# Properties and Applications of a New Extended Gamma Function Involving Confluent Hypergeometric Function

Authors: Abdus Saboor; Gauhar Rahman; Hazrat Ali; Kottakkaran Sooppy Nisar; Thabet Abdeljawad
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, a new confluent hypergeometric gamma function and an associated confluent hypergeometric Pochhammer symbol are introduced. We discuss some properties, for instance, their different integral representations, derivative formulas, and generating function relations. Different special cases are also considered.


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## Body

## 1. Introduction

The classical gamma function was first found by Swiss mathematician Leonhard Euler (1707–1783) in his objective to generalize the factorial to nonintegral values. The classical gamma function and its related Pochhammer symbol are characterized as follows:(1)Γz=∫0∞tz−1e−tdt,ℜz>0,and(2)zρ=Γz+ρΓz,zz+1…z+n−1,ρ=n∈ℕ0,z∈ℂ,1,ρ=0,z∈ℂ∖0.The gamma function belongs to category of the special transcendental functions. It has applications in different fields, e.g., definite integration, asymptotic series, hypergeometric series, etc. In the last three decades, various extended forms of classical gamma functions are introduced by different researchers and mathematicians. Among those, some of the extended gamma functions and their related Pochhammer symbols are defined below.Chaudhry and Zubair [1] introduced an extended gamma function, defined by(3)Γκz=∫0∞tz−1e−t−k/tdt,ℜκ>0,ℜz>0.Srivastava et al. [2] proposed the following extended form of Pochhammer symbol:(4)z;κρ=Γκz+ρΓz,ℜκ>0,z,ρ∈ℂ,zρ,κ=0,z,ρ∈ℂ∖0.Srivastava et al. [3] introduced an extended gamma function involving modified Bessel function and its associated Pochhammer symbol, defined as follows:(5)Γvz;κ=2κπ∫0∞tz−3/2e−tKv+1/2κtdt,and(6)z;κ,vρ=Γvz+ρ;κΓz,ℜκ>0,ℜv>0,z,ρ∈ℂ,z;κρ,v=0,z,ρ∈ℂ∖0,where ℜκ>0, ℜz>0 and Kν. is the modified Bessel function of order v defined by(7)Kνy=∑i=0∞−1iy/2n+2ii!Γn+i+1.Safdar et al. [4] defined an extension of classical gamma function and its associated Pochhammer symbol involving Mittag-Leffler function, defined by(8)Γsz,σ=Γsσz=∫0∞tz−1e−tEσ−stdt,and(9)z;s,σρ=Γsz+ρ;σΓz,ℜs>0,ℜσ>0,z,ρ∈ℂ,z;sρ,σ=1,z,ρ∈ℂ∖0,where ℜρ>0, ℜz>0, and Eσz is the Mittag-Leffler function defined by(10)Eσz=∑j=0∞zjΓjσ+1.Srivastava et al. [5] present the familiar incomplete gamma functions γ (s, x) and Γ (s, x), and they introduce the incomplete Pochhammer symbols that led us to a natural generalization and decomposition of a class of hypergeometric and other related functions which are potentially useful in closed-form representations of definite and semi-infinite integrals of various special functions. Applications of these functions can be found in communication theory, probability distributions, and groundwater pumping modelling.Şahin et al. [6] introduce a new generalization of the Pochhammer symbol by means of the generalization of extended gamma function as follows:(11)Γp,qκ,μz=∫0∞tz−1exp−tκp−qtμdt.Using the generalization of Pochhammer symbol, they give a generalization of the extended hypergeometric functions of one or several variables. Also, they obtain various integral representations, derivative formulas, and certain properties of these functions.The rest of the paper is organized as follows. In Section2, we define a confluent hypergeometric gamma (CHG) function and derive its closed form in terms of Meijer’s G–function, which is built in function of Computational Package Mathematica. In Section 3, a confluent hypergeometric Pochhammer (CHP) symbol is defined and some of its associate properties are also derived. In Section 4, we define an extension of generalized hypergeometric gamma (GHG) function and derive some of its associate properties. In Section 5, we obtain families of generating function relations. In Section 6, we present comparison of confluent hypergeometric gamma (CHG) function with classical and extended gamma functions for certain numerical values using Table 1. Comparison of integral and closed form of confluent hypergeometric gamma (CHG) function is shown in Table 2. Concluding remarks are given in Section 7.Table 1
Comparison between gamma, extended gamma (3), and CHG (12) functions.
zΓzΓpzΓpz;m,b,α0.51.772450.097690.02519110.131520.028631.50.886270.190420.03455210.295680.044262.51.329340.490780.06013320.867560.086543.53.323351.626850.13177463.223620.211974.511.63176.724640.3596652414.71640.64269Table 2
Comparison between integral form of CHG function and its closed form.
zIntegral formClosed form0.50.025190.0251910.028630.028631.50.034550.0345520.044260.044262.50.060130.0601330.086540.086543.50.131770.1317740.211970.211974.50.359660.3596650.642690.64269


## 2. A Confluent Hypergeometric Gamma Function

We propose further generalization of extended gamma function (3) by inducting a confluent hypergeometric function [7] in the integrand of the integral. In Theorem 1, we have derived the closed-form representation of the proposed CHG function using the inverse Mellin transform technique [8] in the form of Meijer’s G–function.Theorem 1.
One has the following closed form forΓpz;m,b,α.(12)Γpz;m,b,α=∫0∞tz−1e−t−p/tF11m;b,−αtdt,(13)=ΓbΓm∑n=0∞pnn!G3,32,2α|1−m,1−z,1−z+n0,1−z,1−b−pzG4,22,2pα|1−m,1−z0,1−z,1−b,−z−n,where ℜp>0,argαt<π/2,ℜz≠0,−1,−2,… and ℜb≠0,−1,−2,…Proof.
From [Rainville [7], p. 102, Theorem 36], (12) becomes(14)Γpz;m,b,α=∫0∞tz−1e−t−p/t12πi∫c−i∞c+i∞αtsΓ−sΓm+sΓb+sdsdt=ΓbΓm2pz/212πi∫c−i∞c+i∞pα−sΓ−sΓλ+sb+skz+s2pds,where kv is Macdonald function [9], defined by(15)kv=π2I−vx−IvxSinvπ,and Ivx is a modified Bessel function of first kind [10], defined by(16)Ivx=∑n=0∞x/2v+2nn!Γv+n+1.
Using (15) and (16) and the residue theorem [11], one has(17)Γpz;m,b,α=ΓbΓm∑n=0∞pnn!×G3,32,2α1,z,bm,z,z−n−pzG4,22,21pα1,z,b,z+n+1m,z,which can be reexpressed as(18)Γpz;m,b,α=ΓbΓm∑n=0∞pnn!G3,32,2α1−z,1−m,−z+n+10,1−z,1−b−pzG4,22,2pα1−z,1−m0,1−z,1−b,−z−n,where ℜp>0 and ℜb,ℜm,ℜz≠0,−1,−2,…Remark 1.
It is also important to mention here that in most of the extended cases, no closed form is provided but only new parameter in its function is inducted in the integrand of the integral of classical gamma function or its any extended gamma forms. Moreover, the proposed CHG function may find applications in statistics and a few associations with other special functions and polynomials.Remark 2.
Now, letm=b and α=0; then, (12) reduces to extended gamma function defined by Zubair and Chaudhry [12], which is defined as(19)Γpz=∫0∞tz−1e−t−p/tdt.
If we considerp=0, then (19) becomes the classical gamma function (1).


### 2.1. Incomplete Confluent Hypergeometric Gamma Functions

Incomplete CHG functions are defined by(20)γpz,y;m,b,α=∫0ytz−1e−t−p/tF11m;b,−αtdt,and(21)Γpz,y;m,b,α=∫y0tz−1e−t−p/tF11m;b,−αtdt.Now if we considerm=b and α=0, then (20) and (21) are reduced to incomplete gamma function defined by Chaudhry and Zubair [12].


## 2.1. Incomplete Confluent Hypergeometric Gamma Functions

Incomplete CHG functions are defined by(20)γpz,y;m,b,α=∫0ytz−1e−t−p/tF11m;b,−αtdt,and(21)Γpz,y;m,b,α=∫y0tz−1e−t−p/tF11m;b,−αtdt.Now if we considerm=b and α=0, then (20) and (21) are reduced to incomplete gamma function defined by Chaudhry and Zubair [12].


## 3. A Confluent Hypergeometric Pochhammer Symbol

The classical Pochhammer symbol is defined by(22)zρ=Γz+ρΓz,zz+1…z+n−1,ρ=n∈ℕ0,z∈ℂ,1,ρ=0,z∈ℂ∖0.Pochhammer symbol introduced by Srivastava et al. [2] is given by(23)z;κρ=Γκz+ρΓz,ℜκ>0,z,ρ∈ℂ,zρ,κ=0,z,ρ∈ℂ∖0.We propose a CHP symbol in following form:(24)z;m,b,α,pρ=Γpz+ρ;m,b,αΓz,ℜm,b,α>0,ℜp>0,z,ρ∈ℂ,z;m,b,αρ,p=0,z,ρ∈ℂ∖0,where Γpz;m,b,α is defined in (12).


### 3.1. Some Results Associated with the Confluent Hypergeometric Pochhammer Symbol

Here, we present an integral representation of CHP symbolz;m,b,α,pρ. Using Equation (12), we define an integral representation of Γpz+ρ;m,b,α as(25)Γpz+ρ;m,b,α=∫0∞tz+ρ−1e−t−p/tF11m;b,−αtdt.Combining (25) with (24), one gets an integral representation of z;m,b,α,pρ in the following form:(26)z;m,b,α,pρ=1Γz∫0∞tz+ρ−1e−t−p/tF11m,b,−αtdt.Remark 3.
Letm=b,α=0, and p=κ; we get the following Pochhammer symbol defined by Srivastava et al. [2]:(27)z;κ,0ρ=1Γz∫0∞tz+ρ−1e−t−k/tdt.Theorem 2.
One obtains the following result for confluent hypergeometric gamma (CHG) function (12).(28)Γpz;m,b,α=2ΓbΓmΓb−mpz/2∫01μm−11−μb−m−11+αμ−z/2×K−z21/μ+α1/pμdμ.Proof.
Using integral form of confluent hypergeometric function, (12) becomes(29)Γpz;m,b,α=ΓbΓmΓb−m∫0∞tz−1e−t−p/t∫01uλ−11−ub−λ−1e−αtududt.
Apply a one-one transformation but without including the boundaries. Letν=ut,μ=t in the above equality and the Jacobian is J=1/μ; then, one gets(30)Γpz;m,b,α=ΓbΓmΓb−m∫0∞∫01νz−1e−v/μ−αν−pμ/vμm−z−1×1−μb−m−1dμdν.
Interchanging the order of integration due to uniform convergence of the integrals and using some basic calculus, one obtains (28).Theorem 3.
For product of two confluent hypergeometric gamma functions, one has the following result:(31)Γpz;m,b,αΓpn;m,b,α=4∫0π/2∫0∞γ2z+n−1cos2z−1θsin2n−1θ×exp−γ2−pγ2cos2θsin2θ×F11m;b;−αγ2cos2θ×F11m;b;−αγ2sin2θdγdθ.Proof.
Substitutingt=σ2 in (12), one gets(32)Γpz;m,b,α=2∫0∞σ2z−1e−σ2−p/σ2F11m;b,−ασ2dσ.
Therefore,(33)Γpz;m,b,αΓpn;m,b,α=4∫0∞∫0∞σ2m−1ω2n−1σ2z−1e−σ2−p/σ2×e−ω2−p/ω2F11m;b;−ασ2F11m;b;−αω2dσdω.
Lettingσ=γcosθ,ω=γsinθ in (33) gives (31).Remark 4.
α=0 in (33) gives the product of extended gamma function (see [13]).Remark 5.
By puttingp=0,α=0 in (33), one obtains the standard relation between gamma and beta function.Theorem 4.
For confluent hypergeometric gamma (CHG) function (12), one has the following recurrence relationship:(34)m−1Γpz;m.b,α=m−bΓpz;m−1,b,α+b−1Γpz;m−1,b−1,α.Proof.
The confluent hypergeometric functionsF11ξ−1;τ;z, F11ξ+1;τ;z, F11ξ;τ−1;z, and F11ξ;τ+1;z in which one parameter is increased or decreased by unity are said to be contiguous to F11ξ;τ;z. The following relations exist between F11 and two of its contiguous functions [14].(35)τF11ξ:τ;z=τF11ξ−1;τ;z+z1F1ξ;τ+1;z,(36)1+ξ−τF11ξ;τ;z=ξ1F1ξ+1;τ;z−τ−1F11ξ;τ−1;z.
Substituting the functionF11 appearing in (36) and (12) and replacing ξ by m, τ by b, and z by −αt, one obtains (34).Theorem 5.
Letz,ρ,μ∈ℂ.Then,(37)z;m,b,α,pρ+μ=zρz+ρ;m,b,α,pμ.Proof.
From (22)–(24), we have(38)z;m,b,α,pρ+μ=Γpz+ρ+μ;m,b,αΓz=Γz+ρΓz+ρΓpz+ρ+μ;m,b,αΓz=zρz+ρ;m,b,α,p.Remark 6.
From equation (37), one can write(39)z;m,b,α,pr+n=zrz+r;m,b,α,pn,z∈ℂ,r,n∈ℕ0.
Now, we can derive the following properties of CHP symbolz;m,b,α,pn through familiar properties of classical and extended Pochhammer symbols zn and z;κn (see, for instance, [2, 7, 15]).Corollary 1.
Forl,r,k,n∈ℕ0 and R ∈ℕ, we can derive following properties:(40)z;m,b,α,pn+r+l=znz+nrz+n+r;m,b,α,pl,z;m,b,α,pn−r+l=−1rzn1−z−nrz+n−r;m,b,α,pl,z;m,b,α,p2r+l=22rz2rz+12rz+2r;m,b,αl,z;m,b,α,pRr+l=RRrzRrz+1Rr…z+R−1Rrz+Rr;m,b,α,pl,z+n;m,b,α,pn+l=z+nnz+2n;m,b,α,pl=z2nznz+2n;m,b,α,pl,(41)z+r;m,b,α,pn+l=znz+nrzrz+n+r;m,b,α,pl,z+kr;m,b,α,pkn+l=zkn+krzkrz+kn+kr;m,b,α,pl,z−n;m,b,α,pn+l=−1n1−znz;m,b,α,pl,z−n;m,b,α,pn+l=1−zrzn1−z−nrz+n−r;m,b,α,pl,z−kr;m,b,α,pkn+l=−1krzkn−kr1−zkrz+kn−kr;m,b,α,pl,z+r;m,b,α,pn−r+l=znzrz+n;m,b,α,pl,z−r;m,b,α,pn−r+l=−1r1−zrzn1−z−n2rz+n−2r;m,b,α,pl,and−z;m,b,α,pn+l=−1rz−n+1n−z+n;m,b,α,pl.


## 3.1. Some Results Associated with the Confluent Hypergeometric Pochhammer Symbol

Here, we present an integral representation of CHP symbolz;m,b,α,pρ. Using Equation (12), we define an integral representation of Γpz+ρ;m,b,α as(25)Γpz+ρ;m,b,α=∫0∞tz+ρ−1e−t−p/tF11m;b,−αtdt.Combining (25) with (24), one gets an integral representation of z;m,b,α,pρ in the following form:(26)z;m,b,α,pρ=1Γz∫0∞tz+ρ−1e−t−p/tF11m,b,−αtdt.Remark 3.
Letm=b,α=0, and p=κ; we get the following Pochhammer symbol defined by Srivastava et al. [2]:(27)z;κ,0ρ=1Γz∫0∞tz+ρ−1e−t−k/tdt.Theorem 2.
One obtains the following result for confluent hypergeometric gamma (CHG) function (12).(28)Γpz;m,b,α=2ΓbΓmΓb−mpz/2∫01μm−11−μb−m−11+αμ−z/2×K−z21/μ+α1/pμdμ.Proof.
Using integral form of confluent hypergeometric function, (12) becomes(29)Γpz;m,b,α=ΓbΓmΓb−m∫0∞tz−1e−t−p/t∫01uλ−11−ub−λ−1e−αtududt.
Apply a one-one transformation but without including the boundaries. Letν=ut,μ=t in the above equality and the Jacobian is J=1/μ; then, one gets(30)Γpz;m,b,α=ΓbΓmΓb−m∫0∞∫01νz−1e−v/μ−αν−pμ/vμm−z−1×1−μb−m−1dμdν.
Interchanging the order of integration due to uniform convergence of the integrals and using some basic calculus, one obtains (28).Theorem 3.
For product of two confluent hypergeometric gamma functions, one has the following result:(31)Γpz;m,b,αΓpn;m,b,α=4∫0π/2∫0∞γ2z+n−1cos2z−1θsin2n−1θ×exp−γ2−pγ2cos2θsin2θ×F11m;b;−αγ2cos2θ×F11m;b;−αγ2sin2θdγdθ.Proof.
Substitutingt=σ2 in (12), one gets(32)Γpz;m,b,α=2∫0∞σ2z−1e−σ2−p/σ2F11m;b,−ασ2dσ.
Therefore,(33)Γpz;m,b,αΓpn;m,b,α=4∫0∞∫0∞σ2m−1ω2n−1σ2z−1e−σ2−p/σ2×e−ω2−p/ω2F11m;b;−ασ2F11m;b;−αω2dσdω.
Lettingσ=γcosθ,ω=γsinθ in (33) gives (31).Remark 4.
α=0 in (33) gives the product of extended gamma function (see [13]).Remark 5.
By puttingp=0,α=0 in (33), one obtains the standard relation between gamma and beta function.Theorem 4.
For confluent hypergeometric gamma (CHG) function (12), one has the following recurrence relationship:(34)m−1Γpz;m.b,α=m−bΓpz;m−1,b,α+b−1Γpz;m−1,b−1,α.Proof.
The confluent hypergeometric functionsF11ξ−1;τ;z, F11ξ+1;τ;z, F11ξ;τ−1;z, and F11ξ;τ+1;z in which one parameter is increased or decreased by unity are said to be contiguous to F11ξ;τ;z. The following relations exist between F11 and two of its contiguous functions [14].(35)τF11ξ:τ;z=τF11ξ−1;τ;z+z1F1ξ;τ+1;z,(36)1+ξ−τF11ξ;τ;z=ξ1F1ξ+1;τ;z−τ−1F11ξ;τ−1;z.
Substituting the functionF11 appearing in (36) and (12) and replacing ξ by m, τ by b, and z by −αt, one obtains (34).Theorem 5.
Letz,ρ,μ∈ℂ.Then,(37)z;m,b,α,pρ+μ=zρz+ρ;m,b,α,pμ.Proof.
From (22)–(24), we have(38)z;m,b,α,pρ+μ=Γpz+ρ+μ;m,b,αΓz=Γz+ρΓz+ρΓpz+ρ+μ;m,b,αΓz=zρz+ρ;m,b,α,p.Remark 6.
From equation (37), one can write(39)z;m,b,α,pr+n=zrz+r;m,b,α,pn,z∈ℂ,r,n∈ℕ0.
Now, we can derive the following properties of CHP symbolz;m,b,α,pn through familiar properties of classical and extended Pochhammer symbols zn and z;κn (see, for instance, [2, 7, 15]).Corollary 1.
Forl,r,k,n∈ℕ0 and R ∈ℕ, we can derive following properties:(40)z;m,b,α,pn+r+l=znz+nrz+n+r;m,b,α,pl,z;m,b,α,pn−r+l=−1rzn1−z−nrz+n−r;m,b,α,pl,z;m,b,α,p2r+l=22rz2rz+12rz+2r;m,b,αl,z;m,b,α,pRr+l=RRrzRrz+1Rr…z+R−1Rrz+Rr;m,b,α,pl,z+n;m,b,α,pn+l=z+nnz+2n;m,b,α,pl=z2nznz+2n;m,b,α,pl,(41)z+r;m,b,α,pn+l=znz+nrzrz+n+r;m,b,α,pl,z+kr;m,b,α,pkn+l=zkn+krzkrz+kn+kr;m,b,α,pl,z−n;m,b,α,pn+l=−1n1−znz;m,b,α,pl,z−n;m,b,α,pn+l=1−zrzn1−z−nrz+n−r;m,b,α,pl,z−kr;m,b,α,pkn+l=−1krzkn−kr1−zkrz+kn−kr;m,b,α,pl,z+r;m,b,α,pn−r+l=znzrz+n;m,b,α,pl,z−r;m,b,α,pn−r+l=−1r1−zrzn1−z−n2rz+n−2r;m,b,α,pl,and−z;m,b,α,pn+l=−1rz−n+1n−z+n;m,b,α,pl.


## 4. A Generalized Hypergeometric Gamma Function

We define a GHG functionFqr. in the following form:(42)Fqrη1;m,b,α,pη2,…,ηr;β1′,…βq′;z=∑n=0∞η1;m,b,α,pnη2n…ηrnβ1′n…βq′nznn!,where ηi∈ℂ for i = 1, 2, …r, βj′∈ℂ for j = 1,2, …, q, and βj′≠0,−1,−2,….New extensions of Gauss hypergeometric and confluent hypergeometric functions, respectively, are defined by(43)F12η1;m,b,α,p,β′;γ;z=∑n=0∞η1;m,b,α,pnβ′nγnznn!,F11η1;m,b,α,p;γ;z=∑n=0∞η1;m,b,α,pnγnznn!.Remark 7.
Let us assumem=b, α=0, and p=κ in (42); one will get the following generalized hypergeometric function defined by Srivastava et al. [2]:(44)Fqrη1;κ,η2,…,δr;β1,…βq;z=∑n=0∞η1;κnη2n…ηrnβ1n…βqnznn!,where ηi∈ℂ for i = 1, 2, …, r, βj′∈ℂ for j = 1, 2, …, q, and βj′≠ 0,-1,-2, ….Remark 8.
If we assumeκ=0 in (44), we then obtain the classical generalized hypergeometric function [7]:(45)Fqrη1,η2,…,ηr;β1′,…βq;z=∑n=0∞η1nη2n…ηrnβ1′n…βq′n,where ηi∈ℂ for i=1,2,⋯r,βj′∈ℂ for j=1,2,…,q, and βj′≠0,−1,−2,⋯.
Now in the following theorems, different integral representations of GHG function (42) are derived.Theorem 6.
Ifηi∈ℂ for i = 1, 2, …r, βj′∈ℂ for j = 1, 2, …, q, and βj′≠0,−1,−2,…., then(46)Fqrη1;m,b,α,p,η2,…,ηr;β1,…βq;z=1Γη1∫0∞tη1+ρ−1e−t−ptF11m;b,−αt×r−1Fqη2,…,ηr;β1′,…βq′;zt.Proof.
Applying integral representation of extended CHP symbol (26) in the definition of extended hypergeometric function (42), we have(47)Fqrη1;m,b,α,p,η2,…,ηr;β1′,…βq′;z=1Γη1∑n=0∞∫0∞tη1+ρ−1e−t−pt×1F1m;b,−αtη2n…δrnβ1′n…βq′nznn!dt.
Exchanging integration and summation order and then by applying generalized hypergeometric function (45), we obtained the desired result.Theorem 7.
Ifℜγ>ℜβ′>0;ℜm,b,α≥0, then the beta-type integral associated with the GHG function is defined by(48)Fqrη1;m,b,α,p,η2,…,ηr−1,β′;β1′,…βq−1′,γ;z=1Bβ′,γ−β′∫01tβ′−11−tγ−β′−1×r−1Fq−1δ2,…,δr;β1′,…βq−1′;ztdt.Proof.
From (42), we have(49)Fqrη1;m,b,α,pη2,…,δr;β1′,…,βq′;z=∑n=0∞η1;m,b,α,pnη2n…ηr−1β′nη1′n…βq−1′nγnznn!.
We know that(50)β′nγn=Bβ′+n,γ−β′Bβ′,γ−β′=1Bβ′,γ−β′∫01tβ′+n−11−tγ−β′−1dt.
Combining (49) and (45) solves the theorem.Theorem 8.
Ifℜm,b,α≥0,n∈ℕ0, then the nth order derivative of GHG function (42) is given by(51)dndznFqrη1;m,b,α,p,η2,…,ηr;β1′,…βq′;z=η1nη2n…ηrnβ1′n…βq′n×rFqη1+n;m,b,α,p,η2+n,…,ηr+n;β1′,…βq′;z.Proof.
The proof of (51) is minimal for n∈0. Consider if n≠0, then(52)ddzFqrη1;m,b,α,p,η2,…,ηr;β1′,…βq′;z=∑n=1∞η1;m,b,α,pnη2n…ηrnβ1′n…βq′nZn−1n−1!=∑n=0∞η1;m,b,α,pn+1η2n+1,…,ηrn+1β1′n+1,…,βq′n+1Znn!.
In light of (37) and applying xn+1=xx+1n in (54), it results in(53)ddzFqrη1;m,b,α,p,η2,…,ηr;β1′,…βq′;z=η1η2…ηrβ1′…βq′Fqη1;m,b,α,p,η2+1,…,ηr+1;β1′+1,…βq′+1;z.
Differentiatingn times proves the theorem.Corollary 2.
If in (40), we consider the casesr = q = 1 and r = 2, q = 1, then the nth derivative of the new extended confluent and Gauss hypergeometric functions become, respectively,(54)dndznF11η;m,b,α,p;γ;z=ηnγnF11η+n;m,b,α,p;γ+n;z,dndznF12η;m,b,α,p;β′;γ;z=ηnβ′nγn×2F1η+n;m,b,α,p;β′+n;γ+n;z.


## 5. Families of Generating Function Relations

LetΔR,z indicate the below R parameters array.(55)zR,zR+1,…,z+R−1R,z∈ℂ;R∈ℕ.IfR=0, then the ΔR,z array will be null (see [16–18]).Theorem 9.
The following confluent hypergeometric gamma (CHG) generating function is defined by(56)∑n=0∞znn!Fqr+RΔR,z+n,η1;m,b,α,p,η2…,ηr;β1′…,βq′;zxn=1−x−zFqr+RΔR,z,η1;m,b,α,p,η2…,ηr;β1′…,βq′;z1−xR,x<1,z∈ℂ,R∈ℕ,provided every element of (56) exists.Proof.
Using the identity(57)∑n=0∞znn!zn=1−z−z,z<1;z∈ℂto generalize hypergeometric function extension (GHF) (42) results in required assertion (56) of the Theorem 5.Theorem 10.
The below generating function exists:(58)∑n=0∞znn!Fqr+RΔR,−n,η1;m,b,α,p,η2,…,ηr;β1′…,βq′;zxn=1−x−zr+RFqΔR,z,η1;m,b,α,p,z2…,ηr;β1′…,βq′;z−x1−xR,x<1,z∈ℂ,R∈ℕ,(59)∑n=0∞znn!Fqr+2RΔR,−n,ΔR,z+n,η1;m,b,α,p,η2…,ηr;β1′…,βq′;zxn=1−xFqr+2RΔ2R,z,η1;m,b,α,p,δ2…,ηr;β1′…,βq′;z−4x1−x2R,x<1,z∈ℂ,R∈ℕ,and(60)∑n=0∞znn!Fq+Rr+RΔR,−n,η1;m,b,α,p,η2…,ηr;β1′…,βq′;zxn=1−x−zFRr+2RΔ2R,z,δ1;m,b,α,p,δ2…,δr;ΔR,1−z−n,β1′…,βq′;zxR,x<1,z∈ℂ,R∈ℕ,provided every element of (58)–(60) exists.Proof.
The proof of (58)–(60) is similar to that of Theorem 6.Corollary 3.
The following generating function relations exist.(61)∑n=0∞znn!Fqr+1z+n,η1;m,b,α,p,η2…,ηr;β1′…,βq′;zxn=1−x−zFqr+1z,η1;m,b,α,p,η2⋯,ηr;β1′…,βq′;z1−x,x<1,z∈ℂ,(62)∑n=0∞znn!Fqr+1−n,η1;m,b,α,p,η2…,ηr;β1′…,βq′;zxn=1−x−zFqr+1z,η1;m,b,α,p,η2…,ηr;β1′…,βq′;z−x1−x,x<1,z∈ℂ,(63)∑n=0∞znn!Fqr+2−n,η1;m,b,α,p,η2…,ηr;β1′⋯,βq′;zxn=1−x−zFqr+2Δ2,z,η1;m,b,α,p,η2…,ηr;β1′…,βq′;z−4x1−x2,x<1,z∈ℂ,N∈ℕ,and(64)∑n=0∞znn!Fq+1r+1−n,η1;m,b,α,p,η2…,ηr;1−z−n,β1′…,βq′;zxn=1−x−zFqrΔ2,z,η1;m,b,α,p,η2…,ηr;β1′…,βq′;zx,x<1,z∈ℂ,provided that each number of (61)–(64) exists.Proof.
AssumingR=1 in Theorems 9 and 10, respectively, results in required assertions (61)–(64) of Corollary 3.


## 6. Numerical Comparisons

In Table1, we compared classical gamma, extended gamma (3), and CHG (12) functions for different numerical values of z in Computational Package Mathematica. We fixed values of m=1.9,b=2.3,α=1.1,p=2.1. For different numerical values of z, extended gamma function (3) gives lesser values than gamma function while CHG (12) gives the least values. We also compared integral and closed forms of CHG functions in Table 2. We observed that results for both integral and closed forms are the same, but for the closed form, numerical values are obtained more quickly compared to the integral form.


## 7. Conclusion

In this paper, further generalization of the extended gamma function of Chaudhry and Zubair [12] is introduced. An associated confluent hypergeometric Pochhammer (CHP) symbol is also defined. Using confluent hypergeometric gamma (CHG) function, further generalizations of hypergeometric functions are also discussed. The new extended gamma function involving confluent hypergeometric was found to be useful in a variety of heat conduction problems. Additionally, we introduced a few families of generating functions for generalized hypergeometric function. Some special cases are also derived. We may develop new extensions of various special functions by applying confluent hypergeometric Pochhammer (CHP) symbol as shown in (24).



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