓ−a′F11σ;δ;κℓ1−ℓdℓ.
Hence,(62)Fσ,δ;κMCa′,b′;c′;ϑ=1−ϑ−a′Fσ,δ;κMCa′,c′−b′;b′;ϑϑ−1.
Replacingϑ⟶1−1/ϑ in equation (57), we obtain the result given in equation (58), and again replacing ϑ by ϑ/1+ϑ in equation (57), we get equation (59).


## 2.2.1. Generalised Gauss Hypergeometric and Confluent Hypergeometric Function

Here, we define the generalized hypergeometric function and confluent hypergeometric function with the help of new generalization of classical beta function given in equation (8).(38)Fσ,δ;κMCa′,b′;c′;ϑ=∑n=0∞a′nBσ,δ;κMCb′+n,c′−b′Bb′,c′−b′ϑnn!,(39)φσ,δ;κMCb′;c′;ϑ=∑n=0∞Bσ,δ;κMCb′+n,c′−b′Bb′,c′−b′ϑnn!,where ϑ<1,ℜea′>0,ℜec′>ℜeb′>0, then(40)Fσ,σ;κMCa′,b′;c′;ϑ=FκMCa′,b′;c′;ϑ,Fσ,σ;0MCa′,b′;c′;ϑ=F21a′,b′;c′;ϑ,φσ,σ;κMCb′;c′;ϑ=φκMCb′;c′;ϑ,φσ,σ;0MCb′;c′;ϑ=φ11b′;c′;ϑ.


## 2.2.2. Integral Representation of Gauss and Confluent Hypergeometric Function by Using New Generalization of Beta Function

Theorem 6.
Ifarg1−ϑ<π, ℜea′>0,ℜec′>ℜeb′>0, then for the new generalized Gauss hypergeometric function Fσ,δ;κMCa′,b′;c′;ϑ, we have the subsequent integral representation demarcated as(41)Fσ,δ;κMCa′,b′;c′;ϑ=1Bb′,c′−b′∫01ℓb′−11−ℓc′−b′−1F11σ;δ;κℓ1−ℓ1−ϑℓ−a′dℓ,(42)Fσ,δ;κMCa′,b′;c′;ϑ=1Bb′,c′−b′∫0∞ςb′−11+ςa′−c′F11σ;δ;κς1+ς21+ς1−ϑ−a′dς,(43)Fσ,δ;κMCa′,b′;c′;ϑ=2Bb′,c′−b′∫0π/2sin2b′−1θcos2c′−2b′−1θ1−ϑsin2θ−a′F11σ;δ;κsin2θcos2θdθ.Proof: .
Considering the left hand side of equation (41) by means of equations (38) and (8), we yield(44)Fσ,δ;κMCa′,b′;c′;ϑ=1Bb′,c′−b′∑n=0∞a′n∫01ℓb′+n−11−ℓc′−b′−1F11σ;δ;κℓ1−ℓϑnn!dℓ.
Changing the order of integration and summation with some simplifications, we attain the right hand side of equation (41).
Further, puttingς=ℓ/1−ℓ in (41), we get(45)=1Bb′,c′−b′∫0∞ςb′−11+ςa′−c′F11σ;δ;κς1+ς21+ς1−ϑ−a′dς.
Again, if we chooseℓ=sin2θ in equation (41), we have(46)Fσ,δ;κMCa′,b′;c′;ϑ=2Bb′,c′−b′∫0π/2sin2b′−1θcos2c′−2b′−1θ1−ϑsin2θ−a′F11σ;δ;κsin2θcos2θdθ.Remark 2.
By choosingϑ=1 in equation (41), we have the subsequent relation between new generalization of Gauss hypergeometric function(47)Fσ,δ;κMCa′,b′;c′;1=1Bb′,c′−b′∫01ℓb′−11−ℓc′−a′−b′−1F11σ;δ;κℓ1−ℓdℓ=1Bb′,c′−b′Bσ,δ;κMCb′,c′−a′−b′.Theorem 7.
Ifℜec′>ℜeb′>0, then new generalization of confluent hypergeometric function φσ,δ;κMCb′;c′;ϑ is defined as follows:(48)φσ,δ;κMCb′;c′;ϑ=1Bb′,c′−b′∫01ℓb′−11−ℓc′−b′−1expϑℓF11σ;δ;κℓ1−ℓdℓ,(49)φσ,δ;κMCb′;c′;ϑ=1Bb′,c′−b′∫0∞ςb′−11+ς−c′expκς1+ςF11σ;δ;κς1+ς2dς,(50)φσ,δ;κMCb′;c′;ϑ=2Bb′,c′−b′∫0π/2sin2b′−1θcos2c′−2b′−1θexpϑsin2θF11σ;δ;κsin2θcos2θdθ.Proof: .
Considering the left hand side of (48), by using equation (39) and equation (8), we yield(51)φσ,δ;κMCb′;c′;ϑ=1Bb′,c′−b′∑n=0∞∫01ℓb′+n−11−ℓc′−b′−1F11σ;δ;κℓ1−ℓϑnn!dℓ.
Changing the order of integration and summation, we obtain the required result (48). By replacing ℓ=ς/1+ς and ℓ=sin2θ separately in equation (48) and simplifying it, we obtain equations (49) and (50).


## 2.2.3. Differentiation Formulas on New Extension of Beta Function

To represent the new generalization of Gauss hypergeometric function and confluent hypergeometric function, we use the following conditions:(52)Bb′,c′−b′=c′b′Bb′+1,c′−b′,a′n+1=a′a′+1n.Theorem 8.
Ifℜea′>0,ℜec′>ℜeb′>0, then we have the following differential formula:(53)dndϑnFσ,δ;κMCa′,b′;c′;ϑ=a′nb′nc′nFσ,δ;κMCa′+n,b′+n;c′+n;ϑ.Proof: .
Taking the derivative ofFσ,δ;κMCa′,b′;c′;ϑ with respect to ϑ, we have(54)ddϑFσ,δ;κMCa′,b′;c′;ϑ=∑n=0∞a′nBσ,δ;κMCb′+n,c′−b′Bb′,c′−b′ddϑϑnn!=∑n=1∞a′nBσ,δ;κMCb′+n,c′−b′Bb′,c′−b′ϑn−1n−1!.
Replacingn⟶n+1 in the above equation (54), then it will reduce to(55)ddϑFσ,δ;κMCa′,b′;c′;ϑ=a′b′c′∑n=0∞a′+1nBσ,δ;κMCb′+n+1,c′−b′Bb′+1,c′−b′ϑnn!=a′b′c′Fσ,δ;κMCa+1′,b+1′;c+1′;ϑ.
Recursive application of this technique provides us the general form of equation (53).Theorem 9.
For confluent hypergeometric function, we have the following differential formula:(56)dndϑnφσ,δ;κMCb′;c′;ϑ=b′nc′nφσ,δ;κMCb′+n;c′+n;ϑ.Proof: .
Proof is same as in the above theorem.


## 2.2.4. Transformation Formula on New Extension of Beta Function

Theorem 10.
For the new generalization of Gauss hypergeometric function with the conditionϑ<π;ℜec′>ℜeb′>0, we have the subsequent transformation formula:(57)Fσ,δ;κMCa′,b′;c′;ϑ=1−ϑ−a′Fσ,δ;κMCa′,c′−b′;b′;ϑϑ−1,(58)Fσ,δ;κMCa′,b′;c′;1−1ϑ=ϑa′Fσ,δ;κMCa′,c′−b′;b′;1−ϑ,(59)Fσ,δ;κMCa′,b′;c′;ϑ1+ϑ=1+ϑa′Fσ,δ;κMCa′,c′−b′;b′;ϑ.Proof: .
Using the expression,(60)1−ϑ1−ℓ−a′=1−ϑ−a′1+ϑ1−ϑℓ−a′,and replacing ℓ⟶1−ℓ in equation (41), we obtain(61)Fσ,δ;κMCa′,b′;c′;ϑ=1Bb′,c′−b′∫01ℓb′−11−ℓc′−b′−11−ϑ1−ℓ−a′F11σ;δ;κℓ1−ℓdℓ=1−ϑ−a′Bb′,c′−b′∫01tb′−11−ℓc′−b′−11−ϑϑ−1ℓ−a′F11σ;δ;κℓ1−ℓdℓ.
Hence,(62)Fσ,δ;κMCa′,b′;c′;ϑ=1−ϑ−a′Fσ,δ;κMCa′,c′−b′;b′;ϑϑ−1.
Replacingϑ⟶1−1/ϑ in equation (57), we obtain the result given in equation (58), and again replacing ϑ by ϑ/1+ϑ in equation (57), we get equation (59).


## 3. New Extension of Riemann–Liouville (R-L) Fractional Derivative Operator

In the past years, different kinds of extensions of R-L fractional operators were performed by many authors (see [19–22]). Here, we announce novel extension of R-L fractional derivative operator using new type of extended beta function and this function is also reducible into classical Riemann–Liouville fractional derivative operator with certain conditions.The classical Riemann fractional derivative offϑ of order ξ defined demarcated by(63)Dϑξfϑ=1Γ−ξ∫0ϑϑ−ℓ−ξ−1fℓdℓ,ℜeξ>0,where the path of integration is a line from 0 to ϑ in the complex ℓ−plane when ℜeξ≥0.Letn∈N be the smallest integer greater than ℜeξ and n−1≤ℜeξ<n. Then,(64)Dϑξ,nfϑ=dndϑnDϑξ−nfϑ=dndϑn1Γn−ξ∫0ϑϑ−ℓn−ξ−1fℓdℓ.Here, we defined new definition of extended Riemann–Liouville (R-L) fractional derivative offϑ of order ξ which is given by(65)Dσ,δ;κMC;ξfϑ=1Γ−ξ∫0ϑϑ−ℓ−ξ−1fℓF11σ;δ;κℓϑ−ℓϑ2dℓ,where {ℜeσ>0,ℜeδ>0,ℜeξ>0;κ<K;K∈ℜ+ and less than 2.0335} and for n−1≤ℜeξ<n,n∈N,κ<K;K∈ℜ+ and less than 2.0335, ℜeσ>0,ℜeδ>0, then(66)Dσ,δ;κMC;ξfϑ=dndϑn1Γn−ξ∫0ϑϑ−ℓn−ξ−1fℓF11σ;δ;κℓϑ−ℓϑ2dℓ.If we putκ=0 in the above equations (65) and (66), then it will become the classical Riemann–Liouville (R-L) fractional derivative equations (63) and (64).Theorem 11.
Letn−1≤ℜeξ<n, for some n∈N,κ<K and ℜeξ<ℜeμ. Then, we have(67)Dσ,δ;κMC;ξϑμ=Bσ,δ;κMCμ+1,−ξΓ−ξϑμ−ξ.Proof: .
We have from equation (65),(68)Dσ,δ:κMC;ξϑμ=1Γ−ξ∫0ϑϑ−ℓ−ξ−1ℓμF11σ;δ;κℓϑ−ℓϑ2dℓ,and putting ℓ=ϑς in the above equation, then we have(69)Dσ,δ;κMC;ξϑμ=ϑ−ξ+μΓ−ξ∫011−ς−ξ−1ςμF11σ;δ;κς1−ςdς=Bσ,δ:κMCμ+1,−ξΓ−ξϑμ−ξ.Theorem 12.
Letn−1≤ℜeμ−ξ<n<ℜeμ,n∈N,ϑ<1,τ∈C,κ<K;K∈ℜ+ and less than 2.0335, ℜeσ>0,ℜeδ>0, then(70)Dσ,δ;κMC;μ−ξϑμ−11−ϑ−τ=ϑξ−1Γξ−μ∫01ςμ−11−ςϑ−τ1−ςξ−μ−1F11σ;δ;κς1−ςdς.Proof: .
Taking the left hand side and using equation (65), we have(71)=1Γ−μ+ξ∫0ϑϑ−ℓ−μ+ξ−1ℓμ−11−ℓ−τF11σ;δ;κℓϑ−ℓϑ2dℓ,and putting ℓ=ϑς, then it becomes(72)Dσ,δ;κMC;λ−ξϑμ−11−ϑ−τ=ϑξ−1Γξ−μ∫01ςμ−11−ςϑ−τ1−ςξ−μ−1F11σ;δ;κς1−ςdς.


### 3.1. Extended Riemann–Liouville (R-L) Derivative to a Functionfϑ Analytic at the Origin

Theorem 13.
Let functionfϑ be analytic at the origin with its power series given by fϑ=∑n=0∞a′nϑn,ϑ<ρ for some ρ∈R+, then(73)Dσ,δ;κMC;ξfϑ=∑n=0∞a′nDσ,δ;κMC;ξϑn.Proof: .
By the definition of extended Riemann–Liouville (R-L) derivative operator and usingfϑ=∑n=0∞a′nϑn in it, we obtain(74)Dσ,δ;κMC;ξfϑ=1Γ−ξ∫0ϑϑ−ℓ−ξ−1∑n=0∞a′nℓnF11σ;δ;κℓϑ−ℓϑ2dℓ.
Changing the order of summation and integration, we have(75)=∑n=0∞a′n1Γ−ξ∫0ϑϑ−ℓ−ξ−1ℓnF11σ;δ;κℓϑ−ℓϑ2dℓ,=∑n=0∞a′nDσ,δ;κMC;ξϑn.Theorem 14.
Let functionfϑ be analytic at the origin with its power series expansion fϑ=∑n=0∞a′nϑn,ϑ<ρ for some ρ∈R+. Then, we have(76)Dσ,δ;κMC;ξϑμ−1fϑ=ϑμ−ξ−1Γ−ξ∑n=0∞a′nϑn∫01ςμ+n−11−ς−ξ−1F11σ;δ;κς1−ςdς.Proof.
By using equation (73), in L.H.S. of the above equation, we have(77)Dσ,δ;κMC;ξϑμ−1fϑ=∑n=0∞a′nDσ,δ;κMC;ξϑμ−1ϑn=∑n=0∞a′n1Γ−ξ∫0ϑϑ−ℓ−ξ−1ℓμ+n−1F11σ;δ;κℓϑ−ℓϑ2dℓ,and putting ℓ=ϑς, then(78)=1Γ−ξ∫01ϑ−ξ+μ−11−ς−ξ−1ςμ−1F11σ;δ;κς1−ς∑n=0∞a′nϑςndς.
The series given in equation (78), ∑n=0∞a′nϑςn, is uniformly convergent in disc ϑ<ρ for 0≤ς≤1 and the integral ∫01ςμ−11−ς−ξ−1F11σ;δ;κς1−ςdς is convergent provided that ℜeμ>0,ℜeξ<0 and κ∈C,κ<K;K∈R+ and less than 2.033 5. Now, interchanging the order of summation and integration, we get the result given in equation (76).


## 3.1. Extended Riemann–Liouville (R-L) Derivative to a Functionfϑ Analytic at the Origin

Theorem 13.
Let functionfϑ be analytic at the origin with its power series given by fϑ=∑n=0∞a′nϑn,ϑ<ρ for some ρ∈R+, then(73)Dσ,δ;κMC;ξfϑ=∑n=0∞a′nDσ,δ;κMC;ξϑn.Proof: .
By the definition of extended Riemann–Liouville (R-L) derivative operator and usingfϑ=∑n=0∞a′nϑn in it, we obtain(74)Dσ,δ;κMC;ξfϑ=1Γ−ξ∫0ϑϑ−ℓ−ξ−1∑n=0∞a′nℓnF11σ;δ;κℓϑ−ℓϑ2dℓ.
Changing the order of summation and integration, we have(75)=∑n=0∞a′n1Γ−ξ∫0ϑϑ−ℓ−ξ−1ℓnF11σ;δ;κℓϑ−ℓϑ2dℓ,=∑n=0∞a′nDσ,δ;κMC;ξϑn.Theorem 14.
Let functionfϑ be analytic at the origin with its power series expansion fϑ=∑n=0∞a′nϑn,ϑ<ρ for some ρ∈R+. Then, we have(76)Dσ,δ;κMC;ξϑμ−1fϑ=ϑμ−ξ−1Γ−ξ∑n=0∞a′nϑn∫01ςμ+n−11−ς−ξ−1F11σ;δ;κς1−ςdς.Proof.
By using equation (73), in L.H.S. of the above equation, we have(77)Dσ,δ;κMC;ξϑμ−1fϑ=∑n=0∞a′nDσ,δ;κMC;ξϑμ−1ϑn=∑n=0∞a′n1Γ−ξ∫0ϑϑ−ℓ−ξ−1ℓμ+n−1F11σ;δ;κℓϑ−ℓϑ2dℓ,and putting ℓ=ϑς, then(78)=1Γ−ξ∫01ϑ−ξ+μ−11−ς−ξ−1ςμ−1F11σ;δ;κς1−ς∑n=0∞a′nϑςndς.
The series given in equation (78), ∑n=0∞a′nϑςn, is uniformly convergent in disc ϑ<ρ for 0≤ς≤1 and the integral ∫01ςμ−11−ς−ξ−1F11σ;δ;κς1−ςdς is convergent provided that ℜeμ>0,ℜeξ<0 and κ∈C,κ<K;K∈R+ and less than 2.033 5. Now, interchanging the order of summation and integration, we get the result given in equation (76).



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