# A study on certain properties of generalized special functions defined by Fox-Wright function

Enes Ata 1  and İ. Onur Kıymaz 1
• 1 Dept. of Mathematics, Faculty of Science, Kırşehir, Turkey
Enes Ata
• Dept. of Mathematics, Faculty of Science, Ahi Evran University, Kırşehir, Turkey
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and İ. Onur Kıymaz
• Corresponding author
• Dept. of Mathematics, Faculty of Science, Ahi Evran University, Kırşehir, Turkey
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## Abstract

In this study, motivated by the frequent use of Fox-Wright function in the theory of special functions, we first introduced new generalizations of gamma and beta functions with the help of Fox-Wright function. Then by using these functions, we defined generalized Gauss hypergeometric function and generalized confluent hypergeometric function. For all the generalized functions we have defined, we obtained their integral representations, summation formulas, transformation formulas, derivative formulas and difference formulas. Also, we calculated the Mellin transformations of these functions.

## 1 Introduction

On the last quarter century, some generalizations of special functions, which frequently used in applied mathematics, have been studied by many scientists [1, 7,8, 9, 10, 11, 12, 13,14, 17, 19,20,21,22,23,24,25,26,27,28,29, 31, 32, 33]. Chaudhry and Zubair  defined the extended gamma function in 1994 as

$Γp(x)=∫0∞tx−1exp[−t−pt]dt,$
where Re(p) > 0. Three years later, Chaudhry et al.  defined the extended beta function as
$Bp(x,y)=∫01tx−1(1−t)y−1exp[−pt(1−t)]dt,$
where Re(p) > 0, Re(x) > 0, Re(y) > 0. It clearly seems that, for p = 0, Γ0(x) = Γ(x) and B0(x,y) = B(x,y), where Γ(x) and B(x,y) are the classical gamma and beta functions .

In 2004, Chaudhry et al.  used Bp(x,y) to extend the Gauss and confluent hypergeometric functions as follows:

$Fp(a,b;c;z)=∑n=0∞(a)nBp(b+n,c−b)B(b,c−b)znn!,$
$Φp(b;c;z)=∑n=0∞Bp(b+n,c−b)B(b,c−b)znn!,$
where p ≥ 0, Re(c) > Re(b) > 0. In the same paper, the authors also gave the integral representations of (1) and (2) as
$Fp(a,b;c;z)​=​1B(b,c​−​b)​∫01​tb−1(1​−​t)c−b−1(1​−​zt)−aexp[−pt(1​−​t)]dt,$
where p > 0, p = 0 and |arg(1 − z)| < π < p, Re(c) > Re(b) > 0, and
$Φp(b;c;z)=1B(b,c−b)∫01tb−1(1−t)c−b−1exp[zt−pt(1−t)]dt,$
where p > 0, p = 0 and Re(c) > Re(b) > 0. Here (a)n is the Pochhammer symbol which defined as
$(a)ν=Γ(a+ν)Γ(a), a,ν∈ℂ$
with the assume (a)0 ≣ 1.

The Fox-Wright function is given in  as

$ξΨη(z)=ξΨη[​​(βi,αi)1,ξ(μj,κj)1,η|z]​=∑n=0∞∏i=1ξΓ(αin+βi)∏j=1ηΓ(κjn+μj)znn!,$
where z,βi, µj ℂ,αi,κj ℝ,i = 1...ξ and j = 1...η. The asymptotic behaviour of the above function was studied by Fox [15, 16] and Wright [34, 35, 36] for the large values of z, considering the condition
$∑j=1ηκj−∑i=1ξαi>−1.$
If these conditions are met, for any z ∈ ℂ the series (3) is convergent. For κ, µ,z ∈ ℂ,Re(κ) > −1, the classic Wright function 
$0Ψ1(z)=0Ψ1[_(μ,κ)|z]=∑n=0∞1Γ(κn+μ)znn!$
can obtained by choosing ξ = 0 and η = 1 in equation (3).

Inspired by the aforementioned studies and motivated by the frequent use of Fox-Wright function in the theory of special functions, we defined two new functions as generalizations of gamma and beta functions.

## 2 Generalized functions and their properties

Throughout the study, we assume that x,y,z ∈ ℂ, k,m,n ∈ ℕ, αi,κj ℝ, βi, µj,a,b,c, p ∈ ℂ, Re(p) > 0, Re(x) > 0, Re(y) > 0, and Re(c) > Re(b) > 0. For the sake of shortness, we did not wrote these conditions for the rest of the article, unless otherwise stated.

Let us defined the new generalizations as

$ΨΓ^p(x):=ΨΓp[​​(βi,αi)1,ξ(μj,κj)1,η|x]=∫0∞tx−1ξΨη(−t−pt)dt$
and
$Ψ​B^p(x,y):=Ψ​Bp[​​(βi,αi)1,ξ(μj,κj)1,η|x,y]=∫01tx−1(1−t)y−1ξΨη(−pt(1−t))dt.$
We called them as ξΨη-gamma and ξΨη-beta functions.

Our first theorem is about the current relationship of the two ξ Ψη-gamma functions.

Theorem 1

The following equality holds true:

$ΨΓ^p(x)ΨΓ^p(y)=4∫0π2∫0∞r2(x+y)−1(cosθ)2x−1(sinθ)2y−1×ξΨη(−r2(cosθ)2−pr2(cosθ)2)×ξΨη(−r2(sinθ)2−pr2(sinθ)2)​drdθ.$

Proof

Substituting t = u2 in (4), we get

$ΨΓ^p(x)=2∫0∞u2x−1ξΨη(−u2−pu2)du.$
Therefore,
$ΨΓ^p(x)ΨΓ^p(y)=4∫0∞∫0∞u2x−1v2y−1ξΨη(−u2−pu2)ξΨη(−v2−pv2)dudv.$
In the above equality, taking u = r(cosθ) and v = r(sinθ) yields
$ΨΓ^p(x)ΨΓ^p(y)=4∫0π2∫0∞r2(x+y)−1(cosθ)2x−1(sinθ)2y−1×ξΨη(−r2(cosθ)2−pr2(cosθ)2)×ξΨη(−r2(sinθ)2−pr2(sinθ)2)drdθ,$
which completes the proof.

Theorem 2

TheξΨη-beta function has the following integral representations:

$Ψ​B^p(x,y)=2∫0π2(sinθ)2x−1(cosθ)2y−1ξΨη(−p(secθ)2(cscθ)2)dθ,Ψ​B^p(x,y)=∫0∞ux−1(1+u)x+yξΨη(−2p−p(u+1u))du,Ψ​B^p(x,y)=(c​−​a)1−x−y​​∫ac​​(u​−​a)x−1(c​−​u)y−1ξΨη​(−p(c​−​a)2(u​−​a)(c​−​u))​du.$

Proof

Taking t = (sinθ)2 in (5), we get

$Ψ​B^p(x,y)=∫01tx−1(1−t)y−1ξΨη(−pt(1−t))dt=2∫0π2(sinθ)2x−1(cosθ)2y−1ξΨη(−p(secθ)2(cscθ)2)dθ.$

Taking $t=u1+u$ in (5), we get

$Ψ​B^p(x,y)=∫0∞​​(​u1​+​u​)x−1​​(​11​+​u​)y−1​​(​11​+​u​)2​​ξΨη​(−p(u1+u)(11+u))​du=∫0∞ux−1(1+u)x+yξΨη(−2p−p(u+1u))du.$
Taking $t​=​u−ac−a$ in (5), we get
$Ψ​B^p(x,y)​=​∫ac(u​−​ac​−​a)x−1​​(1​−​u​−​ac​−​a)y−1​​​1c​−​aξΨη(−p(c​−​a)2(u​−​a)(c​−​u))du=(c​−​a)1−x−y∫ac(u​−​a)x−1(c​−​u)y−1ξΨη(−p(c​−​a)2(u​−​a)(c​−​u))du,$
which gives the result.

Theorem 3

The following derivative formula is provided for Re(x) > m, Re(y) > m:

$dmdpm{Ψ​B^p(x,y)}​=​(−1)mΨ​Bp[​​(αim+βi,αi)1,ξ(κjm+μj,κj)1,η|x−m,y−m].$

Proof

It is done by induction. The first order derivative of (5) is as follows:

$ddp{Ψ​B^p(x,y)}​=​ddp{∫01tx−1(1​−​t)y−1ξΨη(−pt(1−t))dt}=​(−1) Ψ​Bp[​​(αi+βi,αi)1,ξ(κj+μj,κj)1,η|x−1,y−1].$
Let us assume that the k-order derivative of (5) is
$dkdpk{Ψ​B^p(x,y)}​=​(−1)k Ψ​Bp[​​(αik+βi,αi)1,ξ(κjk+μj,κj)1,η|x−k,y−k].$
From the first order derivative of (6), the k + 1-order derivative is found as follows:
$dk+1dpk+1{Ψ​B^p(x,y)}=ddp{dkdpk{Ψ​B^p(x,y)}}=(−1)k+1 Ψ​Bp[​​(αi(k+1)+βi,αi)1,ξ(κj(k+1)+μj,κj)1,η|x−(k+1),y−(k+1)].$
This gives the result.

Theorem 4

The following equality is provided for Re(s) > 0:

$ℳ[Ψ​B^p(x,y)]=B(x+s,y+s)ΨΓ^p(s).$

Proof

If we apply Mellin transformation according to argument p in equation (5), we have

$ℳ[Ψ​B^p(x,y)]=∫0∞​​ps−1​​∫01tx−1(1−t)y−1ξΨη(−pt(1−t))dtdp=​∫01tx−1(1−t)y−1​∫0∞​​ps−1ξΨη(−pt(1​−​t))dpdt.$
Letting $v=pt(1−t)$ in (7), we get
$ℳ[Ψ​B^p(x,y)]=∫01tx+s−1(1−t)y+s−1dt∫0∞vs−1ξΨη(−v)dv.$

Thus, we have

$ℳ[Ψ​B^p(x,y)]=B(x+s,y+s)ΨΓp(s),$
which completes the proof.

Remark 1

By using the inverse Mellin transform, it is easy to see

$Ψ​B^p(x,y)=12πi∫−i∞+i∞B(x+s,y+s)ΨΓp(s)p−sds$
for Re(s) > 0.

Theorem 5

The following equality holds true:

$Ψ​B^p(x,y)=​Ψ​B^p(x​+​1,y)​+​Ψ​B^p(x,y​+​1).$

Proof

Direct calculation yields

$Ψ​B^p(x,y)=∫01tx−1(1​−​t)y−1ξΨη(−pt(1−t))dt=∫01tx(1​−​t)y1t(1​−​t)ξΨη(−pt(1−t))dt=∫01tx(1​−​t)y[(1​−​t)−1​+​t−1]ξΨη(−pt(1−t))dt=∫01[tx(1​−​t)y−1​+​tx−1(1​−​t)y]ξΨη(−pt(1−t))dt=∫01tx(1​−​t)y−1ξΨη(−pt(1−t))dt+​∫01tx−1(1​−​t)yξΨη(−pt(1−t))dt= Ψ​B^p(x​+​1,y)​+​Ψ​B^p(x,y​+​1),$
which is the result.

Theorem 6

The following summation formula is provided for Re(y) < 1:

$Ψ​B^p(x,1−y)=∑n=0∞(y)nn!Ψ​B^p(x+n,1).$

Proof

From the definition of the ξ Ψη-beta function, we obtain

$Ψ​B^p(x,1−y)=∫01tx−1(1−t)−yξΨη(−pt(1−t))dt.$
With the help of the following series expression
$(1−t)−y=∑n=0∞(y)ntnn!, |t|<1,$
we obtain
$Ψ​B^p(x,1−y)=∫01∑n=0∞(y)nn!tx+n−1ξΨη(−pt(1−t))dt=∑n=0∞(y)nn!∫01tx+n−1ξΨη(−pt(1−t))dt=∑n=0∞(y)nn!Ψ​B^p(x+n,1).$
This completes the proof.

Theorem 7

The following equality holds true:

$Ψ​B^p(x,y)​=​∑n=0∞Ψ​B^p(x​+​n,y​+​1).$

Proof

From the definition of the ξΨη-beta function, we get

$Ψ​B^p(x,y)​=​∫01tx−1(1​−​t)y−1ξΨη(−pt(1−t))dt.$
With the help of the following series expression
$(1​−​t)y−1​=​(1​−​t)y∑n=0∞tn, |t|​<​1,$
we obtain
$Ψ​B^p(x,y)=∫01tx−1(1​−​t)y∑n=0∞tnξΨη(−pt(1−t))dt=∑n=0∞∫01tx+n−1(1​−​t)yξΨη(−pt(1−t))dt=∑n=0∞Ψ​B^p(x​+​n,y​+​1),$
which gives the result.

Theorem 8

The following relation is provided for Re(x) > 1,Re(y) > 1:

$xΨ​B^p(x,y​+​1)= yΨ​B^p(x​+​1,y)+2pΨ​Bp[​​(αi+βi,αi)1,ξ(κj+μj,κj)1,η|x,y−1]−pΨ​Bp[​​(αi+βi,αi)1,ξ(κj+μj,κj)1,η|x−1,y−1].$

Proof

(5) equality provides the following equation

$Ψ​B^p(x,y)​=​ℳ[f^(t​:​y;p)​:​x]$
where
$f^(t​:​y;p)​=​(1​−​t)y−1H(1​−​t) ξΨη(−pt(1−t))$
and
$H(1​−​t)​={0,​t​>​1,1,​t​<​1.$
The derivative of $f^(t​:​y;p)$ according to the parameter t provides the following equation:
$ddt{f^(t​:​y;p)}​=​−δ(1​−​t)(1​−​t)y−1ξΨη(−pt(1−t))​−​(y​−​1)(1​−​t)y−2H(1​−​t) ξΨη(−pt(1−t))​+​p(1​−​2t)t2(1​−​t)2(1​−​t)y−1H(1​−​t) ξΨη(−pt(1−t)),$
where $ddtH(1−t)=−δ(1−t)$ and δ represents the Dirac delta δ (1−t) = δ (t −1) = 0 for t ≠ 1. The relationship between the derivative of a function and the Mellin transformation is as follows:
$ℳ[f(x)​:​s]=​F(s)⇒ℳ[f'(x)​:​s]=​−(s​−​1)F(s​−​1).$
From here, by arranging, we find that
$−(x​−​1)Ψ​B^p(x​−​1,y)=​−​(y​−​1)Ψ​B^p(x,y​−​1)​+​pΨ​Bp[​​(αi+βi,αi)1,ξ(κj+μj,κj)1,η|x−2,y−2]​−​2pΨ​Bp[​​(αi+βi,αi)1,ξ(κj+μj,κj)1,η|x−1,y−2].$
Finally if x replaced by x + 1 and y replaced by y + 1 we get (8).

## 3 xiΨη-generalization of Gauss and confluent hypergeometric functions

We used the ξ Ψη-beta function (5) to define the generalizations of Gauss and confluent hypergeometric functions as

$Ψ​F^p(a,b;c;z)​:=​Ψ​Fp[​​(βi,αi)1,ξ(μj,κj)1,η| a,b;c;z]​​=​∑n=0∞(a)nΨ​B^p(b​+​n,c​−​b)B(b,c​−​b)znn!$
and
$ΨΦ^p(b;c;z):=ΨΦp[​​(βi,αi)1,ξ(μj,κj)1,η|b;c;z]​​=​∑n=0∞Ψ​B^p(b+n,c−b)B(b,c−b)znn!,$
respectively. We call $Ψ​F^p(a,b;c;z)$ as ξΨη-Gauss hypergeometric function and $ΨΦ^p(b;c;z)$ as ξΨη-confluent hypergeometric function.

The following two theorems are about the integral representations of ξ Ψη-Gauss and ξ Ψη-confluent hypergeometric functions.

Theorem 9

Theξ Ψη-Gauss hypergeometric function has the following integral representations:

$Ψ​F^p(a,b;c;z)​=1B(b,c−b)∫01tb−1(1−t)c−b−1(1−zt)−aξΨη(−pt(1−t))dt,Ψ​F^p(a,b;c;z)​=1B(b,c−b)∫0∞ub−1(1+u)a−c[1+u(1−z)]−aξΨη​(−2p​−​p(u​+​1u))​du,Ψ​F^p(a,b;c;z)​=2B(b,c​−​b)​∫0π2​​(sinθ)2b−1(cosθ)2c−2b−1​(1​−​z(sinθ)2)−aξΨη(−p(secθ)2(cscθ)2)dθ.$

Proof

Direct calculation yields

$Ψ​F^p(a,b;c;z)=∑n=0∞(a)nΨ​B^p(b+n,c−b)B(b,c−b)znn!=1B(b,c−b)∑n=0∞(a)n∫01tb+n−1(1−t)c−b−1ξΨη(−pt(1−t))znn!dt=1B(b,c−b)∫01tb−1(1−t)c−b−1ξΨη(−pt(1−t))∑n=0∞(a)n(zt)nn!dt=1B(b,c−b)∫01tb−1(1−t)c−b−1ξΨη(−pt(1−t))(1−zt)−adt.$
Setting $u=t1−t$ in (9), we get
$Ψ​F^p(a,b;c;z)=1B(b,c−b)∫0∞ub−1(1+u)a−c[1+u(1−z)]−aξΨη(−2p−p(u+1u))du.$
Besides, substituting t = (sinθ)2 in (9), we have
$Ψ​F^p(a,b;c;z)=2B(b,c−b)∫0π2(sinθ)2b−1(cosθ)2c−2b−1(1−z(sinθ)2)−aξΨη(−p(secθ)2(cscθ)2)dθ.$

Similarly, the ξΨη-confluent hypergeometric function is also performed.

Theorem 10

Theξ Ψη-confluent hypergeometric function has the following integral representations:

$ΨΦ^p(b;c;z)=1B(b,c−b)∫01tb−1(1−t)c−b−1eztξΨη(−pt(1−t))dt,ΨΦ^p(b;c;z)=1B(b,c−b)∫01​​uc−b−1(1−u)b−1ez(1−u)ξΨη(−pu(1−u))du.$

In the following theorems, we obtained the derivative formulas of ξ Ψη-Gauss and ξ Ψη-confluent hypergeometric functions with the help of the following equations:

$B(b,c−b)=cbB(b+1,c−b),(a)n+1=a(a+1)n.$

Theorem 11

The following equality holds true:

$dndzn{Ψ​F^p(a,b;c;z)}=(a)n(b)n(c)n[Ψ​F^p(a+n,b+n;c+n;z)].$

Proof

The derivative of the $Ψ​F^p(a,b;c;z)$ according to the argument z is as follows:

$ddz{Ψ​F^p(a,b;c;z)}=ddz{∑n=0∞(a)nΨ​B^p(b+n,c−b)B(b,c−b)znn!}=∑n=1∞(a)nΨ​B^p(b+n,c−b)B(b,c−b)zn−1(n−1)!.$
Replacing n → n + 1, we get
$ddz{Ψ​F^p(a,b;c;z)}=(a)(b)(c)∑n=0∞(a+1)nΨ​B^p(b+n+1,c−b)B(b+1,c−b)znn!=(a)(b)(c)[Ψ​F^p(a+1,b+1;c+1;z)].$
Thus, the general form of the above equation is
$dndzn{Ψ​F^p(a,b;c;z)}=(a)n(b)n(c)n[Ψ​F^p(a+n,b+n;c+n;z)].$
This completes the proof.

Theorem 12

The following equality is provided for Re(b) > 2,Re(c) > Re(b + 2):

$(b​−​1)B(b​−​1,c​−​b​+​1)Ψ​F^p(a,b​−​1;c;z)=(c​−​b​−​1)B(b,c​−​b​−​1)Ψ​F^p(a,b;c​−​1;z)−azB(b,c​−​b)Ψ​F^p(a​+​1,b;c;z)−pB(b​−​2,c​−​b​−​2)Ψ​Fp[​​(αi+βi,αi)1,ξ(κj+μj,κj)1,η​​| a,b​−​2;c​−​4;z]+2pB(b​−​1,c​−​b​−​2)Ψ​Fp[​​(αi+βi,αi)1,ξ(κj+μj,κj)1,η​​| a,b​−​1;c​−​3;z].$

Proof

Since $B(b,c​−​b)Ψ​F^p(a,b;c;z)$ is the Mellin transform of

$f^a,b,c(t​:​z;p)​=​(1​−​t)c−b−1(1​−​zt)−aH(1​−​t) ξΨη(−pt(1−t)),$
$B(b,c​−​b)Ψ​F^p(a,b;c;z)$ has the Mellin transform formula
$B(b,c​−​b)Ψ​F^p(a,b;c;z)​=​ℳ[f^a,b,c(t​:​z;p)​:​b].$
Differentiating $f^a,b,c(t​:​z;p)$ with respect to t we obtain
$ddt{f^a,b,c(t​:​z;p)}=−(c​−​b​−​1)(1​−​t)c−b−2(1​−​zt)−aH(1​−​t)ξΨη(−pt(1−t))​+​az(1​−​t)c−b−1(1​−​zt)−(a+1)H(1​−​t) ξΨη(−pt(1−t))​+​p1t2(1​−​t)c−b−3(1​−​zt)−aH(1​−​t) ξΨη[(αi+βi,αi)1,ξ(κj+μj,κj)1,η|−pt(1−t)]​−​2p1t(1​−​t)c−b−3(1​−​zt)−aH(1​−​t)ξΨη[(αi+βi,αi)1,ξ(κj+μj,κj)1,η|−pt(1−t)].$
Since
$ℳ{f'(t):b}=−(b−1)ℳ{f(t):b−1},$
we get
$(b​−​1)B(b​−​1,c​−​b​+​1)Ψ​F^p(a,b​−​1;c;z)= (c​−​b​−​1)B(b,c​−​b​−​1)Ψ​F^p(a,b;c​−​1;z)−azB(b,c​−​b)Ψ​F^p(a​+​1,b;c;z)−pB(b​−​2,c​−​b​−​2)Ψ​Fp[​​(αi+βi,αi)1,ξ(κj+μj,κj)1,η​​| a,b​−​2;c​−​4;z]+2pB(b​−​1,c​−​b​−​2)Ψ​Fp[​​(αi+βi,αi)1,ξ(κj+μj,κj)1,η​​| a,b​−​1;c​−​3;z],$
which gives the result.

Theorem 13

The following equality holds true:

$dndzn{ΨΦ^p(b;c;z)}=(b)n(c)n[ΨΦ^p(b+n;c+n;z)].$

Proof

The derivative of the $ΨΦ^p(b;c;z)$ according to argument z is

$ddz{ΨΦ^p(b;c;z)}=ddz{∑n=0∞Ψ​B^p(b+n,c−b)B(b,c−b)znn!}=∑n=1∞Ψ​B^p(b+n,c−b)B(b,c−b)zn−1(n−1)!.$
Replacing n → n + 1, we get
$ddz{ΨΦ^p(b;c;z)}=(b)(c)∑n=0∞Ψ​B^p(b+n+1,c−b)B(b+1,c−b)znn!=(b)(c)[ΨΦ^p(b+1;c+1;z)].$
Thus, the general form of the above equation gives
$dndzn{ΨΦ^p(b;c;z)}=(b)n(c)n[ΨΦ^p(b+n;c+n;z)],$
which is the result.

Theorem 14

The following equality is provided for Re(b) > 2,Re(c) > Re(b + 2):

$(b​−​1)B(b​−​1,c​−​b​+​1)ΨΦ^p(b​−​1;c;z)= (c​−​b​−​1)B(b,c​−​b​−​1)ΨΦ^p(b;c​−​1;z)−zB(b,c​−​b)ΨΦ^p(b;c;z)−pB(b​−​2,c​−​b​−​2)ΨΦp[​​(αi+βi,αi)1,ξ(κj+μj,κj)1,η​​| b​−​2;c​−​4;z]+2pB(b​−​1,c​−​b​−​2)ΨΦp[​​(αi+βi,αi)1,ξ(κj+μj,κj)1,η​​| b​−​1;c​−​3;z].$

Proof

Since $B(b,c​−​b)ΨΦ^p(b;c;z)$ is the Mellin transform of

$f^b,c(t​:​z;p)​=​(1​−​t)c−b−1eztH(1​−​t) ξΨη(−pt(1−t)),$
$B(b,c​−​b)ΨΦ^p(b;c;z)$ has the Mellin transform formula
$B(b,c​−​b)ΨΦ^p(b;c;z)​=​ℳ[f^b,c(t​:​z;p)​:​b].$
Differentiating $f^b,c(t​:​z;p)$ with regard to t obtain
$ddt{f^b,c(t​:​z;p)}=−(c​−​b​−​1)(1​−​t)c−b−2eztH(1​−​t) ξΨη(−pt(1−t))​+​z(1​−​t)c−b−1eztH(1​−​t) ξΨη(−pt(1−t))​+​p1t2(1​−​t)c−b−3eztH(1​−​t) ξΨη[(αi+βi,αi)1,ξ(κj+μj,κj)1,η| −pt(1−t)]​−​2p1t(1​−​t)c−b−3eztH(1​−​t) ξΨη[(αi+βi,αi)1,ξ(κj+μj,κj)1,η| −pt(1−t)].$
Since
$ℳ{f'(t):b}=−(b−1)ℳ{f(t):b−1},$
we get
$(b​−​1)B(b​−​1,c​−​b​+​1)ΨΦ^p(b​−​1;c;z)= (c​−​b​−​1)B(b,c​−​b​−​1)ΨΦ^p(b;c​−​1;z)−zB(b,c​−​b)ΨΦ^p(b;c;z)−pB(b​−​2,c​−​b​−​2)ΨΦp[​​(αi+βi,αi)1,ξ(κj+μj,κj)1,η​​| b​−​2;c​−​4;z]+2pB(b​−​1,c​−​b​−​2)ΨΦp[​​(αi+βi,αi)1,ξ(κj+μj,κj)1,η​​| b​−​1;c​−​3;z],$
which completes the proof.

In the following theorems we obtain the Mellin transform formulas of the ξΨη-Gauss and ξ Ψη-confluent hypergeometric functions.

Theorem 15

The following equality is provided for Re(s) > 0:

$ℳ[Ψ​F^p(a,b;c;z):s]=ΨΓ^(s)B(b+s,c+s−b)B(b,c−b)2F1(a,b+s;c+2s;z).$

Proof

By applying Mellin transformation to equality (9), we get

$ℳ[Ψ​F^p(a,b;c;z):s]=∫0∞ps−1[Ψ​F^p(a,b;c;z)]dp=∫0∞ps−1∑n=0∞Ψ​B^p(b+n,c−b)B(b,c−b)(a)nznn!dp=1B(b,c​−​b)​∫01​​tb−1(1​−​t)c−b−1(1​−​zt)−a​​∫0∞​​​​ps−1ξΨη(−pt(1​−​t))​dpdt.$
Substituting $u=pt(1−t)$ in the above equation gives us
$∫0∞ps−1ξΨη(−pt(1−t))dp=ts(1−t)s ΨΓ^(s).$
Thus, we get
$ℳ[Ψ​F^p(a,b;c;z):s]=ΨΓ^(s)B(b+s,c+s−b)B(b,c−b)2F1(a,b+s;c+2s;z).$

Corollary 16

The following equality is provided for Re(s) > 0:

$Ψ​F^p(a,b;c;z)=12πi∫−i∞+i∞ΨΓ^(s)B(b+s,c+s−b)B(b,c−b)2F1(a,b+s;c+2s;z)p−sds.$

Theorem 17

The following equality is provided for Re(s) > 0:

$ℳ[ΨΦ^p(b;c;z):s]=ΨΓ^(s)B(b+s,c+s−b)B(b,c−b)Φ(b+s;c+2s;z).$

Proof

By applying Mellin transformation to equality (10), we get

$ℳ[ΨΦ^p(b;c;z):s]=∫0∞ps−1[ΨΦ^p(b;c;z)]dp=∫0∞ps−1∑n=0∞Ψ​B^p(b+n,c−b)B(b,c−b)znn!dp=1B(b,c−b)∫01tb−1(1−t)c−b−1ezt[∫0∞ps−1ξΨη(−pt(1​−​t))dp]dt.$
Substituting $u=pt(1−t)$ in the above equation we get
$∫0∞ps−1ξΨη(−pt(1−t))dp=ts(1−t)s ΨΓ^(s).$
Thus, we have
$ℳ[ΨΦ^p(b;c;z):s]=ΨΓ^(s)B(b+s,c+s−b)B(b,c−b)Φ(b+s;c+2s;z).$

Corollary 18

For Re(s) > 0, we have the following equality:

$ΨΦ^p(b;c;z)=12πi∫−i∞+i∞ΨΓ^(s)B(b+s,c+s−b)B(b,c−b)Φ(b+s;c+2s;z)p−sds.$

The following two theorems are about the transformation formulas of ξ Ψη-Gauss and ξΨη-confluent hypergeometric functions.

Theorem 19

The following equality holds true:

$Ψ​F^p(a,b;c;z)=(1−z)−a[Ψ​F^p(a,c−b;b;zz−1)].$

Proof

By writing

$[1−z(1−t)]−a=(1−z)−a(1+zt1−z)−a$
and replacing t → 1 − t in (9), we obtain
$Ψ​F^p(a,b;c;z)=(1−z)−aB(b,c−b)∫01tc−b−1(1−t)b−1(1−ztz−1)−aξΨη(−pt(1−t))dt.$
Then we have
$Ψ​F^p(a,b;c;z)=(1−z)−a[Ψ​F^p(a,c−b;b;zz−1)],$
which is the result.

Theorem 20

The following equality holds true:

$ΨΦ^p(b;c;z)=ez[ΨΦ^p(c−b;b;−z)].$

Proof

From the definition of confluent hypergeometric function, we have

$ΨΦ^p(b;c;z)=∑n=0∞Ψ​B^p(b+n,c−b)B(b,c−b)znn!=1B(b,c−b)∫01tb+n−1(1−t)c−b−1eztξΨη(−pt(1−t))dt.$
Replacing t = 1 − u in (14), we obtain
$ΨΦ^p(b;c;z)=1B(b,c​−​b)​∫01​​uc−b−1(1​−​u)b+n−1ez(1−u)ξΨη(−pu(1​−​u))du=ez[ΨΦ^p(c−b;b;−z)],$
which gives the result.

The following theorems are about the differential and difference relations for ξ Ψη-Gauss hypergeometric and ξ Ψη-confluent hypergeometric functions.

Theorem 21

The following relations hold true:

$Δa[Ψ​F^p(a,b;c;z)]=zbc Ψ​F^p(a​+​1,b​+​1;c​+​1;z)$
$aΔa[Ψ​F^p(a,b;c;z)]=zddz{Ψ​F^p(a,b;c;z)}$
$bΔb[ΨΦ^p(b;c​+​1;z)]=−cΔc[ΨΦ^p(b;c;z)]$
$ddz{ΨΦ^p(b;c;z)}=bc ΨΦ^p(b;c​+​1;z)​−​Δc[ΨΦ^p(b;c;z)]$
where Δαdenotes the difference operator defined by
$Δαf(α,…)=f(α+1,…)−f(α,…).$

Proof

It is seen from (9) and the difference operator Δa that

$ΔaΨ​F^(a,b;c;z)=Ψ​F^(a+1,b;c;z)−Ψ​F^(a,b;c;z)=zB(b,c−b)∫01tb(1−t)c−b−1(1−zt)−a−1ξΨη(−pt(1−t))dt.$
If we write a + 1,b + 1 and c + 1 instead of a,b and c in equation (9), we get the following equation:
$Ψ​F^(a​+​1,b​+​1;c​+​1;z)=1B(b+1,c−b)∫01tb(1−t)c−b−1(1−zt)−a−1ξΨη(−pt(1−t))dt.$
Now using (11) and (20) in (19) we get (15). Using differentiation formula (12) proves (16). Using the difference operator and (10), we obtain (17). Using differentiation formula (13) with n = 1, and considering (17) gives us (18).

## 4 Results and Recommendations

In this study, we introduced new generalizations of gamma, beta, Gauss and confluent hypergeometric functions with the help of Fox-Wright function. We also obtained some of their integral representations, Mellin transformations, derivative formulas, transformation formulas and reduction relations.

When the special cases of these functions are examined, it is seen that these functions are the generalizations of the following predefined functions which can be found in the literature:

For p = 0:

$Γ(x)=ΨΓ0[​​(1,0)1,1(1,0)1,1|x],B(x,y)=Ψ​B0[​​(1,0)1,1(1,0)1,1|x,y],F(a,b;c;z)=Ψ​F0[​​(1,0)1,1(1,0)1,1|a,b;c;z],Φ(b;c;z)=Ψ​Φ0[​​(1,0)1,1(1,0)1,1|b;c;z].$
For p ≠ 0:
$Γp(x)=ΨΓp[​​(1,0)1,1(1,0)1,1|x],Bp(x,y)=Ψ​Bp[​​(1,0)1,1(1,0)1,1|x,y],Fp(a,b;c;z)=Ψ​Fp[​​(1,0)1,1(1,0)1,1|a,b;c;z],Φp(b;c;z)=Ψ​Φp[​​(1,0)1,1(1,0)1,1|b;c;z].$
And also for p ≠ 0:
$Γp(α,β)(x)=Γ(β)Γ(α)ΨΓp[​​(α,1)1,1(β,1)1,1|x],Bp(α,β)(x,y)=Γ(β)Γ(α)Ψ​Bp[​​(α,1)1,1(β,1)1,1|x,y],Fp(α,β)(a,b;c;z)=Γ(β)Γ(α)Ψ​Fp[​​(α,1)1,1(β,1)1,1|a,b;c;z],Φp(α,β)(b;c;z)=Γ(β)Γ(α)Ψ​Φp[​​(α,1)1,1(β,1)1,1|b;c;z],$
where Γ,B,F and Φ are the classic gamma, beta, Gauss and confluent hypergeometric functions; Γp, Bp, Fp and Φp are the functions defined in [7, 8, 10]; $Γp(α,β)$ , $Bp(α,β)$ , $Fp(α,β)$ and $Φp(α,β)$ are the functions defined in .

Besides, the generalized beta function described in this study can be used to define similar generalizations of multivariate hypergeometric functions, which also known as Appell, Lauricella, Horn and Srivastava functions (see [3, 5] and the references therein). Further properties of these functions can be examined and they can also be used in fractional theory (see for example [2, 4, 30] and the references therein).

Acknowledgements

This work was partly presented in the 4th International Conference on Computational Mathematics and Engineering Sciences which organized by Akdeniz University on April 20-22, 2019 in Antalya-Turkey.

## References

• 

P. Agarwal, (2014), Certain properties of the generalized Gauss hypergeometric functions, Applied Mathematics and Information Sciences, 8 (5), pp. 2315–2320.

• 

P. Agarwal, (2017), Some inequalities involving Hadamard-type k-fractional integral operators, Mathematical Methods in the Applied Sciences, 40 (11), pp. 3882–3891.

• 

P. Agarwal, J. Choi and S. Jain, (2015), Extended hypergeometric functions of two and three variables, Commun. Korean Math. Soc, 30 (4), pp. 403–414.

• 

P. Agarwal, S. S. Dragomir, M. Jleli and B. Samet, (2018), Advances in Mathematical Inequalities and Applications, Springer, Singapore.

• 

R. P. Agarwal, M. J. Luo and P. Agarwal, (2017), On the Extended Appell-Lauricella Hypergeometric Functions and their Applications, Filomat, 31 (12), pp. 3693–3713.

• 

G. E. Andrews, R. Askey, and R. Roy, (1999), Special Functions, Cambridge University Press, Cambridge.

• 

M. A. Chaudhry, A. Qadir, M. Rafique and S. M. Zubair, (1997), Extension of Euler’s beta function, Journal of Computational and Applied Mathematics, 78, pp. 19–32.

• 

M. A. Chaudhry, A. Qadir, H. M. Srivastava and R. B. Paris, (2004), Extended hypergeometric and confluent hypergeometric functions, Applied Mathematics and Computation, 159, pp. 589–602.

• 

M. A. Chaudhry, N. M. Temme and E. J. M. Veling, (1996), Asymptotic and closed form of a generalized incomplete gamma function, Journal of Computational and Applied Mathematics, 67, pp. 371–379.

• 

M. A. Chaudhry and S. M. Zubair, (1994), Generalized incomplete gamma functions with applications, Journal of Computational and Applied Mathematics, 55, pp. 99–124.

• 

M. A. Chaudhry and S. M. Zubair, (1995), On the decomposition of generalized incomplete gamma functions with applications to Fourier transforms, Journal of Computational and Applied Mathematics, 59, pp. 253–284.

• 

M. A. Chaudhry and S. M. Zubair, (2002), Extended incomplete gamma functions with applications, Journal of Mathematical Analysis and Applications 274, pp. 725–745.

• 

J. Choi, A. K. Rathie and R. K. Parmar, (2014), Extension of extended beta, hypergeometric and confluent hypergeometric functions, Honam Mathematical Journal, 36, pp. 357–385.

• 

A. Çetinkaya, İ. O. Kıymaz, P. Agarwal and R. Agarwal, (2018), A comparative study on generating function relations for generalized hypergeometric functions via generalized fractional operators, Advances in Difference Equations, 2018:156.

• 

C. Fox, (1928), The asymptotic expansion of generalized hypergeometric functions, Proceedings of the London Mathematical Society (Ser. 2), 27, pp. 389–400.

• 

C. Fox, (1961), The G and H functions as symmetrical Fourier kernels, Transactions of the American Mathematical Society, 98, pp. 395–429.

• 

A. Goswami, S. Jain, P. Agarwal and S. Aracı, (2018), A note on the new extended beta and Gauss hypergeometric functions, Applied Mathematics and Information Sciences, 12, pp. 139–144.

• 

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, (2006), Theory and Applications of Fractional Differential, North-Holland Mathematics Studies 204.

• 

D. M. Lee, A. K. Rathie, R. K. Parmar and Y. S. Kim, (2011), Generalization of extended beta function, hypergeometric and confluent hypergeometric functions, Honam Mathematical Journal, 33, pp. 187–206.

• 

M-J. Luo, G. V. Milovanovic and P. Agarwal, (2014), Some results on the extended beta and extended hypergeometric functions, Applied Mathematics and Computation 248, pp. 631–651.

• 

A. R. Miller, (1998), Remarks on a generalized beta function, Journal of Computational and Applied Mathematics, 100, pp. 23–32.

• 

A. R. Miller, (2000), Reduction of a generalized incomplete gamma function, related Kampe de Feriet functions, and incomplete Weber integrals, The Rocky Mountain Journal of Mathematics, 30, pp. 703–714.

• 

S. Mubeen, G. Rahman, K. S. Nisar, J. Choi and M. Arshad, (2017), An extended beta function and its properties, Far East Journal of Mathematical Sciences, 102, pp. 1545–1557.

• 

F. AL. Musallam and S. L. Kalla, (1998), Further results on a generalized gamma function occurring in diffraction theory, Integral Transforms and Special Functions, 7 (34), pp. 175–190.

• 

E. Özergin, M. A. Özarslan and A. Altın, (2011), Extension of gamma, beta and hypergeometric functions, Journal of Computational and Applied Mathematics, 235, pp. 4601–4610.

• 

R. K. Parmar, (2013), A new generalization of gamma, beta, hypergeometric and confluent hypergeometric functions, Le Matematiche, Vol. LXVIII, pp. 33–52.

• 

P. I. Pucheta, (2017), An new extended beta function, International Journal of Mathematics and its Applications, 5 (3-C), pp. 255–260.

• 

G. Rahman, K. S. Nisar and S. Mubeen, (2018), A new generalization of extended beta and hypergeometric functions, Preprints, DOI:.

• Crossref
• Export Citation
• 

G. Rahman, G. Kanwal, K. S. Nisar and A. Ghaffar, (2018), A new extension of beta and hypergeometric function, Preprints, DOI:.

• Crossref
• Export Citation
• 

M. V. Ruzhansky, Y. Je Cho, P. Agarwal and I. Area, (2017), Advances in Real and Complex Analysis with Applications, Springer, Singapore.

• 

M. Shadab, J. Saime and J. Choi, (2018), An extended beta function and its application, Journal of Mathematical Sciences, 103, pp. 235–251.

• 

H. M. Srivastava, P. Agarwal and S. Jain, (2014), Generating functions for the generalized Gauss hypergeometric functions, Applied Mathematics and Computation, 247, pp. 348–352.

• 

R. Şahin, O. Yağcı, M. B. Yağbasan, İ. O. Kıymaz and A. Çetinkaya, (2018), Further generalizations of gamma, beta and related functions, Journal of Inequalities and Special Functions, 9 (4), pp. 1–7.

• 

E. M. Wright, (1935), The asymptotic expansion of the generalized hypergeometric function, Journal of the London Mathematical Society, 10, pp. 286–293.

• 

E. M. Wright, (1940), The asymptotic expansion of integral functions defined by Taylor Series, Philosophical Transactions of the Royal Society of London. Series A., 238, pp. 423–451.

• 

E. M. Wright, (1940), The asymptotic expansion of the generalized hypergeometric function II, Proceedings of the London Mathematical Society, 46 (2), pp. 389–408.

If the inline PDF is not rendering correctly, you can download the PDF file here.

• 

P. Agarwal, (2014), Certain properties of the generalized Gauss hypergeometric functions, Applied Mathematics and Information Sciences, 8 (5), pp. 2315–2320.

• 

P. Agarwal, (2017), Some inequalities involving Hadamard-type k-fractional integral operators, Mathematical Methods in the Applied Sciences, 40 (11), pp. 3882–3891.

• 

P. Agarwal, J. Choi and S. Jain, (2015), Extended hypergeometric functions of two and three variables, Commun. Korean Math. Soc, 30 (4), pp. 403–414.

• 

P. Agarwal, S. S. Dragomir, M. Jleli and B. Samet, (2018), Advances in Mathematical Inequalities and Applications, Springer, Singapore.

• 

R. P. Agarwal, M. J. Luo and P. Agarwal, (2017), On the Extended Appell-Lauricella Hypergeometric Functions and their Applications, Filomat, 31 (12), pp. 3693–3713.

• 

G. E. Andrews, R. Askey, and R. Roy, (1999), Special Functions, Cambridge University Press, Cambridge.

• 

M. A. Chaudhry, A. Qadir, M. Rafique and S. M. Zubair, (1997), Extension of Euler’s beta function, Journal of Computational and Applied Mathematics, 78, pp. 19–32.

• 

M. A. Chaudhry, A. Qadir, H. M. Srivastava and R. B. Paris, (2004), Extended hypergeometric and confluent hypergeometric functions, Applied Mathematics and Computation, 159, pp. 589–602.

• 

M. A. Chaudhry, N. M. Temme and E. J. M. Veling, (1996), Asymptotic and closed form of a generalized incomplete gamma function, Journal of Computational and Applied Mathematics, 67, pp. 371–379.

• 

M. A. Chaudhry and S. M. Zubair, (1994), Generalized incomplete gamma functions with applications, Journal of Computational and Applied Mathematics, 55, pp. 99–124.

• 

M. A. Chaudhry and S. M. Zubair, (1995), On the decomposition of generalized incomplete gamma functions with applications to Fourier transforms, Journal of Computational and Applied Mathematics, 59, pp. 253–284.

• 

M. A. Chaudhry and S. M. Zubair, (2002), Extended incomplete gamma functions with applications, Journal of Mathematical Analysis and Applications 274, pp. 725–745.

• 

J. Choi, A. K. Rathie and R. K. Parmar, (2014), Extension of extended beta, hypergeometric and confluent hypergeometric functions, Honam Mathematical Journal, 36, pp. 357–385.

• 

A. Çetinkaya, İ. O. Kıymaz, P. Agarwal and R. Agarwal, (2018), A comparative study on generating function relations for generalized hypergeometric functions via generalized fractional operators, Advances in Difference Equations, 2018:156.

• 

C. Fox, (1928), The asymptotic expansion of generalized hypergeometric functions, Proceedings of the London Mathematical Society (Ser. 2), 27, pp. 389–400.

• 

C. Fox, (1961), The G and H functions as symmetrical Fourier kernels, Transactions of the American Mathematical Society, 98, pp. 395–429.

• 

A. Goswami, S. Jain, P. Agarwal and S. Aracı, (2018), A note on the new extended beta and Gauss hypergeometric functions, Applied Mathematics and Information Sciences, 12, pp. 139–144.

• 

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, (2006), Theory and Applications of Fractional Differential, North-Holland Mathematics Studies 204.

• 

D. M. Lee, A. K. Rathie, R. K. Parmar and Y. S. Kim, (2011), Generalization of extended beta function, hypergeometric and confluent hypergeometric functions, Honam Mathematical Journal, 33, pp. 187–206.

• 

M-J. Luo, G. V. Milovanovic and P. Agarwal, (2014), Some results on the extended beta and extended hypergeometric functions, Applied Mathematics and Computation 248, pp. 631–651.

• 

A. R. Miller, (1998), Remarks on a generalized beta function, Journal of Computational and Applied Mathematics, 100, pp. 23–32.

• 

A. R. Miller, (2000), Reduction of a generalized incomplete gamma function, related Kampe de Feriet functions, and incomplete Weber integrals, The Rocky Mountain Journal of Mathematics, 30, pp. 703–714.

• 

S. Mubeen, G. Rahman, K. S. Nisar, J. Choi and M. Arshad, (2017), An extended beta function and its properties, Far East Journal of Mathematical Sciences, 102, pp. 1545–1557.

• 

F. AL. Musallam and S. L. Kalla, (1998), Further results on a generalized gamma function occurring in diffraction theory, Integral Transforms and Special Functions, 7 (34), pp. 175–190.

• 

E. Özergin, M. A. Özarslan and A. Altın, (2011), Extension of gamma, beta and hypergeometric functions, Journal of Computational and Applied Mathematics, 235, pp. 4601–4610.

• 

R. K. Parmar, (2013), A new generalization of gamma, beta, hypergeometric and confluent hypergeometric functions, Le Matematiche, Vol. LXVIII, pp. 33–52.

• 

P. I. Pucheta, (2017), An new extended beta function, International Journal of Mathematics and its Applications, 5 (3-C), pp. 255–260.

• 

G. Rahman, K. S. Nisar and S. Mubeen, (2018), A new generalization of extended beta and hypergeometric functions, Preprints, DOI:.

• Crossref
• Export Citation
• 

G. Rahman, G. Kanwal, K. S. Nisar and A. Ghaffar, (2018), A new extension of beta and hypergeometric function, Preprints, DOI:.

• Crossref
• Export Citation
• 

M. V. Ruzhansky, Y. Je Cho, P. Agarwal and I. Area, (2017), Advances in Real and Complex Analysis with Applications, Springer, Singapore.

• 

M. Shadab, J. Saime and J. Choi, (2018), An extended beta function and its application, Journal of Mathematical Sciences, 103, pp. 235–251.

• 

H. M. Srivastava, P. Agarwal and S. Jain, (2014), Generating functions for the generalized Gauss hypergeometric functions, Applied Mathematics and Computation, 247, pp. 348–352.

• 

R. Şahin, O. Yağcı, M. B. Yağbasan, İ. O. Kıymaz and A. Çetinkaya, (2018), Further generalizations of gamma, beta and related functions, Journal of Inequalities and Special Functions, 9 (4), pp. 1–7.

• 

E. M. Wright, (1935), The asymptotic expansion of the generalized hypergeometric function, Journal of the London Mathematical Society, 10, pp. 286–293.

• 

E. M. Wright, (1940), The asymptotic expansion of integral functions defined by Taylor Series, Philosophical Transactions of the Royal Society of London. Series A., 238, pp. 423–451.

• 

E. M. Wright, (1940), The asymptotic expansion of the generalized hypergeometric function II, Proceedings of the London Mathematical Society, 46 (2), pp. 389–408.

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