# Onq-Operators and Summation of Some q-Series

Authors: Sana Guesmi; Ahmed Fitouhi
Journal: International Journal of Mathematics and Mathematical Sciences
 (2010)

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## Abstract

Using Jackson'sq-derivative and the q-Stirling numbers, we establish some transformation theorems leading to the values of some
convergent q-series.


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## Body

## 1. Introduction

The operator(x(d/dx))n has many assets and plays a central role in arithmetic fields and in computation of some finite or infinite sums. For example, when we try to compute the sum ∑k=0+∞knxk, we use the operators (x(d/dx))n, which give(1.1)∑k=0+∞knxk=(xddx)n(11-x),|x|<1,n=0,1,2….
These operators are intimately related to the Stirling numbers of second kind {nk} by the formula (see [1])(1.2)(xddx)nf(x)=∑k=1n{nk}xkdkfdxk,
where f is a suitable function. We note that the q-analogue of formula (1.2) has been studied by many authors (see [2, 3] and references therein) and has found applications in many fields such as arithmetic partitions and asymptotic expansions.This paper deals with the analogues of the operators(x(d/dx))n in Quantum Calculus and some q-transformation theorems that will be used to establish the sums of some q-series.This paper is organized as follows. In Section2, we present some preliminary notions and notations useful in the sequel. Section 3 gives three applications of a result proved in [2], states a transformation theorem using the q-Stirling numbers, and presents some related applications. Section 4 attempts to give a new q-analogue of formula (1.2) by studying the transformation theorem related to a q-derivative operator.


## 2. Notations and Preliminaries

To make this paper self-containing and easily decipherable, we recall some useful preliminaries about theQuantum Calculus and we select Gasper-Rahman’s book [4], for the notations and for a deep study in this way. Throughout this paper, we fix q∈]0,1[.


### 2.1.q-Shifted Factorials

Fora∈ℂ, the q-shifted factorials are defined by (2.1)(a;q)0=1,(a;q)n=∏k=0n-1(1-aqk),(a;q)∞=∏k=0+∞(1-aqk).
We also write (2.2)(a1,…,ak;q)n=∏j=1k(aj;q)n,n=0,1,…,∞.
We put (2.3)[x]q=1-qx1-q,x∈C,[n]q!=(q;q)n(1-q)n,n∈N.
For a,x∈ℂ and n∈ℕ, we adopt the following notation [5]: (2.4)(x-a)qn={1ifn=0,(x-a)(x-aq)⋯(x-aqn-1)ifn⩾1.
The q-analogue of the Jordan factorial is given by(2.5)[x]k,q=[x]q[x-1]q⋯[x-k+1]q=(1-qx)(1-qx-1)⋯(1-qx-k+1)(1-q)k,
and the q-binomial coefficient is defined by(2.6)[xk]q=[x]k,q[k]q!.


### 2.2. The Jackson'sq-Derivative

Theq-derivative Dqf of a function f is defined by (see [4])(2.7)Dqf(x)=f(qx)-f(x)(q-1)xifx≠0,
and (Dqf)(0)=f′(0) provided f′(0) exists. Note that when f is differentiable, at x, then (Dq)f(x) tends to f′(x) as q tends to 1-.It is easy to see that for suitable functionsf and g, we have(2.8)Dq(fg)(x)=f(qx)Dqg(x)+g(x)Dqf(x),(2.9)Dq(fg)(x)=g(qx)Dqf(x)-f(qx)Dqg(x)g(x)g(qx).


### 2.3. Elementaryq-Special Functions

Twoq-analogues of the exponential function are given by (see [4]) (2.10)eq(z)=∑n=0+∞zn[n]q!=1((1-q)z;q)∞,|z|<(1-q)-1,Eq(z)=∑n=0+∞qn(n-1)/2zn[n]q!=(-(1-q)z;q)∞,z∈C.
They satisfy the relations (2.11)Dqeq(z)=eq(z),DqEq(z)=Eq(qz),eq(z)Eq(-z)=Eq(z)eq(-z)=1,Eq(z)=e1/q(z).
In 1910, F. H. Jackson defined a q-analogue of the Gamma function by (see [4, 6]) (2.12)Γq(x)=(q;q)∞(qx;q)∞(1-q)1-x,x≠0,-1,-2,….
It satisfies the following functional equations: (2.13)Γq(x+1)=[x]qΓq(x),Γq(1)=1,Γq(n+1)=[n]q!,n∈N.


### 2.4.q-Stirling Numbers of Noncentral Type

In [7], Charalambides introduced the so-called noncentral q-Stirling numbers, which are q-analogues of the Stirling numbers and classified into two kinds.The noncentralq-Stirling numbers of the first kind sq(n,k;r) are defined by the following generating relation:(2.14)[t-r]n,q=q-(n2)-rn∑k=0nsq(n,k;r)[t]qk,n=0,1,…,
and they are given by(2.15)sq(n,k;r)=1(1-q)n-k∑j=kn(-1)j-kq(n-j2)+r(n-j)[nj]q(jk).
The noncentral q-Stirling numbers of the second kind Sq(n,k;r) are defined by the following generating relation:(2.16)[t]qn=∑k=0nq(k2)-rkSq(n,k;r)[t-r]k,q,n=0,1,…,
and they are given by(2.17)Sq(n,k;r)=1[k]q!∑j=0k(-1)k-jq(j+12)-(r+j)k[kj]q[r+j]qn=1(1-q)n-k∑j=kn(-1)j-kqr(j-k)(nj)[jk]q.Remark 2.1.
Note that whenr=0, then sq(n,k;r) and Sq(n,k;r) reduce to the q-Stirling numbers, respectively, of the first and the second kind studied by Gould, Carlitz, and Kim (see [8–11]).Properties
The noncentralq-Stirling numbers satisfy the following properties.  (i)
Forn=1,2,… and k=1,2,…,n,
(2.18)sq(n,k;r)=sq(n-1,k-1;r)-[n+r-1]qsq(n-1,k;r),
under the following conditions:
(2.19)sq(0,0;r)=1,sq(n,0;r)=q(n2)+rn[-r]n,q,n>0,sq(0,k;r)=0,k>0,sq(n,k;r)=0,k>n.(ii)
Forn=1,2,… and k=1,2,…,n,
(2.20)Sq(n,k;r)=Sq(n-1,k-1;r)+[r+k]qSq(n-1,k;r),
under the  following conditions:
(2.21)Sq(0,0;r)=1,Sq(n,0;r)=[r]qn,n>0,Sq(0,k;r)=0,k>0,Sq(n,k;r)=0,k>n.


## 2.1.q-Shifted Factorials

Fora∈ℂ, the q-shifted factorials are defined by (2.1)(a;q)0=1,(a;q)n=∏k=0n-1(1-aqk),(a;q)∞=∏k=0+∞(1-aqk).
We also write (2.2)(a1,…,ak;q)n=∏j=1k(aj;q)n,n=0,1,…,∞.
We put (2.3)[x]q=1-qx1-q,x∈C,[n]q!=(q;q)n(1-q)n,n∈N.
For a,x∈ℂ and n∈ℕ, we adopt the following notation [5]: (2.4)(x-a)qn={1ifn=0,(x-a)(x-aq)⋯(x-aqn-1)ifn⩾1.
The q-analogue of the Jordan factorial is given by(2.5)[x]k,q=[x]q[x-1]q⋯[x-k+1]q=(1-qx)(1-qx-1)⋯(1-qx-k+1)(1-q)k,
and the q-binomial coefficient is defined by(2.6)[xk]q=[x]k,q[k]q!.


## 2.2. The Jackson'sq-Derivative

Theq-derivative Dqf of a function f is defined by (see [4])(2.7)Dqf(x)=f(qx)-f(x)(q-1)xifx≠0,
and (Dqf)(0)=f′(0) provided f′(0) exists. Note that when f is differentiable, at x, then (Dq)f(x) tends to f′(x) as q tends to 1-.It is easy to see that for suitable functionsf and g, we have(2.8)Dq(fg)(x)=f(qx)Dqg(x)+g(x)Dqf(x),(2.9)Dq(fg)(x)=g(qx)Dqf(x)-f(qx)Dqg(x)g(x)g(qx).


## 2.3. Elementaryq-Special Functions

Twoq-analogues of the exponential function are given by (see [4]) (2.10)eq(z)=∑n=0+∞zn[n]q!=1((1-q)z;q)∞,|z|<(1-q)-1,Eq(z)=∑n=0+∞qn(n-1)/2zn[n]q!=(-(1-q)z;q)∞,z∈C.
They satisfy the relations (2.11)Dqeq(z)=eq(z),DqEq(z)=Eq(qz),eq(z)Eq(-z)=Eq(z)eq(-z)=1,Eq(z)=e1/q(z).
In 1910, F. H. Jackson defined a q-analogue of the Gamma function by (see [4, 6]) (2.12)Γq(x)=(q;q)∞(qx;q)∞(1-q)1-x,x≠0,-1,-2,….
It satisfies the following functional equations: (2.13)Γq(x+1)=[x]qΓq(x),Γq(1)=1,Γq(n+1)=[n]q!,n∈N.


## 2.4.q-Stirling Numbers of Noncentral Type

In [7], Charalambides introduced the so-called noncentral q-Stirling numbers, which are q-analogues of the Stirling numbers and classified into two kinds.The noncentralq-Stirling numbers of the first kind sq(n,k;r) are defined by the following generating relation:(2.14)[t-r]n,q=q-(n2)-rn∑k=0nsq(n,k;r)[t]qk,n=0,1,…,
and they are given by(2.15)sq(n,k;r)=1(1-q)n-k∑j=kn(-1)j-kq(n-j2)+r(n-j)[nj]q(jk).
The noncentral q-Stirling numbers of the second kind Sq(n,k;r) are defined by the following generating relation:(2.16)[t]qn=∑k=0nq(k2)-rkSq(n,k;r)[t-r]k,q,n=0,1,…,
and they are given by(2.17)Sq(n,k;r)=1[k]q!∑j=0k(-1)k-jq(j+12)-(r+j)k[kj]q[r+j]qn=1(1-q)n-k∑j=kn(-1)j-kqr(j-k)(nj)[jk]q.Remark 2.1.
Note that whenr=0, then sq(n,k;r) and Sq(n,k;r) reduce to the q-Stirling numbers, respectively, of the first and the second kind studied by Gould, Carlitz, and Kim (see [8–11]).Properties
The noncentralq-Stirling numbers satisfy the following properties.  (i)
Forn=1,2,… and k=1,2,…,n,
(2.18)sq(n,k;r)=sq(n-1,k-1;r)-[n+r-1]qsq(n-1,k;r),
under the following conditions:
(2.19)sq(0,0;r)=1,sq(n,0;r)=q(n2)+rn[-r]n,q,n>0,sq(0,k;r)=0,k>0,sq(n,k;r)=0,k>n.(ii)
Forn=1,2,… and k=1,2,…,n,
(2.20)Sq(n,k;r)=Sq(n-1,k-1;r)+[r+k]qSq(n-1,k;r),
under the  following conditions:
(2.21)Sq(0,0;r)=1,Sq(n,0;r)=[r]qn,n>0,Sq(0,k;r)=0,k>0,Sq(n,k;r)=0,k>n.


## 3. The Operator(xDq)m and Some Related Transformations Theorems

As in the classical case (see [1]), the iterate (xDq)m, m∈ℕ, can be expanded in finite terms involving the q-Stirling numbers. This is the purpose of the following result.Lemma 3.1 (see [2, 3]).
Lettingf be a differentiable function, then one has
(3.1)(xDq)mf(x)=∑k=1m{mk}q,1xkDqkf(x),m=1,2,…,
where
(3.2){mk}q,1=qk(k-1)/2Sq(m-1,k-1;1)=1[k-1]q!∑j=0k-1(-1)jq(j2)[k-1j]q[k-j]qm-1.Now, let us give three applications of the previous lemma.Example 3.2 (q-binomial series).
Theq-binomial theorem asserts that
(3.3)Φ10(qa;-;q,x)=∑n=0+∞(qa;q)n(q;q)nxn=(qax;q)∞(x;q)∞,|x|<1.
Using the fact that, for all m∈ℕ,
(3.4)(xDq)mxn=[n]qmxn
and the previous lemma, we deduce that
(3.5)∑n=0+∞(qa;q)n(q;q)n[n]qmxn=∑k=1m{mk}q,1xkDqk((qax;q)∞(x;q)∞),|x|<1.
On the other hand, the definition of q-derivative (2.9) gives
(3.6)Dq((qax;q)∞(x;q)∞)=[a]q(qa+1x;q)∞(x;q)∞,
and by iteration we have
(3.7)Dqk((qax;q)∞(x;q)∞)=Γq(a+k)Γq(a)(qa+kx;q)∞(x;q)∞.
Thus,
(3.8)∑n=0+∞(qa;q)n(q;q)n[n]qmxn=1Γq(a)(x;q)∞∑k=1m{mk}q,1xkΓq(a+k)(qa+kx;q)∞.
So, taking a=1, we obtain
(3.9)∑n=0+∞[n]qmxn=11-x∑k=1m{mk}q,1xk[k]q!(xq;q)k.
Remark that if q tends to 1-, we obtain the formula given in [1, page 366].Example 3.3 (q-Bessel function).
We consider the function(3.10)Cp(x)=∑k=0+∞(-1)kxk[k]q![k+p]q!=(x2)-pJp(1)((1-q)2x,q),
where Jp(1)(·,q) is the first Jackson's q-Bessel function of order p (see [12, 13]).
By application of the operator(xDq)m to C0(x) and the use of relation (3.4), we obtain
(3.11)(xDq)mC0(x)=∑k=0+∞(-1)k[k]qm([k]q!)2xk.
Then, using Lemma 3.1 and the fact that
(3.12)DqCp(x)=-Cp+1(x),
we get
(3.13)∑k=0+∞(-1)k[k]qm([k]q!)2xk=∑k=1m{mk}q,1(-1)kxkCk(x).Example 3.4 (q-polynomial exponential).
Takef(x)=eq(x). From relation (3.4) and Lemma 3.1, we obtain
(3.14)(xDq)meq(x)=eq(x)Φm,q(x)=∑n=1+∞[n]qm[n]q!xn,
where
(3.15)Φm,q(x)=∑k=1m{mk}q,1xk,
which is called the q-polynomial exponential. So,
(3.16)∑k=1m{mk}q,1xk=Eq(-x)∑n=1+∞[n]qm[n]q!xn=∑n=1+∞∑k=1n(-1)kq(k2)[n-k]qm[k]q![n-k]q!xn.
In many mathematical fields there are some transformation theorems using the Stirling numbers leading one to compute certain sums (see [14]). The purpose of the following result is to give a q-analogue context.Theorem 3.5.
Letf(x) and g(x) be two functions satisfying
(3.17)f(x)=∑n=0+∞an[x]qn,g(x)=∑n=0+∞cnxn.
Then
(3.18)∑n=0+∞cnf(n)xn=∑m=1+∞am∑k=1m{mk}q,1xkDqkg(x)
provided the series
(3.19)∑n=0+∞cnf(n)xn
converges absolutely.Proof.
From Lemma3.1 and the properties of the q-Stirling numbers of the second kind (2.21), we obtain
(3.20)∑m=1+∞am∑k=1m{mk}q,1xkDqkg(x)=∑m=1+∞am(xDq)mg(x)=∑m=1+∞am(xDq)m(∑n=0+∞cnxn).
The result follows, then, from relations (3.4) and (3.19).Corollary 3.6.
Letf(x)=∑n=0+∞an[x]qn. Then
(3.21)∑n=0+∞f(n)xn=∑m=1+∞am∑k=1m{mk}q,1xk[k]q!(x;q)k+1
provided the series ∑n=0+∞f(n)xn converges absolutely.Proof.
By takingg(x)=1/(1-x)=∑n=0+∞xn,   |x|<1, in the previous theorem, and by application of relation (3.7), we obtain
(3.22)∑n=0+∞f(n)xn=∑m=1+∞am∑k=1m{mk}q,1xkDqk(11-x)=∑m=1+∞am∑k=1m{mk}q,1xkΓq(k+1)Γq(1)(qk+1x;q)∞(x;q)∞=∑m=1+∞am∑k=1m{mk}q,1xk[k]q!(x;q)k+1.Example 3.7.
Letf(x)=[x]n,q. Then, the fact that
(3.23)[x]n,q=q-(n2)∑k=0nsq(n,k;0)[x]qk,n=0,1,…
and Corollary 3.6 give
(3.24)∑k=0+∞[k]n,qxk=q-(n2)1-x∑m=1nsq(n,m;0)∑l=1m{ml}q,1xl[l]q!(qx;q)l.
Remark that when q tends to 1-, we obtain the formula given in [14, (6.4), page 3863].Some others summation formulas are presented in the following statements.Corollary 3.8.
Forf(x)=∑n=0+∞an[x]qn, the transformation formulas lead to the following:  (1)
∑n=0+∞qn(n-1)/2[αn]qf(n)xn=∑n=1+∞an∑k=1nqk(k-1)/2{nk}q,1[k]q![αk]qxk(1+qkx)qα-k;(2)
∑n=0+∞((1-qα)qn/(1-q)qn)f(n)xn=∑n=1+∞an∑k=1n{nk}q,1[α+k-1k-1]q(xk/(1-x)qα+k);(3)
∑n=0+∞(f(n)/[n]q!)xn=eq(x)∑n=1+∞an∑k=1n{nk}q,1xk=eq(x)∑n=1+∞anΦn,q(x);(4)
∑n=0+∞qn(n-1)(f(n)/[n]q!)xn=∑n=1+∞an∑k=1n{nk}q,1qk(k-1)/2(xk/(-(1-q)qkx;q)∞) provided the series converge absolutely.Proof.
The results are direct consequences of Theorem3.5 by putting the following:  (1)
g(x)=(1+x)qα=∑n=0+∞qn(n-1)/2[αn]qxn and remark that Dqk(1+x)qα=qk(k-1)/2[α]q[α-1]q⋯[α-k+1]q(1+qkx)qα-k;(2)
g(x)=1/(1-x)qα=∑n=0+∞((1-qα)qn/(1-q)qn)xn and remark that Dqk(1/(1-x)qα)=[α]q[α+1]q⋯[α+k-1]q/(1-x)qα+k;(3)
g(x)=eq(x);(4)
g(x)=Eq(x) and remark that DqkEq(x)=qk(k-1)/2Eq(qkx).Remark 3.9.
The last formulas coincide with some of the ones given in [15] when q tends to 1-.


## 4. The Operator((x;q)1Dq)m and Related Transformation Theorem

Lemma 4.1.
For a suitable functionf, one has for m=1,2,…(4.1)[(x;q)1Dq]mf(x)=∑k=1m(-1)m-kSq(m-1,k-1;1)(x;q)kDqkf(x).Proof.
The formula can be obtained by induction with respect tom. Indeed, for m=1, we have
(4.2)[(x;q)1Dq]f(x)=(x;q)1Dqf(x)=Sq(0,0;1)(x;q)1Dqf(x).
Assuming that formula (4.1) is true for m, then
(4.3)[(x;q)1Dq]m+1f(x)=(x;q)1Dq[∑k=1m(-1)m-kSq(m-1,k-1;1)(x;q)kDqkf(x)]=(x;q)1∑k=1m(-1)m-kSq(m-1,k-1;1)(qx;q)kDqk+1f(x)-(x;q)1∑k=1m(-1)m-kSq(m-1,k-1;1)[k]q(qx;q)k-1Dqkf(x)=∑k=2m+1(-1)m-k+1Sq(m-1,k-2;1)(x;q)kDqkf(x)-∑k=1m(-1)m-k[k]qSq(m-1,k-1;1)(x;q)kDqkf(x)=∑k=2m(-1)m-k+1[Sq(m-1,k-2;1)-[k]qSq(m-1,k-1;1)](x;q)kDqkf(x)+Sq(m-1,m-1;1)(x;q)m+1Dqm+1f(x)-(-1)m-1Sq(m-1,0;1)(x;q)1Dqf(x).
The result is easily deduced by formulas (2.20), and (2.21).Theorem 4.2.
Letf(x) and g(x) be two functions defined by
(4.4)f(x)=∑n=0+∞(-1)nαn[x]qn,g(x)=∑n=0+∞cn(x;q)n.
If the series
(4.5)∑n=0+∞cnf(n)(x;q)n
converges absolutely, then
(4.6)∑n=0+∞cnf(n)(x;q)n=∑m=1+∞αm∑k=1m(-1)m-kSq(m-1,k-1;1)(x;q)kDqkg(x).Proof.
From the previous lemma, we obtain form=1,2,…,
(4.7)∑m=0+∞αm[(x;q)1Dq]mg(x)=∑m=1+∞αm∑k=1m(-1)m-kSq(m-1,k-1;1)(x;q)kDqkg(x).
So, the absolute convergence of the series (4.5) and the fact that
(4.8)[(x;q)1Dq]m(x;q)n=(-1)m[n]qm(x;q)n,m∈N
achieve the proof.Corollary 4.3.
Letf(x)=∑m=0+∞(-1)mαm[x]qm. Then
(4.9)∑m=1+∞αm∑k=1m(-1)m-kSq(m-1,k-1;1)(x;q)k[n]k,qxn-k=∑k=0n(-q)ksq(k,0,-n)f(k)(x;q)k.Proof.
Putg(x)=xn.
Using the representation (see [5])
(4.10)xn=∑k=0n(-1)k[nk]qq-(n-1)kq(k2)(x;q)k=∑k=0n(-q)ksq(k,0,-n)(x;q)k,
relation (4.6) and the fact that
(4.11)Dqk(xn)=[n]k,qxn-k
give the desired result.Remark 4.4.
Note that recently Liu in his paper (see [16]) has obtained some interesting q-identities in showing that the solutions of two difference equations involve some series of q-operators Dqn of q-Cauchy type.



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