# q-Analogue of Wright Function

Authors: Moustafa El-Shahed; Ahmed Salem
Journal: Abstract and Applied Analysis
 (2008)

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## Abstract

We introduce aq-analogues of Wright function
and its auxiliary functions as Barnes integral representations and series expansion. The relations
between q-analogues of Wright function
and some other functions are investigated.


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## Body

## 1. Introduction

In the second half of the past century, the
discoveries of new special functions and applications of special functions to
new areas of mathematics have initiated a resurgence of interest in this field
[1]. These discoveries include work in combinatories initiated by
Schutzenberger and Foata. Moreover, in recent years, particular cases of long
familiar special functions have been clearly defined and applied to orthogonal
polynomials [1].The Wright function is one of the special functions,
which plays an important role in the solution of fractional differential
equations. The Wright function is defined by the series representation for all
complex variablez as [2]:(1.1)Wα,β(z)=∑n=0∞znn!Γ(αn+β),α>−1,β∈C.The Barnes integral
representation of Wright function is defined by(1.2)Wα,β(z)=12πi∫DΓ(−s)Γ(β+αs)(−z)sds,where D is a contour in the complex s-plane which runs from s=−i∞ to s=i∞,
so that the points s=n,⁢n=0,1,2,… lie to the right of D.
There are two auxiliary functions of Wright function defined as(1.3)Mα(z)=W−α,1−α(−z)=∑n=0∞(−1)nznn!Γ(1−α(n+1)),0<α<1,Fα(z)=W−α,0(−z)=∑n=1∞(−1)nznn!Γ(−αn),0<α<1.Here, we listed some of special
cases of Wright function and its auxiliary functions [2]:(1.4)W0,1(z)=ez,W−1/2,1(z)=Erfc(−z2),(z2)νW1,1+ν(−z24)=Jν(z),(z2)νW1,1+ν(z24)=Iν(z),W−1,β(z)=1Γ(β)(1+z)β−1,β⁢≠0,⁢−1,−2,…,|z|<1,Fα(z)=αzMα(z),M1/2(z)=1π⁢exp⁡(−z24),M1/3(z)=32/3A(z31/3),where Erfc (z) is the complementary error function, Jν(z),Iν(z) are the Bessel and modified Bessel functions,
and A(z) is Airy's function, defined as(1.5)A(z)=z3[I−1/3(23z3/2)−I1/3(23z3/2)].


## 2. Preliminaries

### 2.1.q-Hypergeometric Series

Aq-hypergeometric series is a power series in
one complex variable z with power series coefficients which depend,
apart from q,
on r complex upper parameters a1,a2,…,ar and s complex lower parameters b1,b2,…,bs as follows [3]:(2.1)ϕsr[b1,b2,…,bsa1,a2,…,ar;q,z]=ϕrs(a1,a2,…,ar;b1,b2,…,bs;q,z),:=∑n=0∞(a1,a2,…,ar,q)n(b1,b2,…,bs,q)n((−1)nq(n2))s−r+1zk,where q⁢≠0 when r>s+1.


### 2.2.q-Exponential Series

Theq-analogues of the exponential functions are
given by [1, 3](2.2)Eqz:=⁢ϕ00(−;−;q,−(1−q)z)=∑n=0∞q(n2)zn[n]q!=(−(1−q)z,q)∞,eqz:=⁢ϕ10(0;−;q,(1−q)z)=∑n=0∞zn[n]q!=1((1−q)z,q)∞.


### 2.3. Jacksonq-Derivative

In 1908, Jackson reintroduced and started a systematic study of theq-difference operator [1, 3]:(2.3)(Dqf)(x):=f(x)−f(qx)(1−q)x,q⁢≠1,x⁢≠0,which is now sometimes referred
to as Euler-Jackson , Jackson q-difference operator, or simply the q-derivative. By definition, the limit as q-approaches to 1 is the ordinary derivative, that is,(2.4)lim⁡q→1(Dqf)(x)=dfdx(x),if f is differentiable at x.
The two exponential functions have the q-derivative:(2.5)DqEqax=aEqqax,Dqeqax=aeqax.


### 2.4. Jacksonq-Integrals

Thomae (1869) and Jackson (1910) introduced theq-integral defined in
[3](2.6)∫0af(x)dqx:=a(1−q)∑n=0∞f(aqn)qn,Jackson also defined an integral
from 0 to ∞ by(2.7)∫0∞f(x)dqx:=(1−q)∑n=−∞∞f(qn)qn,provided the sums converge
absolutely.Theq-Jackson integral in a generic interval [a,b] is given by(2.8)∫abf(x)dqx:=∫0bf(x)dqx−∫0af(x)dqx.The q-integration by parts is given for suitable
functions f and g by [3](2.9)∫abf(x)Dqg(x)dqx:=f(b)g(b)−f(a)g(a)−∫abg(qx)Dqf(x)dqx.


### 2.5.q-Gamma Function

Jackson (1910) defined theq-analogue of the gamma function by(2.10)Γq(x):=(q,q)∞(qx,q)∞(1−q)1−x,0<q<1.Moreover, it has the q-integral representations [1, 3]:(2.11)Γq(x):=∫01/1−qtx−1Eq−qtdqt,0<q<1,ℜ(x)>0.Let k denote a positive integer. In the same spirit
we define(2.12)Γqk(x):=(qk,qk)∞(qkx,qk)∞(1−qk)1−x,0<q<1,x⁢≠0,⁢−1,−2,…,:=∫01/1−qktx−1Eqk−qktdqkt,0<q<1,ℜ(x)>0.It is obvious from (2.10) that Γq(z) has simple poles at z=0,−1,−2,….
The residue at z=−n is [3](2.13)lim⁡z→−n(z+n)Γq(z)=(−1)n+1qn(n+1)/2(1−q)n+1(q,q)nlog⁡q.A q-analogue of Legendre's duplication formula
can be easily derived by(2.14)Γq(2z)Γq2(12)=(1+q)2z−1Γq2(z)Γq2(z+12).Similarly, it can be shown that
Gauss multiplication formula has a q-analogue of the form(2.15)Γq(kz)Γqk(1k)Γqk(2k)⋯Γqk(k−1k)=(1+q+q2+⋯+qk−1)kz−1Γqk(z)Γqk(z+1k)Γqk(z+2k)⋯Γqk(z+k−1k).Lemma 2.1.
Letk be a positive integer and n a nonnegative integer, then one has(2.16)Γqk(lk+n)=(ql,qk)n(1−qk)nΓqk(lk)l=1,2,…,Γqk(lk−n)=(1−qk)nq(n/2)(kn+k−2l)Γqk(l/k)(−1)n(qk−l,qk)n,l=1,2,…,k⁢≠l,Γq(kn+l)=(1+q+q2+⋯+qk−1)kn+l−1(ql,qk)n(ql+1,qk)n⋯(qk+l−1,qk)nΓqk(l/k)Γqk(2/k)⋯Γqk((k−l)/k)(1−qk)nk×Γqk(lk)Γqk(l+1k)⋯Γqk(k+lk),l=1,2,….The proof of this lemma follows
from the definition ofq-analogue of gamma function.


### 2.6. Jackson'sq-Bessel Function

Jackson introduced in 1905 the followingq-analogues of the Bessel functions [3]:(2.17)Jν(1)(z;q):=(qν+1,q)∞(q,q)∞(z2)νϕ12(0,0;qν+1;q,−z24),|z|<2,Jν(2)(z;q):=(qν+1,q)∞(q,q)∞(z2)νϕ10(−;qν+1;q,−qν+1z24),Jν(3)(z;q):=(qν+1,q)∞(q,q)∞(z2)νϕ11(0;qν+1;q,qz24).


## 2.1.q-Hypergeometric Series

Aq-hypergeometric series is a power series in
one complex variable z with power series coefficients which depend,
apart from q,
on r complex upper parameters a1,a2,…,ar and s complex lower parameters b1,b2,…,bs as follows [3]:(2.1)ϕsr[b1,b2,…,bsa1,a2,…,ar;q,z]=ϕrs(a1,a2,…,ar;b1,b2,…,bs;q,z),:=∑n=0∞(a1,a2,…,ar,q)n(b1,b2,…,bs,q)n((−1)nq(n2))s−r+1zk,where q⁢≠0 when r>s+1.


## 2.2.q-Exponential Series

Theq-analogues of the exponential functions are
given by [1, 3](2.2)Eqz:=⁢ϕ00(−;−;q,−(1−q)z)=∑n=0∞q(n2)zn[n]q!=(−(1−q)z,q)∞,eqz:=⁢ϕ10(0;−;q,(1−q)z)=∑n=0∞zn[n]q!=1((1−q)z,q)∞.


## 2.3. Jacksonq-Derivative

In 1908, Jackson reintroduced and started a systematic study of theq-difference operator [1, 3]:(2.3)(Dqf)(x):=f(x)−f(qx)(1−q)x,q⁢≠1,x⁢≠0,which is now sometimes referred
to as Euler-Jackson , Jackson q-difference operator, or simply the q-derivative. By definition, the limit as q-approaches to 1 is the ordinary derivative, that is,(2.4)lim⁡q→1(Dqf)(x)=dfdx(x),if f is differentiable at x.
The two exponential functions have the q-derivative:(2.5)DqEqax=aEqqax,Dqeqax=aeqax.


## 2.4. Jacksonq-Integrals

Thomae (1869) and Jackson (1910) introduced theq-integral defined in
[3](2.6)∫0af(x)dqx:=a(1−q)∑n=0∞f(aqn)qn,Jackson also defined an integral
from 0 to ∞ by(2.7)∫0∞f(x)dqx:=(1−q)∑n=−∞∞f(qn)qn,provided the sums converge
absolutely.Theq-Jackson integral in a generic interval [a,b] is given by(2.8)∫abf(x)dqx:=∫0bf(x)dqx−∫0af(x)dqx.The q-integration by parts is given for suitable
functions f and g by [3](2.9)∫abf(x)Dqg(x)dqx:=f(b)g(b)−f(a)g(a)−∫abg(qx)Dqf(x)dqx.


## 2.5.q-Gamma Function

Jackson (1910) defined theq-analogue of the gamma function by(2.10)Γq(x):=(q,q)∞(qx,q)∞(1−q)1−x,0<q<1.Moreover, it has the q-integral representations [1, 3]:(2.11)Γq(x):=∫01/1−qtx−1Eq−qtdqt,0<q<1,ℜ(x)>0.Let k denote a positive integer. In the same spirit
we define(2.12)Γqk(x):=(qk,qk)∞(qkx,qk)∞(1−qk)1−x,0<q<1,x⁢≠0,⁢−1,−2,…,:=∫01/1−qktx−1Eqk−qktdqkt,0<q<1,ℜ(x)>0.It is obvious from (2.10) that Γq(z) has simple poles at z=0,−1,−2,….
The residue at z=−n is [3](2.13)lim⁡z→−n(z+n)Γq(z)=(−1)n+1qn(n+1)/2(1−q)n+1(q,q)nlog⁡q.A q-analogue of Legendre's duplication formula
can be easily derived by(2.14)Γq(2z)Γq2(12)=(1+q)2z−1Γq2(z)Γq2(z+12).Similarly, it can be shown that
Gauss multiplication formula has a q-analogue of the form(2.15)Γq(kz)Γqk(1k)Γqk(2k)⋯Γqk(k−1k)=(1+q+q2+⋯+qk−1)kz−1Γqk(z)Γqk(z+1k)Γqk(z+2k)⋯Γqk(z+k−1k).Lemma 2.1.
Letk be a positive integer and n a nonnegative integer, then one has(2.16)Γqk(lk+n)=(ql,qk)n(1−qk)nΓqk(lk)l=1,2,…,Γqk(lk−n)=(1−qk)nq(n/2)(kn+k−2l)Γqk(l/k)(−1)n(qk−l,qk)n,l=1,2,…,k⁢≠l,Γq(kn+l)=(1+q+q2+⋯+qk−1)kn+l−1(ql,qk)n(ql+1,qk)n⋯(qk+l−1,qk)nΓqk(l/k)Γqk(2/k)⋯Γqk((k−l)/k)(1−qk)nk×Γqk(lk)Γqk(l+1k)⋯Γqk(k+lk),l=1,2,….The proof of this lemma follows
from the definition ofq-analogue of gamma function.


## 2.6. Jackson'sq-Bessel Function

Jackson introduced in 1905 the followingq-analogues of the Bessel functions [3]:(2.17)Jν(1)(z;q):=(qν+1,q)∞(q,q)∞(z2)νϕ12(0,0;qν+1;q,−z24),|z|<2,Jν(2)(z;q):=(qν+1,q)∞(q,q)∞(z2)νϕ10(−;qν+1;q,−qν+1z24),Jν(3)(z;q):=(qν+1,q)∞(q,q)∞(z2)νϕ11(0;qν+1;q,qz24).


## 3. Theq-Analogue of Error Functions

Definition 3.1.
One defines theq-analogues of error function and complementary
error function, respectively, as(3.1)Erfq(x)=1+qΓq2(1/2)∫0xEq2−q2t2dqt,(3.2)Erfcq(x)=1+qΓq2(1/2)∫x1/1−q2Eq2−q2t2dqt.Remark 3.2.
Ifx→1/1−q2 or 0 in the above definitions, respectively, then
we have(3.3)Erfq(11−q2)=Erfcq(0)=1,and we can deduce that(3.4)Erfq(x)=1−Erfcq(x).Proof.
(3.5)Erfq(11−q2)=1+qΓq2(1/2)∫01/1−q2Eq2−q2t2dqt=1+qΓq2(1/2)1−q1−q2∑n=0∞qn(q2n+2,q2)∞=1−q2Γq2(1/2)(q2,q2)∞∑n=0∞qn(q2,q2)n=1−q2Γq2(1/2)(q2,q2)∞(q,q2)∞=1.The series representations of
the q-error function are as follows:(3.6)erfq(x)=1+qΓq2(1/2)∑n=0∞(−1)nqn(n+1)x2n+1[n]q2![2n+1]q=1+qΓq2(1/2)∑n=0∞(−1)nqn(n+1)(q,q2)n(1−q2)nx2n+1(q2,q2)n(q3,q2)n=(1+q)xΓq2(1/2)ϕ11(q;q3;q2,q2(1−q2)x2).


## 4.q-Analogue of Wright Function

In this section,
we introduce a definition of aq-analogue of Wright function (Wα,β(z,qk)) as a Barnes integral representations.Definition 4.1.
According to standard
notation, one defines aq-analogue of Wright function as(4.1)Wα,β(z,qk)=log⁡q2πi(q−1)∫CΓq(−s)Γqk(β+αs)(−z)sds,where k=1,2,…, 0<q<1, z is not equal to zero and(4.2)(−z)s=exp⁡[s(log⁡|z|+i⁢arg⁡(−z))],where C is a suitable path in the complex s-plane that runs from s=−i∞ to s=i∞,
so the points s=n,⁢n=0,1,2,…, lie to the right of the contour C.


### 4.1. Existence and Representation ofq-Wright Function

Firstly, we rewrite the
definition ofq-Wright function as(4.3)Wα,β(z,qk)=(q,q)∞(1−q)(qk,qk)∞(1−qk)1−βlog⁡q2πi(q−1)∫C(qkβ+ksα,qk)∞(q−s,q)∞(1−qk)sα(1−q)−s(−z)sds.Next, we consider a q-analogue of Wright function in the case that 0<q<1.
Let q=e−ω⁢(ω>0);
using the triangle inequality, we get(4.4)|1−|a|e−ωℜ(s)|≤|1−aqs|≤1+|a|e−ωℜ(s),|(qkβ+ksα,qk)∞(q−s,q)∞|≤∏r=0∞1+e−π[r+k(ℜ(β)+αℜ(s))]ω1−e−π[r−ℜ(s)]ω,which is bounded on the contour C.Theorem 4.2.
Letk be a positive integer, and let β be a complex number, then the q-Wright function is absolutely convergent for
all complex variables z; if α>−log⁡(1−q)/log⁡(1−qk) and if α=−log⁡(1−q)/log⁡(1−qk), then |z|<1.Proof.
Consider the integral in (4.1) with the
contour C 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 C is bounded away from the poles.
Lets=Reiθ, then we have(4.5)|Γq(−s)Γqk(β+αs)(−z)s|=ABeR⁢cos⁢⁡θ(log⁡(1−q)+αlog⁡(1−qk))|z|R⁢cos⁢⁡θe−Rsin⁡θarg⁡(−z),where(4.6)A=(q,q)∞(1−q)(qk,qk)∞(1−qk)1−ℜ(β),B=|(qkβ+ksα,qk)∞(q−s,q)∞|,as R→∞ on CR,
it follows that from (4.5) the integral (4.1) with C replaced by CR tends to zero as R→∞ if and only if α>−log⁡(1−q)/log⁡(1−qk) for all complex variable z and if α=−log⁡(1−q)/log⁡(1−qk), then |z|<1.Theorem 4.3 (explicit power
series expansion).
Letk be a positive integer, let β be a complex number,  and let either α>−log⁡(1−q)/log⁡(1−qk) and z⁢≠0 or α=−log⁡(1−q)/log⁡(1−qk) and |z|<1.
Then the q-Wright function (4.1) has the power series
expansion(4.7)Wα,β(z,qk)=∑n=0∞qn(n+1)/2zn[n]q!Γqk(αn+β).Proof.
From the existence theorem after replacings by −s, we can apply Cauchy's theorem (residues
theorem) to the closed contour which is consisting of the contour CR and that part of C terminated above and below by CR as R→∞,
we obtain that the q-analogue of Wright function (4.1) equals the sum of the
residues of the integrand at s=−n,⁢n=0,1,2,….
This completes the proof.Remark 4.4.
Ifq→1, then α=−log⁡(1−q)/log⁡(1−qk)→−1, and the q-Wright function (4.7) tends to the
classical case (1.1).Definition 4.5 (the auxiliary functions ofq-Wright function).
We introduce two (q-Wright-type) auxiliary functions Mα(z,qk)=W−α,1−α(−z,qk) and Fα(z,qk)=W−α,0(−z,qk) with 0<α<log⁡(1−q)/log⁡(1−qk).
The two functions can be define for any order α∈(0,log⁡(1−q)/log⁡(1−qk)) and for all complex variable z⁢≠0 by(4.8)Mα(z,qk)=W−α,1−α(−z,qk)=∑n=0∞(−1)nqn(n+1)/2zn[n]q!Γqk(1−α(n+1)),(4.9)Fα(z,qk)=W−α,0(−z,qk)=∑n=1∞(−1)nqn(n+1)/2zn[n]q!Γqk(−αn).Remark 4.6.
An important relationship between auxiliary functions ofq-Wright function as(4.10)Fα(z,qk)=z(1−q)q1−αk1−qk∑n=0∞(−1)nqn(n+1)/2(q1−αkz)n[n]q!Γqk(1−α(n+1))1−qαk(n+1)1−qn+1;when αk=1,
we get(4.11)F1/k(z,qk)=z1+q+⋯+qk−1M1/k(z,qk).


## 4.1. Existence and Representation ofq-Wright Function

Firstly, we rewrite the
definition ofq-Wright function as(4.3)Wα,β(z,qk)=(q,q)∞(1−q)(qk,qk)∞(1−qk)1−βlog⁡q2πi(q−1)∫C(qkβ+ksα,qk)∞(q−s,q)∞(1−qk)sα(1−q)−s(−z)sds.Next, we consider a q-analogue of Wright function in the case that 0<q<1.
Let q=e−ω⁢(ω>0);
using the triangle inequality, we get(4.4)|1−|a|e−ωℜ(s)|≤|1−aqs|≤1+|a|e−ωℜ(s),|(qkβ+ksα,qk)∞(q−s,q)∞|≤∏r=0∞1+e−π[r+k(ℜ(β)+αℜ(s))]ω1−e−π[r−ℜ(s)]ω,which is bounded on the contour C.Theorem 4.2.
Letk be a positive integer, and let β be a complex number, then the q-Wright function is absolutely convergent for
all complex variables z; if α>−log⁡(1−q)/log⁡(1−qk) and if α=−log⁡(1−q)/log⁡(1−qk), then |z|<1.Proof.
Consider the integral in (4.1) with the
contour C 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 C is bounded away from the poles.
Lets=Reiθ, then we have(4.5)|Γq(−s)Γqk(β+αs)(−z)s|=ABeR⁢cos⁢⁡θ(log⁡(1−q)+αlog⁡(1−qk))|z|R⁢cos⁢⁡θe−Rsin⁡θarg⁡(−z),where(4.6)A=(q,q)∞(1−q)(qk,qk)∞(1−qk)1−ℜ(β),B=|(qkβ+ksα,qk)∞(q−s,q)∞|,as R→∞ on CR,
it follows that from (4.5) the integral (4.1) with C replaced by CR tends to zero as R→∞ if and only if α>−log⁡(1−q)/log⁡(1−qk) for all complex variable z and if α=−log⁡(1−q)/log⁡(1−qk), then |z|<1.Theorem 4.3 (explicit power
series expansion).
Letk be a positive integer, let β be a complex number,  and let either α>−log⁡(1−q)/log⁡(1−qk) and z⁢≠0 or α=−log⁡(1−q)/log⁡(1−qk) and |z|<1.
Then the q-Wright function (4.1) has the power series
expansion(4.7)Wα,β(z,qk)=∑n=0∞qn(n+1)/2zn[n]q!Γqk(αn+β).Proof.
From the existence theorem after replacings by −s, we can apply Cauchy's theorem (residues
theorem) to the closed contour which is consisting of the contour CR and that part of C terminated above and below by CR as R→∞,
we obtain that the q-analogue of Wright function (4.1) equals the sum of the
residues of the integrand at s=−n,⁢n=0,1,2,….
This completes the proof.Remark 4.4.
Ifq→1, then α=−log⁡(1−q)/log⁡(1−qk)→−1, and the q-Wright function (4.7) tends to the
classical case (1.1).Definition 4.5 (the auxiliary functions ofq-Wright function).
We introduce two (q-Wright-type) auxiliary functions Mα(z,qk)=W−α,1−α(−z,qk) and Fα(z,qk)=W−α,0(−z,qk) with 0<α<log⁡(1−q)/log⁡(1−qk).
The two functions can be define for any order α∈(0,log⁡(1−q)/log⁡(1−qk)) and for all complex variable z⁢≠0 by(4.8)Mα(z,qk)=W−α,1−α(−z,qk)=∑n=0∞(−1)nqn(n+1)/2zn[n]q!Γqk(1−α(n+1)),(4.9)Fα(z,qk)=W−α,0(−z,qk)=∑n=1∞(−1)nqn(n+1)/2zn[n]q!Γqk(−αn).Remark 4.6.
An important relationship between auxiliary functions ofq-Wright function as(4.10)Fα(z,qk)=z(1−q)q1−αk1−qk∑n=0∞(−1)nqn(n+1)/2(q1−αkz)n[n]q!Γqk(1−α(n+1))1−qαk(n+1)1−qn+1;when αk=1,
we get(4.11)F1/k(z,qk)=z1+q+⋯+qk−1M1/k(z,qk).


## 5. Relation with Some Known Special Functions

It follows from the definition of theq-analogue of Wright function as a series expansion (4.7) that(5.1)W0,1(z,qk)=Eqqz.The Jackson's third q-Bessel
function and modified third q-Bessel function can be expressed in terms of
q-Wright function as(5.2)(z2)νW1,ν+1(−z24,q)=Jν(3)(z(1−q),q),(z2)νW1,ν+1(z24,q)=Iν(3)(z(1−q),q)The q-error function complement
can also be expressed a particular case of q-Wright function as(5.3)W−1/2,1(z,q2)=Erfcq(−qz1+q),where Erfcq(z) denotes the q-error function complement which is defined as
in (3.2). To prove this formula, we use the definition of q-Wright function (4.7) and the identities of
the q-gamma function:(5.4)W−1/2,1(z,q2)=∑n=0∞qn(n+1)/2zn[n]q!Γq2(1−n/2)=1+∑n=0∞q(2n+1)(n+1)z2n+1[2n+1]q!Γq2(1/2−n)=1−1+qΓq2(1/2)∑n=0∞(−1)nqn(n+1)[n]q2![2n+1]q(−qz1+q)2n+1=1−Erfq(−qz1+q)=Erfcq(−qz1+q).Taking α=−1 and k=1 in the definition of q-Wright function, then
we obtain(5.5)W−1,β(z,q)=∑n=0∞qn(n+1)/2zn[n]q!Γq(β−n)=1Γq(β)∑n=0∞(−1)nzn(q1−β,q)n(q,q)n=1Γq(β)(−zq1−β,q)∞(−z,q)∞,|z|<1.Explicit expressions of F1/k(z,qk) and M1/k(z,qk) in terms of known functions are expected for
some particular values of k≥2.
In the particular case k=2,
we find(5.6)M1/2(z,q2)=1Γq2(1/2)Eq2−q2z2/(1+q)2,F1/2(z,q2)=z(1+q)Γq2(1/2)Eq2−q2z2/(1+q)2.To prove the first formula and
the second formula, we use the relationship between them (4.11). Using the
definition of q-auxiliary Wright function (4.8) and the
identities of the q-gamma function (2.16), we obtain the following:(5.7)M1/2(z,q2)=∑n=0∞(−1)nqn(n+1)/2zn[n]q!Γq2((1−n)/2)=∑n=0∞qn(2n+1)/2znΓq(2n+1)Γq2(1/2−n)=1Γq2(1/2)∑n=0∞(−1)nqn(n−1)[n]q2!(q2z2(1+q)2)n=1Γq2(1/2)Eq2−q2z2/(1+q)2.In the case of k=3,
we can deduce that(5.8)M1/3(z,q3)=(1+q+q2)2/3Aq(z(1+q+q2)1/3),where Aq(z) is the q-analogue of the Airy function which
is defined as(5.9)Aq(z)=qz1+q+q2[I−1/3(2)(21+q+q2(1−q3)(qz)3/2)−I1/3(2)(21+q+q2(1−q3)(qz)3/2)],where Iν(2)(z) is Jackson's modified second q-Bessel. To prove this formula, we use the
definition of q-Wright function and the identities of the gamma function
(2.16):(5.10)M1/3(z,q3)=∑n=0∞(−1)nqn(n+1)/2zn[n]q!Γq3((2−n)/3)=∑n=0∞(−1)nq3n(3n+1)/2z3n[3n]q!Γq3(2/3−n)−∑n=0∞(−1)nq(3n+1)(3n+2)/2z3n[3n+1]q!Γq3(1/3−n)=1Γq3(2/3)∑n=0∞q3n(n−1)+5nz3n(1−q)3n(q2,q3)n(q3,q3)n(1−q3)n−qz(1+q+q2)Γq3(4/3)∑n=0∞q3n(n−1)+7nz3n(1−q)3n(q4,q3)n(q3,q3)n(1−q3)n=1Γq3(2/3)ϕ10(q2;q3,q5z3(1−q)31−q3)−qz0ϕ1(q4;q3,q7z3(1−q)3/(1−q3))(1+q+q2)Γq3(4/3)=qz(1+q+q2)1/2[I−1/3(2)(2(1−q)(qz)3/2(1+q+q2)1/2,q3)−I1/3(2)(2(1−q)(qz)3/2(1+q+q2)1/2,q3)]=(1+q+q2)2/3Aq(z(1+q+q2)1/3);when q→1,
we cover the classical results about the Wright function and its auxiliary
functions.



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*Source: 962849-2008-05-12.xml*