A MULTIPLE q-EXPONENTIAL DIFFERENTIAL OPERATIONAL IDENTITY∗

(刘治国)

School of Mathematical Sciences, Key Laboratory of MEA (Ministry of Education)& Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, China

E-mail: zgliu@math.ecnu.edu.cn; liuzg@hotmail.com

Dedicated to Professor George Andrews on the occasion of his 85th birthday

Abstract Using Hartogs’ fundamental theorem for analytic functions in several complex variables and q-partial differential equations,we establish a multiple q-exponential differential formula for analytic functions in several variables.With this identity,we give new proofs of a variety of important classical formulas including Bailey’s6ψ6 series summation formula and the Atakishiyev integral.A new transformation formula for a double q-series with several interesting special cases is given.A new transformation formula for a3ψ3 series is proved.

Key words q-hypergeometric series; q-exponential differential operator;Bailey’s6ψ6 summation;double q-hypergeometric series; q-partial differential equation

1 Introduction and Preliminaries

Throughout the paper,we shall use the standardq-notations.Unless stated otherwise,it is assumed that 0

It is obvious that (a;q)0=1,and by a simple calculation we find that,for any positive integern,

For simplicity,we also adopt the following compact notation for multipleq-shifted factorials:

Herenis an integer or∞.

The basic hypergeometric series or theq-hypergeometric seriesr+1φr(·) is given by

and the bilateral basic hypergeometric series or the bilateralq-hypergeometric seriesrψr(·) is defined as

One of the most important results in theq-series is theq-binomial theorem(see,for example[11,eq.(1.3.2)]),which states that for|q|<1 and|x|<1,

Theq-Gauss summation theorem (see,for example [11,1.5.1]) was first proposed by E.Heine in 1847,and states that for|q|<1 and|c/ab|<1,

For any complex function off(x),theq-derivative operatorDqis defined by

Theq-derivative was introduced by Schendel [29]in 1878 and Jackson [12]in 1908,and is aq-analog of the ordinary derivative.

By a direct computation,we deduce thatxn=0 form>n,and that

The Gaussian polynomials,also called theq-binomial coefficients,areq-analogs of the binomial coefficients,and are given by (see,for example [11,p.24])

Now we introduce the definition of the Rogers–Szeg˝o polynomials,which were first studied by Rogers [28]and then by Szeg˝o [30].

Definition 1.1The Rogers–Szeg˝o polynomials are defined by

For recent research on the Rogers–Szegpolynomials,please refer to [24]and [25].

Ifqis replaced byq-1in the Rogers-Szeg˝o polynomials,we can obtain the Stieltjes–Wigert polynomials (see,for example,[7,30]).

Definition 1.2The Stieltjes–Wigert polynomials are defined by

By multiplying two copies of theq-binomial theorem together,we get the following proposition.

Proposition 1.3Ifgn(x,y|q) are the Stieltjes–Wigert polynomials,then we have that

Now we give the definitions of theq-partial derivative and theq-partial differential equations[20].

Definition 1.4Aq-partial derivative of a function of several variables is itsq-derivative with respect to one of those variables,regarding other variables as constants.Theq-partial derivative of a functionfwith respect to the variablexis denoted by∂q,xf.

Definition 1.5Aq-partial differential equation is an equation that contains unknown multivariable functions and theirq-partial derivatives.

Based on∂q,x,we can construct theq-exponential differential operatorT(y∂q,x) as follows:

One of the most important results in the theory ofq-exponential differential operator is the following operator identity [15,eq.(3.1)]:

Proposition 1.6For max{|as|,|at|,|au|,|bs|,|bt|,|bu|,|abstu/v|}<1,we have that

Theq-exponential differential operatorT(y∂q-1,x) is obtained from theq-exponential differential operatorT(y∂q,x) by replacingqwithq-1,namely,

We cannot simply transfer the properties ofT(y∂q,x) toT(y∂q-1,x) through the transformationq →q-1.In this paper we will focus on the study ofT(y∂q-1,x).

We [15,Theorems 1 and 2](see also [17,Lemma 13.2]) proved the following generalqexponential differential operational identities by using some basic properties of analytic functions in two complex variables:

Theorem 1.7Suppose thatf(x,y)is a two-variable analytic function near(x,y)=(0,0).Then we have thatf(x,y)=T(y∂q-1,x)f(x,0) if and only if∂q-1,xf=∂q-1,yf.

The first principal result of this paper is the following theorem:

Theorem 1.8Suppose thatf(x) is an analytic function nearx=0 and that its power series is

If,for sufficiently largen,an=O(qn(n-1)/2),thenT(y∂q-1,x)f(x) is a two-variable analytic function ofxandyat (x,y)=(0,0),and we also have that

Fornbeing a non-negative integern,by takingf(x)=xnin the theorem above,we immediately find that

Using mathematical induction we can extend Theorem 1.8 to

Theorem 1.9Suppose thatf(x1,x2,···,xk) is analytic at (0,0,···,0)∈Ck,and that its Maclaurin series is given by

If,for sufficiently largen1,···,nk,

then,at (x1,y1,x2,y2,···,xk,yk)=(0,0,···,0)∈C2kwe have

The second principal result of this paper is.

Theorem 1.10Suppose thatf(x1,y1,···,xk,yk) is a 2k-variable analytic function at(0,0,···,0)∈C2kwhich satisfies theq-partial differential equations

Then we have that

This theorem reveals the deep relationship between analytic functions in several complex variables,q-partial differential equations and theq-series.It tells us that if there is an analytic functionf(x1,y1,···,xk,yk) in 2kvariables which satisfies a system ofq-partial differential equations in the theorem,then we can recoverf(x1,y1,···,xk,yk) from its special casef(x1,0,x2,0,···,xk,0) by using someq-exponential differential operators.

In order to prove Theorems 1.9 and 1.10 we need Hartogs’ theorem in the theory of several complex variables (see,for example,[31,p.28]),which is a fundamental result in the theory of several complex variables.

Theorem 1.11(Hartogs’theorem) If a complex valued functionf(z1,z2,···,zn)is holomorphic (analytic) in each variable separately in a domainU ∈Cn,then it is holomorphic(analytic) inU.

We also need the following fundamental property of several complex variables (see,for example,[26,p5,Proposition 1],[27,p90]):

Theorem 1.12Iff(x1,x2,···,xk)is analytic at the origin(0,0,···,0)∈Ck,thenfcan be expanded in an absolutely convergent power series as

The rest of this paper is organized as follows: Section 2 is devoted to the proofs of Theorems 1.8,1.9 and 1.10.In Section 3,we will prove amongst other things the followingq-exponential differential operational identity:

Theorem 1.13Ifnis a non-negative integer then we have that

Settingn=0 in (1.8) and noting that (1;q)n=δ0n,we immediately arrive at [9,Theorem 2.11]

Section 4 is devoted to the proofs of Bailey’s6ψ6series summation formula and a3ψ3series transformation due to Chen and Liu,and a new transformation formula for a3ψ3series is given.In Section 5,we will provide a new proof of the Atakishiyev integral.In Section 6 we will derive the following new doubleq-series transformation formula:

Theorem 1.14For|αab/q|<1,we have that

Some interesting special cases of Theorem 1.14 and their application to Rogers–Hecke type series will be discussed in Section 7.

2 The Proofs of Theorems 1.8,1.9 and 1.10

Proof of Theorem 1.8Settingan=qn(n-1)/2bnand noting that,for sufficiently largen,an=O(qn(n-1)/2),we find that there exists a positive constantMsuch that|bn|≤Mfor alln ∈N.Keeping the definition ofgn(x,y|q) in mind and using the triangle inequality,we find that for 0

It is easy to show that for any non-negative integernand 0

Thus we have that

Settingx=y=1 in the generating function ofgn(x,y|q),we deduce that

Since the radius of convergence of the above power series is∞,by the Cauchy root test we have that

Using the triangle inequality,we conclude that,for max{|x|,|y|}≤1,

Since|bn|≤M,we find that

Thus we conclude that the seriesconverges absolutely and uniformly for max{|x|,|y|}≤1,sincegn(x,y|q) is analytic for max{|x|,|y|}≤1.Hence

is a two-variable analytic function ofxandyfor max{|x|,|y|} ≤1.With the help of∂q-1,xgn(x,y|q)=∂q-1,ygn(x,y|q),we find that∂q-1,xf(x,y)=∂q-1,yf(x,y) .It follows that

This completes the proof of Theorem 1.8.

Proof of Theorem 1.9We use mathematical induction and Theorem 1.8 to prove Theorem 1.9.By Theorem 1.8 we know that Theorem 1.9 holds for the case whenk=1.

Now assume that the theorem has been proven for the casek-1.Now we consider the casek.Suppose thatf(x1,x2,···,xk) is analytic at (0,0,···,0)∈Ck,and its Maclaurin series is given by

If we regard the functionf(x1,x2,···,xk) as a function ofx1,thenfis analytic atx1=0.Keeping the fact that,for sufficiently largen1,an1,n2,···,nk=O(qn1(n1-1)/2) in mind,by Theorem 1.8,we have that

By Theorem 1.8 we know that the left-hand side of the above equation is an analytic function ofx1,x2,···,xkandy1at (0,0,···,0)∈Ck+1.Using Hartogs’s theorem,we know that this is also analytic function ofx2,···,xkat (0,0,···,0)∈Ck-1.Thus,by the induction hypothesis,we can useT(y2∂q-1,x2)···T(yk∂q-1,xk) to act on both sides of the above equation to obtain that

This completes the proof of Theorem 1.9.

Proof of Theorem 1.10Letf(x1,y1,···,xk,yk) be the given function.Then,by Hartogs’s theorem,we know thatf(x1,y1,···,xk,yk) is also an analytic function ofy1aty1=0,so we can assume that

Substituting this into theq-partial differential equation,∂q-1,x1f=∂q-1,y1f,we deduce that

Equating the coefficients ofyields that

By iteration,we easily conclude that

By settingy1=0 in the Maclaurin expansion offabouty1,we have that

It follows that

Thus we have that

By using the same argument as above,we can find that

Combining the above two equations gives that

Repeating the above process completes the proof of Theorem 1.10.

3 The Proof of Theorem 1.13

We begin this section with the following proposition:

Proposition 3.1For max{|av|,|bt|}<1,we have that

Settingv=sand noting that (1;q)k=δk,0,we immediately arrive at

Lettingv →0 in (3.1),and appealing to theq-binomial theorem we deduce that

ProofFor max{|av|,|bt|}<1,we now introduce the functionf(a,b) by

By the ratio test we can verify thatf(a,b) is analytic at (a,b)=(0,0).A straightforward calculation shows that

Hence,by Theorem 1.7,we find thatf(a,b)=T(b∂q-1,a)f(a,0),which is the same as (3.1).

Proof of Theorem 1.13Now we introduce the two-variable complex functiong(t,u)by

The series on the right-hand side is a finite series whose terms are all analytic at (t,u)=(0,0).It follows thatg(t,u) is analytic at (t,u)=(0,0).By a direct computation,we find that

Thus,by Theorem 1.7,we deduce thatg(t,u)=T(u∂q-1,t)g(t,0).Combining (3.1) and(3.5),we conclude that

It follows that

Using the operational identity in (3.2),we have that

Combining the above two equations gives that

Recall the Sears transformation (see,for example [14,Theorem 3]):

Settinga3=q-nin the equation above,and then in the resulting equation making the change that

Using the equation above we deduce that

Combining the equation above with (3.5) and (3.6) completes the proof of Theorem 1.13.

Theorem 2 in [16]needs to be supplemented by a conditions=vq-n,wherenis a nonnegative integer.

4 Bailey’s6ψ6 Summation Formula and Bailey’s2ψ2 Transformation

4.1 Bailey’s6ψ6 Summation Formula

Theq-exponential differential operator is used to give a proof of Bailey’s6ψ6summation formula [15].In this section we will improve the proof to make it more concise.We begin with the following proposition for the bilateral basic hypergeometric series:

Proposition 4.1Let{An} be a sequence independent ofa,b,canddand let the series on the left hand side of the equation below be an analytic function of four variablesa,b,canddat (0,0,0,0)∈C4.Then we have that

ProofUsing Hartogs’ theorem and theq-binomial theorem,it is easily seen that,for any integern,

is analytic at (0,0,0,0)∈C4,and by a direct computation we find thatfn(a,b,c,d) satisfies the following twoq-partial differential equations:

It is obvious that the left-hand side of the equation in Proposition 4.1 is a linear combination offn(a,b,c,d).If we denote it byf(a,b,c,d),then we have that

Under the hypotheses of Proposition 4.1,f(a,b,c,d)is analytic at(0,0,0,0)∈C4.Thus we can use the case of whenk=2 from Theorem 1.10 to obtain that

Using the case of whenk=1 from Theorem 1.10,we can conclude that

Combining the two equations completes the proof of Proposition 4.1.

By some elementary calculations and Jacobi’s triple product identity,we can find the following lemma [15,Lemma 2].

Lemma 4.2For 0<|q|<1 andα≠0,we have that

For completeness,we will repeat the proof of this lemma.

ProofIt is easy to show that the series on the left hand side of the above equation converges to a functionf(α,a),which is an analytic function ofawhen|a|<∞.

Using the simple identity 1-αq2n=(1-αaqn-1)-αq2n(1-aq-n-1),we find that

Making the index changen →n+1 in the first summation and then combining it with the second summation,we readily find that

Similarly,replacingn+1 bynin the second summation in (4.2),we find that

Combining the two equations above yields thatf(α,a)=f(α,aq),which gives thatf(α,a)=f(α,aqn).Lettingn →∞and noticing thatf(α,a) is an analytic function ofa,we have that

by the Jacobi triple product identity.This completes the proof of Lemma 4.2.

Theorem 4.3(Bailey’s6ψ6summation) Fora,b,c,d ∈C with|α2abcd/q3|<1,we have that

ProofTakingAn=(1-αq2n)α2nq2n2-nin Proposition 4.1 and keeping Lemma 4.2 in mind,we find that

Noting thatT(c∂q-1,a) is a linear operator abouta,we immediately deduce that

Substituting this equation into the left-hand side of (4.4),we conclude that

By making use of (3.2) and (3.3) in Proposition 3.1,we have that

Substituting the equation above into the left-hand side of (4.5),we deduce that

which is the same as Bailey’s6ψ6summation.This completes the proof of Theorem 4.3.

4.2 Bailey’s2ψ2 Transformation

We begin with the following lemma:

Lemma 4.4For 0

ProofFor any integerm,settingc=d=0 andb=qm+1in (4.6),we deduce that

Multiplying both sides of the above equation byαmqm2(cq-m,dq-m;q)∞and then summing the resulting equation aboutmfrom-∞to∞yields that

Now we begin to compute the inner series on the left-hand side of (4.8).From the definition of aq-shifted factorial we have that

Settingm-n=kand noting that 1/(q;q)k=0 fork<0,we conclude that

Replacingcbyαq2k+1andabyqn+1/candbbyqn+1/din theq-Gauss summation in (1.3),we deduce that

Combining the above two equations,we conclude that

Substituting the equation above into (4.9) gives that

Combining the equation above with (4.8) completes the proof of Lemma 4.4.

Based on Lemma 4.4,we can prove the following transformation formula for a3ψ3series[9,Theorem 5.3],which includes Bailey’s2ψ2transformation formula [5]as a special case:

Theorem 4.5For|αcd/q|<1,we have that

ProofFor the sake of brevity,we temporarily denote that

Applying theq-exponential differential operatorT(b∂q-1,a) to act on both sides of (4.7),and using (1.9) in the resulting equation,we deduce that

Multiplying both sides of the equation above by (αab/q;q)∞,we find that

Applying theq-exponential differential operatorT(u∂q-1,a) to act on both sides of the above equation,and making using of (1.9) in the resulting equation,we conclude that

Noting the definition ofAnin (4.10) completes the proof of Theorem 4.5.

Since the left-hand side of the equation above is symmetric aboutdandu,so must be the right-hand side.It follows that

By simplifying the above equation,we can obtain the following theorem,which seems to be new:

Theorem 4.6For|αcd/q|<1 and|αcu/q|<1 we have that

5 Ramanujan-type Representation for the Askey–Wilson Integral

Forx=cosθ,we define the notationh(x;a|q) andh(x;a1,a2,···,am|q) as

Using the the modular transformation property for the Jacobi theta functions and the Askey–Wilson integral evaluation,N.M.Atakishiyev discovered the Ramanujan-type representation for the Askey–Wilson integral evaluation admitting the transformationq →q-1.In this section we will use our method to give a new proof of it.Our method does not need to know the Askey–Wilson integral evaluation in advance,and we also do not need the Jacobi theta functions [4].

Theorem 5.1(Atakishiyev) Ifαis a real number andq=exp(-2α2),then we have that

ProofIt is easy to verify that

Ifgn(a,b|q) is the Stieltjes–Wigert polynomials,then we definegn(sinhαx|q) as

Using the generating function for the Stieltjes–Wigert polynomials in Proposition 1.3,we easily find that

To compute the integral in Theorem 5.1,we begin with the integral

which can be calculated using the following well-known integral formula in the calculus:

Using this integral formula and the definition ofgn(sinhαx|q) in (5.3),we easily deduce that

Using the finite form of theq-binomial theorem,and through careful examination,we find that

Denoting the left hand side of(5.2)byK(a,b,c,d),we can show that it is an analytic function ofa,b,c,dat(0,0,0,0).Multiplying both sides of the above equation with(-ia)nqn(n-1)/2/(q;q)nand then sum overnfrom 0 to∞,we find that

It is easy to check thatf1(a,c):=K(a,0,c,0)/(ac/q;q)∞satisfies theq-partial differential equation

By making use of the case of whenk=1 for Theorem 1.10,we deduce that

Setting thatf(a,b,c,d):=K(a,b,c,d)/(ab/q,cd/q;q)∞,we can show that

Employing the case of whenk=2 for Theorem 1.10,we conclude that

Keeping (5.6) in mind and making use of (3.2) and (3.3) from Proposition 3.1,we have the following:

Substituting the equation above into (5.7) completes the proof of Theorem 5.1.

In the same way,we can derive the following theorem of Askey [2]:

Theorem 5.2For|abcd/q3|<1,we have the integral formula

6 The Proof of Theorem 1.14

We begin this section with the following proposition,which is stated in [17,Proposition 13.10]without proof.

Proposition 6.1Let{fn(x)} be a sequence of analytic functions nearx=0 such that the seriesconverges uniformly to an analytic functionf(x) nearx=0,and the seriesconverges uniformly to an analytic functionf(x,y) near (x,y)=(0,0).Then we have thatf(x,y)=T(y∂q-1,x)f(x),or that

ProofIf we usefn(x,y) to denoteT(y∂q-1,x)fn(x),then it is easy to verify that

Sincef(x,y) is a linear combination ofT(y∂q-1,x)fn(x),it follows that∂q-1,xf(x,y)=∂q-1,yf(x,y).Thus,by Theorem 1.7,we have that

which is the same as

We also need the following proposition,which can be found in [17,Proposition 13.9]:

Proposition 6.2Let{fn(x)} be a sequence of analytic functions nearx=0 such that the seriesconverges uniformly to an analytic functionf(x) nearx=0,and the seriesconverges uniformly to an analytic functionf(x,y)near(x,y)=(0,0).Then we have thatf(x,y)=T(y∂q,x)f(x),or that

The following proposition is a special case of Watson’sq-analogue of Whipple’s theorem,which can be found in [18,Theorem 1.8]:

Proposition 6.3For|αab/q|<1,we have theq-transformation formula

Using Watson’sq-analogue of Whipple’s theorem,the Rogers6φ5summation formula and Tannery’s theorem,one can prove the following (see,for example [16,Lemma 3]):

Proposition 6.4Asn →∞,we have the asymptotic formula

Proof of Theorem 1.14Using theq-exponential differential operator identities (1.8)and (1.9) we can prove that

For brevity,we will temporarily denote that

Settingd=0 in Proposition 6.3 and keeping the definition ofAn(a,b) in mind,we conclude that

Whenγ=0,Proposition 6.4 yields that

Using this fact and (6.1),and the ratio test,we can show that the series

converges to an analytic function ofcanddat (0,0).Thus,by Proposition 6.1,we have that

Combining the equation above with (6.4) and using Proposition 6.1 again,we deduce that

Substituting (6.1) and (6.2) into the above equation and simplifying,we conclude that

Lettingv=uin Proposition 1.6,we immediately arrive at the operational identity

For notational clarity,we denote that

so (6.8) can be written as

Propositions 6.2 and 6.4 allow us to use the exponential differential operatorT(γ∂q,β) to act on both sides of the equation above to obtain that

With the help of (6.9),we immediately deduce that

Combining the above three equations and noting the definition ofCncompletes the proof of Theorem 1.14.

7 Some Special Cases of Theorem 1.14 and the Rogers–Hecke Type Series

Theorem 1.14 can be used to recover some identities of the Rogers–Hecke type series,and sometimes this theorem is more effective.For this purpose,we first discuss some special cases of the theorem.

7.1 Some Special Cases of Theorem 1.14

Lettingγ=0 in Theorem 1.14,we immediately get that

Proposition 7.1For|αab/q|<1,we have

Using Proposition 7.1 and the Sears4φ3transformation formula we can prove the following proposition,which is similar to [19,Theorem 1.12]:

Proposition 7.2For|αab/q|<1,we have that

ProofThe Sears4φ3transformation can be restated as (see,for example [11,p.71])

Settingγ=0 in the equation above,and then replacing (c,d) by (q/d,q/c),we deduce that

Substituting the equation above into the left-hand side of (7.1) completes the proof of Proposition 7.2.

Based on Proposition 7.1 we can also prove that.

Proposition 7.3For|αβabc/q2|<1,we have that

ProofLettingd →∞in (7.1) and then substituting the limits

into the resulting equation completes the proof of Proposition 7.3.

Proposition 7.3 includes Rogers’6φ5summation formula [11,p44]as a special case;this is stated in the following proposition:

Proposition 7.4For|αabc/q2|<1,we have

ProofLetδmnbe de the Kronecker delta.Takingβ=1 in Proposition 7.3 and noting that (1;q)k=δk0,we find that the3φ2series in (7.2) have a value of 1,and thus we find that

Using theq-Gauss summation in (1.3),we deduce that

Substituting the equation above into the right-hand side of (7.5) completes the proof of Proposition 7.4.

7.2 Rogers–Hecke Type Series

Denote the finite theta seriesTn(q) by

The following Theorem first appeared in [19,Theorem 4.9]without proof,and now we will use Proposition 7.1 to prove it:

Theorem 7.5For|ab/q|<1,we have that

ProofDividing both sides of (7.1) by 1-αand then lettingα →1,and finally setting thatc=q1/2,d=-q-1/2andβ=-q,we find that

To simplify the3φ2series we need the followingq-identity,which can be found in [1,eq.(5.3)]and [19,p2087]:

Substituting the equation above into (7.7) completes the proof of Theorem 7.5.

Setting thata=b=0 in Theorem 7.5 and simplifying,we arrive at the Andrews identity[1,eq.(1.10)]

Taking thatb=0 in Theorem 7.5 and then multiplying both sides of the resulting equation by(1-a),and finally lettinga →1,we find that

Letting thatb=0 in Theorem 7.5,and replacingqbyq2,and finally putting thata=q,we find that

Dividing both sides of (7.1) by 1-α,and then lettingα →1,and finally setting thatc=q1/2,d=-q-1/2andβ=qand using theq-identity,

We can now find the following Theorem[19,Theorem 4.8],which has been used to derive several identities of the Rogers–Hecke type series:

Theorem 7.6For|ab/q|<1,we have that

Using Proposition 7.2 we can prove the following theorem [19,Theorem 1.9]which can be used to prove some identities of the Rogers–Hecke type,especially the Andrews–Dyson–Hickerson identity [19,p2703]:

Theorem 7.7For|ab|<1,we have that

ProofLettingc →∞in Proposition 7.2,and making a simple calculation,we easily find that

Setting thatα=β=qandd=-1,we find that

The followingq-identity can be found in [19,eq.(4.1)]:

Combining the above two equations completes the proof of the theorem.

For recent research work on Rogers–Hecke type series,please refer to [8,10,32,34,35].

8 Remarks

In a series papers [15–17,20–23]we started research on the applications ofq-partial differential equations and the analytic functions of several variables inq-analysis;this led us to develop a new method for derivingq-formulas.Aslan and Ismail [3]call this method “Liu’s calculus”.This method can not only be used to reprove existingq-series identities,but can also help us to find newq-series identities.Some results derived from this method cannot be simply proven by the usual methods.Wang and Ma [33]used a matrix inversion formula to give an extension of ourq-rational interpolation formula [13]found in 2002;we have been unable to prove the results presented here using similar techniques.Bhatnagar and Rai[6]extended some of our expansion formulas to the context of multiple series over root systems by using Bailey’s transformation,and their method is quite different from ours.

AcknowledgementsI am grateful to the anonymous referees and to Dandan Chen and Dunkun Yang for careful reading of the original manuscript,and for proposing some corrections and many constructive and helpful comments that resulted in substantial improvements to the paper.

Conflict of InterestThe author declares no conflict of interest.