# An Extension of Wright Function and Its Properties

Authors: Moustafa El-Shahed; Ahmed Salem
Journal: Journal of Mathematics
 (2015)

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## Abstract

The paper is devoted to the study of the functionWα,βγ,δ(z), which is an extension of the classical Wright function and Kummer confluent hypergeometric function. The properties of Wα,βγ,δ(z) including its auxiliary functions and the integral representations are proven.


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## Body

## 1. Introduction

The special functions of mathematical physics are found to be very useful for finding solutions of initial and/or boundary-value problems governed by partial differential equations and fractional differential equations. Special functions have widespread applications in other areas of mathematics and often new perspectives in special functions are motivated by such connections. Several special functions, called recently special functions of fractional calculus, play a very important and interesting role as solutions of fractional order differential equations, such as the Mittag-Leffler function, Wright function with its auxiliary functions, and Fox’sH-function.The Wright function is one of the special functions which plays an important role in the solution of linear partial fractional differential equations. It was introduced for the first time in [1, 2] in connection with a problem in the number theory regarding the asymptotic of the number of some special partitions of the natural numbers. Recently this function has appeared in papers related to partial differential equations of fractional order. Considering the boundary-value problems for the fractional diffusion-wave equation, that is, the linear partial integrodifferential equation obtained from the classical diffusion or wave equation by replacing the first- or second-order time derivative by a fractional derivative of order α with 0<α<2, it was found that the corresponding Green functions can be represented in terms of the Wright function. Furthermore, extending the methods of Lie groups in partial differential equations to the partial differential equations of fractional order, it was shown that some of the group-invariant solutions of these equations can be given in terms of the Wright and the generalized Wright functions [3]. A list of formulas concerning this function can be found in the handbook of Bateman Project, Erdélyi et al. 1953 [4]; see also [3, 5–11].The Wright function is defined by the series representation, valid in the whole complex plane(1)Wα,βz=∑n=0∞znn!Γαn+β,α>-1,β∈C. It is an entire function of order 1/(1+α), which has been known also as generalized Bessel (or Bessel Maitland) function [12, 13]. There are two auxiliary functions of Wright function defined as(2)Mαz=W-α,1-α-z=∑n=0∞-1nznn!Γ1-αn+10<α<1,Fαz=W-α,0-z=∑n=1∞-1nznn!Γ-αn0<α<1,where the function Mα(z) is recently known as the Mainardi function [12, 13]. In a continuation of this study, we investigate the generalized Wright function Wα,βγ,δ(z) which is defined for real α and β,γ,δ∈C;α>-1,δ≠0,-1,-2,… with z∈C and |z|<1 with α=-1 as (3)Wα,βγ,δz=∑n=0∞γnδnΓαn+βznn!, where (γ)n=Γγ+n/Γγ=γγ+1γ+2⋯(γ+n-1) is a Pochhammer symbol and Γ(·) is a gamma function. The function Wα,βγ,δ(z) is an entire function of order 1/(1+α) and generalized Wright function (1).By analogy with (2), one can also introduce the two (generalized Wright type) auxiliary functions for any order α∈(0,1) and for all complex variable z≠0 by(4)Mαγ,δz≔W-α,1-αγ,δ-z=∑n=0∞γnδnΓ1-αn+1-1nznn!,(5)Fαγ,δz≔W-α,0γ,δ-z=∑n=1∞γnδnΓ-αn-1nznn!.


## 2. Basic Definitions

In this section, the definitions of some special functions and their properties necessary for our studies on generalized Wright function (3) and its auxiliary functions (4) and (5) will be presented to provide convenient references.


### 2.1. Wright Function

The Wright function which is defined by (1) has the Hankel contour integral representation and special case(6)Wα,βz=12πi∫Hat-βet+ztαdt,(7)W-1/2,1-z=erfc⁡z2,where Ha is the Hankel contour and erfc(z) is the complementary error function which is defined as (8)erfc⁡z=2π∫z∞e-t2dt=1-erf⁡z. The important relation between two auxiliary functions (2) of Wright function (1) and some special cases are(9)Fαz=αzMαz,(10)M1/2z=1πe-z2/4,(11)M1/3z=32/3Aiz31/3,where Ai(z) is Airy’s function which is the one of the linearly independent solutions of the ordinary differential equation y′′-zy=0 and has the following properties [14, 15]:(12)Aix=13xI-1/323x3/2-I1/323x3/2(13)=132/3Γ2/3F0123;x39-x31/3Γ1/3F0143;x39,(14)Ai0=132/3Γ2/3,Ai′0=-131/3Γ1/3.Airy’s function has, from its differential equation, the following indefinite integrals:(15)∫xAixdx=Ai′x,∫x2Aixdx=xAi′x-Aix,∫xn+3Aixdx=xn+2Ai′x-n+2xn+1Aix+n+1n+2∫xnAixdx,n=0,1,2,….


### 2.2. Mittag-Leffler Function

A two-parameter function of the Mittag-Leffler type is defined by the series expansion [12, 13] (16)Eα,βz=∑k=0∞zkΓαk+βα>0,β∈C.It follows from the definition that (17)E1,mz=1zm-1ez-∑k=0m-2zkk!. Let m>1 be an integer, let ρ1,ρ2,…,ρm>0, and let μ1,…,μm be arbitrary real numbers. By means of “multi-index” (ρi),(μi), we introduce the so-called multi-index Mittag-Leffler functions [16, 17]: (18)E1/ρi,μiz=∑k=0∞ϕkzk=∑k=0∞zkΓμ1+k/ρ1⋯Γμm+k/ρm.


### 2.3. Fox-Wright Function

The Fox-WrightΨpq(z) function has series representation [5, 12, 18–20]: (19)Ψpqzb1,β1,…,bq,βq;a1,α1,…,ap,αp;=∑n=0∞∏i=1pΓai+αin∏j=1qΓbj+βjnznn!. The Fox-Wright Ψ-function is special case of Fox’s H-function: (20)Ψpqzb1,β1,…,bq,βq;a1,α1,…,ap,αp;=Hpq+11p-z0,1,1-b1,β1,…,1-bq,βq1-a1,α1⋯1-ap,αp, where Hpqmnz(bk,Bk)1q(ak,Ak)1p denotes the Fox H-function [21] which can be expressed as Mellin-Barnes contour integral representation in the form(21)Hpqmnz=Hpqmnzbk,Bk1qak,Ak1p=12πi∫L∏k=1mΓbk-Bks∏j=1nΓ1-aj+sAj∏k=m+1qΓ1-bk+sBk∏j=n+1pΓaj-sAjzsds,where L is a suitable contour in the complex s-plane, the orders (m,n,p,q) are integers 1≤m≤q, 0≤n≤p, and the parameters aj∈R, Aj>0, j=1,2,…,p, and bk∈R, Bk>0, k=1,2,…,q, are such that Aj(bk+i)≠Bk(aj-i-1),i=0,1,2,….


### 2.4. MeijerG-Function

In 1936, C. S. Meijer introduced theG-function as [14](22)Gpqmnxcqap=Gpqmnxc1,c2,…,cqa1,a2,…,ap=Hpqmnxbk,11qak,11p=∑k=1m∏j=1mΓcj-ck∏j=1nΓ1+ck-ajxck∏j=m+1qΓ1+ck-cj∏j=n+1pΓaj-ckFpq-11+ck-a1,…,1+ck-ap;1+ck-c1,…,1+ck-cq;-1p-m-nx,where 1≤m≤q, 0≤n≤p≤q-1, and Fpq(a1,…,ap;c1,…,cq;x) is the generalized hypergeometric function which is defined as (23)Fpqa1,…,ap;c1,…,cq;x=∑n=0∞∏r=1parn∏r=1qbrnxnn!. If p=q=1, we have the confluent hypergeometric function which has the integral representation with the property (24)F11a;b;x=ΓbΓaΓb-a∫01ua-11-ub-a-1exudu=exF11b-a;b;-x.


### 2.5. Gamma Function and Incomplete Gamma Function

The contour integral representation for the reciprocal gamma function, multiplication formula, and important property [22] are(25)1Γα=12πi∫Haett-αdt,(26)Γx=mx-1/22π1-m/2∏r=1mΓx+r-1m,(27)Γa-n=-1nΓa1-an,n=0,1,2,….The upper and lower incomplete gamma function, respectively, are(28)Γs,x=∫x∞ts-1e-tdt,Rs>0,(29)γs,x=∫0xts-1e-tdt,Rs>0.The two types of incomplete gamma function have the properties(30)Γs,x+γs,x=Γs,γ12,z2=πerf⁡z,γn,z=n-1!1-e-z∑k=0n-1zkk!,n=1,2,…,γn+12,z2=12nπerf⁡z-e-z2∑k=1nz2k-11/2k,n=1,2,….


## 2.1. Wright Function

The Wright function which is defined by (1) has the Hankel contour integral representation and special case(6)Wα,βz=12πi∫Hat-βet+ztαdt,(7)W-1/2,1-z=erfc⁡z2,where Ha is the Hankel contour and erfc(z) is the complementary error function which is defined as (8)erfc⁡z=2π∫z∞e-t2dt=1-erf⁡z. The important relation between two auxiliary functions (2) of Wright function (1) and some special cases are(9)Fαz=αzMαz,(10)M1/2z=1πe-z2/4,(11)M1/3z=32/3Aiz31/3,where Ai(z) is Airy’s function which is the one of the linearly independent solutions of the ordinary differential equation y′′-zy=0 and has the following properties [14, 15]:(12)Aix=13xI-1/323x3/2-I1/323x3/2(13)=132/3Γ2/3F0123;x39-x31/3Γ1/3F0143;x39,(14)Ai0=132/3Γ2/3,Ai′0=-131/3Γ1/3.Airy’s function has, from its differential equation, the following indefinite integrals:(15)∫xAixdx=Ai′x,∫x2Aixdx=xAi′x-Aix,∫xn+3Aixdx=xn+2Ai′x-n+2xn+1Aix+n+1n+2∫xnAixdx,n=0,1,2,….


## 2.2. Mittag-Leffler Function

A two-parameter function of the Mittag-Leffler type is defined by the series expansion [12, 13] (16)Eα,βz=∑k=0∞zkΓαk+βα>0,β∈C.It follows from the definition that (17)E1,mz=1zm-1ez-∑k=0m-2zkk!. Let m>1 be an integer, let ρ1,ρ2,…,ρm>0, and let μ1,…,μm be arbitrary real numbers. By means of “multi-index” (ρi),(μi), we introduce the so-called multi-index Mittag-Leffler functions [16, 17]: (18)E1/ρi,μiz=∑k=0∞ϕkzk=∑k=0∞zkΓμ1+k/ρ1⋯Γμm+k/ρm.


## 2.3. Fox-Wright Function

The Fox-WrightΨpq(z) function has series representation [5, 12, 18–20]: (19)Ψpqzb1,β1,…,bq,βq;a1,α1,…,ap,αp;=∑n=0∞∏i=1pΓai+αin∏j=1qΓbj+βjnznn!. The Fox-Wright Ψ-function is special case of Fox’s H-function: (20)Ψpqzb1,β1,…,bq,βq;a1,α1,…,ap,αp;=Hpq+11p-z0,1,1-b1,β1,…,1-bq,βq1-a1,α1⋯1-ap,αp, where Hpqmnz(bk,Bk)1q(ak,Ak)1p denotes the Fox H-function [21] which can be expressed as Mellin-Barnes contour integral representation in the form(21)Hpqmnz=Hpqmnzbk,Bk1qak,Ak1p=12πi∫L∏k=1mΓbk-Bks∏j=1nΓ1-aj+sAj∏k=m+1qΓ1-bk+sBk∏j=n+1pΓaj-sAjzsds,where L is a suitable contour in the complex s-plane, the orders (m,n,p,q) are integers 1≤m≤q, 0≤n≤p, and the parameters aj∈R, Aj>0, j=1,2,…,p, and bk∈R, Bk>0, k=1,2,…,q, are such that Aj(bk+i)≠Bk(aj-i-1),i=0,1,2,….


## 2.4. MeijerG-Function

In 1936, C. S. Meijer introduced theG-function as [14](22)Gpqmnxcqap=Gpqmnxc1,c2,…,cqa1,a2,…,ap=Hpqmnxbk,11qak,11p=∑k=1m∏j=1mΓcj-ck∏j=1nΓ1+ck-ajxck∏j=m+1qΓ1+ck-cj∏j=n+1pΓaj-ckFpq-11+ck-a1,…,1+ck-ap;1+ck-c1,…,1+ck-cq;-1p-m-nx,where 1≤m≤q, 0≤n≤p≤q-1, and Fpq(a1,…,ap;c1,…,cq;x) is the generalized hypergeometric function which is defined as (23)Fpqa1,…,ap;c1,…,cq;x=∑n=0∞∏r=1parn∏r=1qbrnxnn!. If p=q=1, we have the confluent hypergeometric function which has the integral representation with the property (24)F11a;b;x=ΓbΓaΓb-a∫01ua-11-ub-a-1exudu=exF11b-a;b;-x.


## 2.5. Gamma Function and Incomplete Gamma Function

The contour integral representation for the reciprocal gamma function, multiplication formula, and important property [22] are(25)1Γα=12πi∫Haett-αdt,(26)Γx=mx-1/22π1-m/2∏r=1mΓx+r-1m,(27)Γa-n=-1nΓa1-an,n=0,1,2,….The upper and lower incomplete gamma function, respectively, are(28)Γs,x=∫x∞ts-1e-tdt,Rs>0,(29)γs,x=∫0xts-1e-tdt,Rs>0.The two types of incomplete gamma function have the properties(30)Γs,x+γs,x=Γs,γ12,z2=πerf⁡z,γn,z=n-1!1-e-z∑k=0n-1zkk!,n=1,2,…,γn+12,z2=12nπerf⁡z-e-z2∑k=1nz2k-11/2k,n=1,2,….


## 3. Integral Representations of the Generalized Wright Function

Here, we introduce the Mellin-Barnes contour integral representation and definite integral representation of function (3).Theorem 1 (Mellin-Barnes contour integral representation).
Letα>-1;γ,δ,β∈C, and δ≠0,-1,-2,…. Then function (3) can be represented by the Mellin-Barnes contour integral as (31)Wα,βγ,δz=ΓδΓγ12πi∫LΓγ+sΓδ+sΓβ+αsΓ-s-zsds, where z is not equal to zero and (32)-zs=exp⁡slog⁡z+iarg⁡-z, in which log denotes the natural logarithm and arg(-z) is not necessarily the principal value, and L is a suitable path in the complex s-plane which runs from s=-i∞ to s=i∞, so the points s=n, n=0,1,2,…, lie to the right of a contour L and the points s=-γ-n, n=0,1,2,…, lie to its left.Proof.
Consider the integral in (31) with the contour L replaced by the contour CR consisting of a large clockwise-oriented semicircle of radius R and the center of the origin which lies to the right of the contour L and is bounded away from the poles. We can apply Cauchy’s theorem (Residues Theorem) to the closed contour which is consisting of the contour CR and that part of L terminated above and below by CR as R→∞; we obtain(33)12πi∫LΓγ+sΓδ+sΓβ+αsΓ-s-zsds=∑n=0∞Ress=nΓγ+sΓδ+sΓβ+αsΓ-s-zs=∑n=0∞lims→n⁡Γγ+sΓδ+sΓβ+αss-nΓ-s-zs=∑n=0∞Γγ+nΓδ+nΓβ+αnznn!=ΓγΓδWα,βγ,δz.Theorem 2.
Letα>-1;γ,δ,β∈C, and R(δ)>R(γ)>0. Then function (3) can be represented as (34)Wα,βγ,δz=ΓδΓγΓδ-γ∫01uγ-11-uδ-γ-1Wα,βzudu, and consequently one can get(35)Mαγ,δz=ΓδΓγΓδ-γ∫01uγ-11-uδ-γ-1Mαzudu,(36)Fαγ,δz=ΓδΓγΓδ-γ∫01uγ-11-uδ-γ-1Fαzudu.Proof.
Using the contour integral representation for reciprocal gamma function (25), the integral representation of confluent hypergeometric function (23), and the Hankel contour integral representation of Wright function (6), then(37)Wα,βγ,δz=∑n=0∞γnδnΓαn+βznn!=12πi∑n=0∞γnδnznn!∫Hat-αn-βetdt=12πi∫Hat-βet∑n=0∞γnδnt-αnznn!dt=12πi∫Hat-βetF11γ;δ;zt-αdt=12πi∫Hat-βetΓδΓγΓδ-γ∫01uγ-11-uδ-γ-1ezut-αdudt=ΓδΓγΓδ-γ∫01uγ-11-uδ-γ-112πi∫Hat-βet+zut-αdtdu=ΓδΓγΓδ-γ∫01uγ-11-uδ-γ-1Wα,βzudu.


## 4. Relationship with Some Known Special Functions

We devote this section to studying the relationship between generalized Wright function (3) and some known special functions like Fox H-function, Fox-Wright function, Meijer G-function, Mittag-Leffler function, and generalized hypergeometric function.


### 4.1. Relation with FoxH-Function

Using (31) and the definition of Fox H-function, we obtain (38)Wα,βγ,δ-z=ΓδΓγH1311z0,1,1-β,α,1-δ,11-γ,1.


### 4.2. Relation with Fox Wright Function

Generalized Wright function (3) can be represented by Fox Wright function (19) as (39)Wα,βγ,δz=ΓδΓγΨ12zδ,1,β,α;γ,1;.


### 4.3. Relation with MeijerG-Function

From the definitions of generalized Wright function (3) and Meijer G-function (22) with α=1, we obtain (40)W1,βγ,δ-z=ΓδΓγG1311z0,1-β,1-δ1-γ.Theorem 3.
Letα=p/q be a rational number with p,q∈N∪{0},q≠0;γ,δ,β∈C and δ≠0,-1,-2,…; then one has(41)Wp/q,βγ,δ-z=p1/2-βqγ-δ+1/22πp-q/2ΓδΓγGq2q+pqqzqqqpp0,1/q,…,q-1/q;1-δ/q,1-δ+1/q,…,1-δ+q-1/q;1-β/p,1-β+1/p,…,1-β+p-1/p1-γ/q,1-γ+1/q,…,1-γ+q-1/q.In particular, if γ=δ, (42)Wp/q,β-z=2πp-q/2p1/2-βq1/2G0p+qq0zqqqpp0,1/q,…,q-1/q;1-β/p,1-β+1/p,…,1-β+p-1/p----------.Proof.
Using the contour integral representation of generalized Wright function (31) with change of the variable from s to qs and multiplication formula (26), we get(43)Wp/q,βγ,δ-z=ΓδΓγq2πi∫LΓγ+qsΓδ+qsΓβ+psΓ-qszqsds=p1/2-βqγ-δ+1/22πp-q/2ΓδΓγ12πi∫L∏r=1qΓr-1/q-s∏r=1qΓγ+r-1/q+s∏r=1qΓδ+r-1/q+s∏r=1pΓβ+r-1/p+szqqqppsds=p1/2-βqγ-δ+1/22πp-q/2ΓδΓγGq2q+pqqzqqqpp0,1/q,…,q-1/q;1-δ/q,1-δ+1/q,…,1-δ+q-1/q;1-β/p,1-β+1/p,…,1-β+p-1/p1-γ/q,1-γ+1/q,…,1-γ+q-1/q.


### 4.4. Relation with Mittag-Leffler Function

Letα=0,γ=1,β,δ∈C,R(δ)>0; from definition (3), we have (44)W0,β1,δz=ΓδΓβE1,δz, where Eα,β(z) is Mittag-Leffler function (16). If we put δ=m∈N with using formula (17), then we obtain (45)W0,β1,mz=ΓmΓβE1,mz=ΓmΓβzm-1ez-∑k=0m-2zkk!. In the same way, if we put γ=1 and α>0, then (46)Wα,β1,δz=ΓδEα,1,β,δz,where E(αi),(βi)(z) is the multiple Mittag-Leffler function which is defined by formula (18).


### 4.5. Relation withFpq(z)

Theorem 4.
Letα=0,β≠0,-1,-2,…,δ≠0,-1,-2,…, where β,γ,δ∈C; then one has (47)W0,βγ,δz=ezΓβF11δ-γ;δ;-z,Rδ>Rγ>0.Proof.
From the definitions of generalized Wright function (3) and confluent hypergeometric function ((23); (24)), we have (48)W0,βγ,δz=1Γβ∑n=0∞γnznδnn!=1ΓβF11γ;δ;z=ezΓβF11δ-γ;δ;-z.Theorem 5.
Letα=1,β≠0,-1,-2,…,δ≠0,-1,-2,…, where β,γ,δ∈C; then one has (49)W1,βγ,δz=1ΓβF12γ;δ,β;z,Rδ>Rγ>0.Proof.
From the definitions of generalized Wright function (3) and generalized hypergeometric function, we have (50)W1,βγ,δz=1Γβ∑n=0∞γnznδnβnn!=1ΓβF12γ;δ,β;z.Theorem 6.
Letα=-1 with |z|<1;γ,β∈C, and δ=1-β, whereβ is not integer number; then(51a)W-1,βγ,1-βz=1Γβ1+z-γ,and, in particular, if γ=1-β, (51b)W-1,βz=1Γβ1+zβ-1.Proof.
From the definitions of generalized Wright function (3) and formula (27), we have(52)W-1,βγ,1-βz=∑n=0∞γn1-βnΓβ-nznn!=∑n=0∞γn1-βn1-βnΓβ-1nznn!=1Γβ∑n=0∞γn-1nznn!=1ΓβF10γ;-;-z=1Γβ1+z-γ.Theorem 7.
Letα=-1/2,β=1; γ,δ∈C;R(γ)>0,R(δ)>R(γ), and R(z)>0; then one can deduce that (53)W-1/2,1γ,δ-z=1-γzδπF3312,1+γ2,2+γ2;32,1+δ2,2+δ2;-z24.Proof.
From the integral representation of generalized Wright function (34) and formula (7), we can deduce that(54)W-1/2,1γ,δ-z=ΓδΓγΓδ-γ∫01uγ-11-uδ-γ-1erfc⁡zu2du=ΓδΓγΓδ-γ∫01uγ-11-uδ-γ-11-2π∑n=0∞-1nzu/22n+1n!2n+1du=1-2γδπ∑n=0∞-1n1/2nz/22n+1n!3/2n1+γ/2n2+γ/2n1+δ/2n2+δ/2n=1-γzδπF3312,1+γ2,2+γ2;32,1+δ2,2+δ2;-z24.Theorem 8.
Letα=1/2;γ,δ∈C;R(γ)>0,R(δ)>R(γ), and R(z)>0; then one can deduce that (55)M1/2γ,δz=1πF22γ2,1+γ2;δ2,1+δ2;-z24.Proof.
From the integral representation of the auxiliary functions of generalized Wright function (35) and formula (10), we can deduce that (56)M1/2γ,δz=ΓδπΓγΓδ-γ∫01uγ-11-uδ-γ-1e-z2u2/4du=ΓδπΓγΓδ-γ∫01uγ-11-uδ-γ-1∑n=0∞-1nzu/22nn!du=1π∑n=0∞-1nz/22nn!γ2nδ2n=1π∑n=0∞-1nz/22nn!γ/2n1+γ/2nδ/2n1+δ/2n=1πF22γ2,1+γ2;δ2,1+δ2;-z24.Theorem 9.
Letα=1/3;γ,δ∈C;R(γ)>0,R(δ)>R(γ), and R(z)>0; then one can deduce that(57a)M1/3γ,δz=1Γ2/3F34γ3,1+γ3,2+γ3;23,δ3,1+δ3,2+δ3;z327-γzδΓ1/3F341+γ3,2+γ3,3+γ3;43,1+δ3,2+δ3,3+δ3;z327.In particular, if δ=γ+1, (57b)M1/3γ,γ+1z=1Γ2/3F12γ3;23,3+γ3;z327-γzγ+1Γ1/3F121+γ3;43,4+γ3;z327. Finally, if γ=1, (57c)M1/31,2z=1Γ2/3F1213;23,43;z327-z2Γ1/3F1223;43,53;z327.Proof.
From the integral representation of the auxiliary functions of generalized Wright function (35) and formula (10), we can deduce that(58)M1/3γ,δz=32/3ΓδΓγΓδ-γ∫01uγ-11-uδ-γ-1Aizu32/3du=ΓδΓγΓδ-γ∫01uγ-11-uδ-γ-11Γ2/3F0123;z327u3-zuΓ1/3F0143;z327u3du=1Γ2/3∑n=0∞γ/3n1+γ/3n2+γ/3n2/3nδ/3n1+δ/3n2+δ/3nz/33nn!-γzδΓ1/3∑n=0∞1+γ/3n2+γ/3n3+γ/3n4/3n1+δ/3n2+δ/3n3+δ/3nz/33nn!=1Γ2/3F34γ3,1+γ3,2+γ3;23,δ3,1+δ3,2+δ3;z327-γzδΓ1/3F341+γ3,2+γ3,3+γ3;43,1+δ3,2+δ3,3+δ3;z327.


## 4.1. Relation with FoxH-Function

Using (31) and the definition of Fox H-function, we obtain (38)Wα,βγ,δ-z=ΓδΓγH1311z0,1,1-β,α,1-δ,11-γ,1.


## 4.2. Relation with Fox Wright Function

Generalized Wright function (3) can be represented by Fox Wright function (19) as (39)Wα,βγ,δz=ΓδΓγΨ12zδ,1,β,α;γ,1;.


## 4.3. Relation with MeijerG-Function

From the definitions of generalized Wright function (3) and Meijer G-function (22) with α=1, we obtain (40)W1,βγ,δ-z=ΓδΓγG1311z0,1-β,1-δ1-γ.Theorem 3.
Letα=p/q be a rational number with p,q∈N∪{0},q≠0;γ,δ,β∈C and δ≠0,-1,-2,…; then one has(41)Wp/q,βγ,δ-z=p1/2-βqγ-δ+1/22πp-q/2ΓδΓγGq2q+pqqzqqqpp0,1/q,…,q-1/q;1-δ/q,1-δ+1/q,…,1-δ+q-1/q;1-β/p,1-β+1/p,…,1-β+p-1/p1-γ/q,1-γ+1/q,…,1-γ+q-1/q.In particular, if γ=δ, (42)Wp/q,β-z=2πp-q/2p1/2-βq1/2G0p+qq0zqqqpp0,1/q,…,q-1/q;1-β/p,1-β+1/p,…,1-β+p-1/p----------.Proof.
Using the contour integral representation of generalized Wright function (31) with change of the variable from s to qs and multiplication formula (26), we get(43)Wp/q,βγ,δ-z=ΓδΓγq2πi∫LΓγ+qsΓδ+qsΓβ+psΓ-qszqsds=p1/2-βqγ-δ+1/22πp-q/2ΓδΓγ12πi∫L∏r=1qΓr-1/q-s∏r=1qΓγ+r-1/q+s∏r=1qΓδ+r-1/q+s∏r=1pΓβ+r-1/p+szqqqppsds=p1/2-βqγ-δ+1/22πp-q/2ΓδΓγGq2q+pqqzqqqpp0,1/q,…,q-1/q;1-δ/q,1-δ+1/q,…,1-δ+q-1/q;1-β/p,1-β+1/p,…,1-β+p-1/p1-γ/q,1-γ+1/q,…,1-γ+q-1/q.


## 4.4. Relation with Mittag-Leffler Function

Letα=0,γ=1,β,δ∈C,R(δ)>0; from definition (3), we have (44)W0,β1,δz=ΓδΓβE1,δz, where Eα,β(z) is Mittag-Leffler function (16). If we put δ=m∈N with using formula (17), then we obtain (45)W0,β1,mz=ΓmΓβE1,mz=ΓmΓβzm-1ez-∑k=0m-2zkk!. In the same way, if we put γ=1 and α>0, then (46)Wα,β1,δz=ΓδEα,1,β,δz,where E(αi),(βi)(z) is the multiple Mittag-Leffler function which is defined by formula (18).


## 4.5. Relation withFpq(z)

Theorem 4.
Letα=0,β≠0,-1,-2,…,δ≠0,-1,-2,…, where β,γ,δ∈C; then one has (47)W0,βγ,δz=ezΓβF11δ-γ;δ;-z,Rδ>Rγ>0.Proof.
From the definitions of generalized Wright function (3) and confluent hypergeometric function ((23); (24)), we have (48)W0,βγ,δz=1Γβ∑n=0∞γnznδnn!=1ΓβF11γ;δ;z=ezΓβF11δ-γ;δ;-z.Theorem 5.
Letα=1,β≠0,-1,-2,…,δ≠0,-1,-2,…, where β,γ,δ∈C; then one has (49)W1,βγ,δz=1ΓβF12γ;δ,β;z,Rδ>Rγ>0.Proof.
From the definitions of generalized Wright function (3) and generalized hypergeometric function, we have (50)W1,βγ,δz=1Γβ∑n=0∞γnznδnβnn!=1ΓβF12γ;δ,β;z.Theorem 6.
Letα=-1 with |z|<1;γ,β∈C, and δ=1-β, whereβ is not integer number; then(51a)W-1,βγ,1-βz=1Γβ1+z-γ,and, in particular, if γ=1-β, (51b)W-1,βz=1Γβ1+zβ-1.Proof.
From the definitions of generalized Wright function (3) and formula (27), we have(52)W-1,βγ,1-βz=∑n=0∞γn1-βnΓβ-nznn!=∑n=0∞γn1-βn1-βnΓβ-1nznn!=1Γβ∑n=0∞γn-1nznn!=1ΓβF10γ;-;-z=1Γβ1+z-γ.Theorem 7.
Letα=-1/2,β=1; γ,δ∈C;R(γ)>0,R(δ)>R(γ), and R(z)>0; then one can deduce that (53)W-1/2,1γ,δ-z=1-γzδπF3312,1+γ2,2+γ2;32,1+δ2,2+δ2;-z24.Proof.
From the integral representation of generalized Wright function (34) and formula (7), we can deduce that(54)W-1/2,1γ,δ-z=ΓδΓγΓδ-γ∫01uγ-11-uδ-γ-1erfc⁡zu2du=ΓδΓγΓδ-γ∫01uγ-11-uδ-γ-11-2π∑n=0∞-1nzu/22n+1n!2n+1du=1-2γδπ∑n=0∞-1n1/2nz/22n+1n!3/2n1+γ/2n2+γ/2n1+δ/2n2+δ/2n=1-γzδπF3312,1+γ2,2+γ2;32,1+δ2,2+δ2;-z24.Theorem 8.
Letα=1/2;γ,δ∈C;R(γ)>0,R(δ)>R(γ), and R(z)>0; then one can deduce that (55)M1/2γ,δz=1πF22γ2,1+γ2;δ2,1+δ2;-z24.Proof.
From the integral representation of the auxiliary functions of generalized Wright function (35) and formula (10), we can deduce that (56)M1/2γ,δz=ΓδπΓγΓδ-γ∫01uγ-11-uδ-γ-1e-z2u2/4du=ΓδπΓγΓδ-γ∫01uγ-11-uδ-γ-1∑n=0∞-1nzu/22nn!du=1π∑n=0∞-1nz/22nn!γ2nδ2n=1π∑n=0∞-1nz/22nn!γ/2n1+γ/2nδ/2n1+δ/2n=1πF22γ2,1+γ2;δ2,1+δ2;-z24.Theorem 9.
Letα=1/3;γ,δ∈C;R(γ)>0,R(δ)>R(γ), and R(z)>0; then one can deduce that(57a)M1/3γ,δz=1Γ2/3F34γ3,1+γ3,2+γ3;23,δ3,1+δ3,2+δ3;z327-γzδΓ1/3F341+γ3,2+γ3,3+γ3;43,1+δ3,2+δ3,3+δ3;z327.In particular, if δ=γ+1, (57b)M1/3γ,γ+1z=1Γ2/3F12γ3;23,3+γ3;z327-γzγ+1Γ1/3F121+γ3;43,4+γ3;z327. Finally, if γ=1, (57c)M1/31,2z=1Γ2/3F1213;23,43;z327-z2Γ1/3F1223;43,53;z327.Proof.
From the integral representation of the auxiliary functions of generalized Wright function (35) and formula (10), we can deduce that(58)M1/3γ,δz=32/3ΓδΓγΓδ-γ∫01uγ-11-uδ-γ-1Aizu32/3du=ΓδΓγΓδ-γ∫01uγ-11-uδ-γ-11Γ2/3F0123;z327u3-zuΓ1/3F0143;z327u3du=1Γ2/3∑n=0∞γ/3n1+γ/3n2+γ/3n2/3nδ/3n1+δ/3n2+δ/3nz/33nn!-γzδΓ1/3∑n=0∞1+γ/3n2+γ/3n3+γ/3n4/3n1+δ/3n2+δ/3n3+δ/3nz/33nn!=1Γ2/3F34γ3,1+γ3,2+γ3;23,δ3,1+δ3,2+δ3;z327-γzδΓ1/3F341+γ3,2+γ3,3+γ3;43,1+δ3,2+δ3,3+δ3;z327.


## 5. Basic Properties of the Generalized Wright Function

Theorem 10.
The important relation between the two auxiliary functions ((4); (5)) of generalized Wright function (3) is (59)Fαγ,δz=αγzδMαγ+1,δ+1z.Proof.
By using the integral representations of two auxiliary functions ((35); (36)) and formula (9), we have(60)Fαγ,δz=ΓδΓγΓδ-γ∫01uγ-11-uδ-γ-1Fαzudu=αΓδzΓγΓδ-γ∫01uγ1-uδ-γ-1Mαzudu=αΓδzΓγΓδ-γΓγ+1Γδ-γΓδ+1Mαγ+1,δ+1z=αγzδMαγ+1,δ+1z.Theorem 11.
Letα=-1/2, β=1, and γ∈C; R(γ)>0 and R(z)>0; then one can arrive at (61)W-1/2,1γ,γ+1-z=erfc⁡z2+2γπzγγ1+γ2,z24, where γ(a,x) is incomplete gamma function (29) and erf⁡(x) is error function (8).Proof.
From the integral representation of generalized Wright function (34) and formula (7), we have(62)W-1/2,1γ,γ+1-z=γ∫01uγ-1W-1/2,1-zudu=γ∫01uγ-1erfc⁡zu2du=2γπ∫01uγ-1du∫zu/2∞e-t2dt=zγπ∫01uγdu∫1∞e-z2u2t2/4dt=γ2γzγπ∫1∞t-γ-1γ1+γ2,t2z24dt=erfc⁡z2+2γπzγγ1+γ2,z24.Remark 12.
In the caseγ=2n+1, n=0,1,2,…, we have (63)W-1/2,12n+1,2n+2-z=erfc⁡z2+22n+1πz2n+1n!1-e-z2/4∑k=0nz2kk!22k, and in the case γ=2n, n=1,2,…, we have (64)W-1/2,12n,2n+1-z=erfc⁡z2+22nπz2n12nπerf⁡z2-e-z2/4∑k=1nz2k-11/2k22k-1.Theorem 13.
Letα=1/2, γ∈C; R(γ)>0 and R(z)>0; then one has (65)M1/2γ,γ+1z=γ2γ-1πzγγγ2,z24, where γ(a,x) is the incomplete gamma function which is defined by (29).Proof.
From integral representation (35) of the auxiliary function Mαγ,δ(z) with using relation (10) and changing the variable z2u2/4=t, we have(66)M1/2γ,γ+1z=γ∫01uγ-1M1/2zudu=γπ∫01uγ-1e-z2u2/4du=γ2γ-1zγπ∫0z2/4tγ/2-1e-tdt=γ2γ-1πzγγγ2,z24.Corollary 14.
Letγ=2n+1, n=0,1,2,…, be an odd integer; one has (67)M1/22n+1,2n+2z=2z2n+112n+1erf⁡z2-e-z2/4π∑k=1nz/22k-11/2k, and if γ=2n, n=1,2,…, is an even integer, one has (68)M1/22n,2n+1z=22nn!πz2n1-e-z2/4∑k=0n-1z2kk!22k.Theorem 15.
Letα=1/3 and R(z)>0; then first auxiliary function (4) of the generalized Wright function has the forms(69)M1/32,3z=6z231/3Ai′z31/3+1Γ1/3,M1/33,4z=9z331/3zAi′z31/3-32/3Aiz31/3+1Γ2/3,M1/3n+4,n+5z=32/3n+4z2Ai′z31/3-3n+2n+3z3Aiz31/3+3n+1n+2n+4z3∫01tnAiz31/3tdt,n=0,1,2,….Proof.
From the integral representation ofM1/3γ,δ(z) (35) and relation (11), we can deduce that (70)M1/3γ,γ+1z=γ∫01uγ-1M1/3zudu=γ32/3∫01uγ-1Aizu31/3du. In the above relation if we change the variable zu/31/3=v and using the relations ((14)-(15)) and integral of Airy’s function with putting γ=2, γ=3, and γ=n+4, n=0,1,2,…, then the proof is completed.


## 6. Recurrence Relations

Theorem 16.
Letα>-1; z,β,γ,δ∈C; R(γ)>0, R(δ)>R(γ); R(z)>0 and n=0,1,2,…, then one can deduce the following recurrence relation: (71)Wα,βγ,γ+n+1z=γn+1∑k=0n-1kk!n-k!γ+kWα,βγ+k,γ+k+1z, and, in particular, if n=1, (72)Wα,βγ,γ+2z+γWα,βγ+1,γ+2z=γ+1Wα,βγ,γ+1z.Proof.
Consider(73)Wα,βγ,γ+n+1z=Γγ+n+1ΓγΓn+1∫01uγ-11-unWα,βzudu=Γγ+n+1ΓγΓn+1∑k=0nnk-1k∫01uγ+k-1Wα,βzudu=γn+1∑k=0n-1kk!n-k!γ+kWα,βγ+k,γ+k+1z.Theorem 17.
Letα>-1; z,β,γ,δ∈C; R(γ)>0, R(δ)>R(γ), and R(z)>0; then one can deduce that (74)Wα,β-1γ,δz+1-βWα,βγ,δz=αγzδWα,α+βγ+1,δ+1z.Remark 18.
The above theorem can be returned to Theorem10 if we substitute α=-α,β=1, and z=-z.


## 7. Integral Transforms of the Generalized Wright Function

In this section, we will introduce the Euler transform, Laplace transform, and Mellin transform of the functionWα,βγ,δ(z).


### 7.1. Euler Transform

The Euler transform of the functionWα,βγ,δ(z) follows from the beta function (75)∫01ta-11-tb-1Wα,βγ,δxtσdt=ΓδΓbΓγΨ23xδ,1,β,α,a+b,σ;γ,1,a,σ;.


### 7.2. Laplace Transform

Consider(76)∫0∞ta-1e-stWα,βγ,δxtσdt=ΓδsaΓγΨ22xsσδ,1,β,α;γ,1,a,σ;.


### 7.3. Mellin Transform

Consider(77)∫0∞ts-1Wα,βγ,δ-tdt=ΓsΓγ-sΓδ-sΓβ-αs.


## 7.1. Euler Transform

The Euler transform of the functionWα,βγ,δ(z) follows from the beta function (75)∫01ta-11-tb-1Wα,βγ,δxtσdt=ΓδΓbΓγΨ23xδ,1,β,α,a+b,σ;γ,1,a,σ;.


## 7.2. Laplace Transform

Consider(76)∫0∞ta-1e-stWα,βγ,δxtσdt=ΓδsaΓγΨ22xsσδ,1,β,α;γ,1,a,σ;.


## 7.3. Mellin Transform

Consider(77)∫0∞ts-1Wα,βγ,δ-tdt=ΓsΓγ-sΓδ-sΓβ-αs.


## 8. Derivative of the Generalized Wright Function

Theorem 19.
Letz,β,γ,δ∈C; α>-1 and δ≠0,-1,-2,…; then the first derivative of generalized Wright function (3) has the form (78)ddzWα,βγ,δz=γδWα,α+βγ+1,δ+1z. And the higher derivative is (79)dmdzmWα,βγ,δz=γmδmWα,mα+βγ+m,δ+mz,m=0,1,2,….Proof.
From definition (3), we have(80)ddzWα,βγ,δz=ddz∑n=0∞γnδnΓαn+βznn!=∑n=1∞γnδnΓαn+βzn-1n-1!=γδ∑n=0∞γ+1nδ+1nΓαn+α+βznn!=γδWα,α+βγ+1,δ+1z. By repeating this process m-times, we arrive at the second requisition.Theorem 20.
Letz,β,γ,δ∈C; q∈N and R(γ)>0, R(δ)>R(γ), and R(z)>0; then the (q-1)th derivative of auxiliary function (4) of generalized Wright function (3) has the form (81)dq-1dzq-1M1/qγ,δz=-1q-1γqqδqzM1/qγ+q,δ+qz,q=1,2,…, and, for h=0,1,2,…,q-2 at z=0, one obtains (82)dhdzhM1/qγ,δzz=0=-1hγhΓh+1/qπδhsin⁡πh+1q.Proof.
From integral representation (35), we have(83)dq-1dzq-1M1/qγ,δz=dq-1dzq-1ΓδΓγΓδ-γ∫01uγ-11-uδ-γ-1M1/qzudu=-1q-1ΓδΓγΓδ-γ∫01uγ+q-21-uδ-γ-112πi∫Haet-zut1/qdtdu=-1q-1ΓδΓγΓδ-γ∫01uγ+q-21-uδ-γ-1F1/qzudu=-1q-1γq-1δq-1F1/qγ+q-1,δ+q-1z,and one can use the relation between two auxiliary functions of generalized Wright function (59) to complete the proof of the first requisition. To obtain the second requisition, we start with definition (4) and reflection formula of gamma function, as(84)dhdzhM1/qγ,δz=dhdzh∑n=0∞γnδnΓ1-n+1/q-1nznn!=∑n=h∞γnδnΓ1-n+1/q-1nzn-hn-h!, when z=0; then we get (85)dhdzhM1/qγ,δzz=0=-1hγhδhΓ1-h+1/q=-1hγhΓh+1/qπδhsin⁡πh+1q.Theorem 21.
Letγ,δ∈C; n∈N∪{0}, 0<α<1, R(γ)>0, and R(δ)>R(γ); then one has (86)∫0∞rnMαγ,δrdr=Γn+1Γαn+11-δn+11-γn+1. In particular, if n=0, then one obtains (87)∫0∞Mαγ,δrdr=1-δ1-γ.Proof.
⁡ Consider(88)∫0∞rnMαγ,δrdr=ΓδΓγΓδ-γ∫01uγ-11-uδ-γ-1∫0∞rnMαrudrdu=n!ΓδΓγΓδ-γ∫01uγ-n-21-uδ-γ-112πi∫Hat-αn-1etdtdu=n!ΓδΓγΓδ-γΓαn+1∫01uγ-n-21-uδ-γ-1du=n!ΓδΓγΓδ-γΓαn+1Γγ-n-1Γδ-γΓδ-n-1=Γn+1Γαn+11-δn+11-γn+1.


## 9. Conclusion

In this paper, we generalize the definition of Wright function (1) and its auxiliary functions (2) to be the function Wα,βγ,δ(z) defined as in (3) and its auxiliary functions (4) and (5). The properties of Wα,βγ,δ(z) including its auxiliary functions and the integral representations are provided and proven. The relationship with some known special functions like Fox H-function, Fox-Wright function, Meijer G-function, Mittag-Leffler function, and generalized hypergeometric function are given. The Euler transform, Laplace transform, and Mellin transform of function (3) are introduced.



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