Volume 2010, Article ID 785949,12pages doi:10.1155/2010/785949

*Research Article*

**On**

*q*

**-Operators and Summation of Some**

*q*

**-Series**

**Sana Guesmi and Ahmed Fitouhi**

*Facult´e des Sciences de Tunis, Tunis 1060, Tunisia*

Correspondence should be addressed to Ahmed Fitouhi,[email protected]

Received 14 October 2010; Accepted 11 December 2010

Academic Editor: Naseer Shahzad

Copyrightq2010 S. Guesmi and A. Fitouhi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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

**1. Introduction**

The operator*xd/dxn*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*xd/dx*

*n*_{, which give}

∞

*k*0
*knxk*

*x* *d*

*dx*
n

1
1−*x*

*,* |*x*|*<*1, n0,1,2*. . . .* 1.1

These operators are intimately related to the Stirling numbers of second kind {*n _{k}*} by the
formulasee1

*x* *d*

*dx*
*n*

*fx*
*n*

*k*1

*n*
*k*

*xkd*
*k _{f}*

*dxk,* 1.2

where*f*is a suitable function. We note that the*q-analogue of formula*1.2has been studied
by many authorssee2,3and references thereinand has found applications in many fields
such as arithmetic partitions and asymptotic expansions.

This paper is organized as follows. In Section2, we present some preliminary notions
and notations useful in the sequel. Section 3gives three applications of a result proved in
2, states a transformation theorem using the*q-Stirling numbers, and presents some related*
applications. Section4attempts to give a new *q-analogue of formula*1.2by 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 the Quantum 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.**

**2.1.**

*q*

**-Shifted Factorials**

**-Shifted Factorials**

For*a*∈ , the*q-shifted factorials are defined by*

*a;q* _{0}1, *a;q* _{n}*n*−1

*k*0

1−*aqk,* *a;q* _{∞}

∞

*k*0

1−*aqk.* 2.1

We also write

*a*1*, . . . , ak*;*q* _{n}*k*

*j*1

*aj*;*q* _{n},*n*0,1, . . . ,∞*.* 2.2

We put

*x _{q}* 1−

*q*

*x*

1−*q* *,* *x*∈ *,*

*n _{q}*!

*q;q* _{n}

1−*q* *n,* *n*∈
*.*

2.3

For*a, x*∈ and*n*∈, we adopt the following notation5:

*x*−*an _{q}*
⎧
⎨
⎩

1 if *n*0,

*x*−*ax*−*aq* · · ·*x*−*aqn*−1 _{if} _{n}_{1.} 2.4

The*q-analogue of the Jordan factorial is given by*

*x _{k,q}*

*x*−1

_{q}x*· · ·*

_{q}*x*−

*k*1

_{q}

1−*qx* _{1}_{−}* _{q}x*−1

_{· · ·}

_{1}

_{−}

*−*

_{q}x*k*1

1−*q* *k* *,*

and the*q-binomial coe*ﬃcient is defined by

*x*
*k*

*q*

*xk,q*

*k _{q}*!

*.*2.6

**2.2. The Jackson’s**

**2.2. The Jackson’s**

*q*

**-Derivative**

**-Derivative**

The*q-derivativeDqf*of a function*f*is defined bysee4

*Dqfx*

*fqx* −*fx*

*q*−1 *x* if*x /*0, 2.7

and *Dqf*0 *f*0 provided *f*0exists. Note that when *f* is diﬀerentiable, at*x, then*
*Dqfx*tends to*fx*as*q*tends to 1−.

It is easy to see that for suitable functions*f*and*g*, we have

*Dqfg* *x* *fqx* *Dqgx* *gxDqfx,* 2.8

*Dq*
_{f}

*g*

*x* *g*

*qx* *Dqfx*−*fqx* *Dqgx*

*gxgqx* *.* 2.9

**2.3. Elementary**

**2.3. Elementary**

*q*

**-Special Functions**

**-Special Functions**

Two*q-analogues of the exponential function are given by*see4

*eqz*

∞

*n*0
*zn*
*n _{q}*!

1

1−*q* *z;q* _{∞}*,* |*z*|*<*

1−*q* −1*,*

*Eqz*

∞

*n*0

*qnn*−1*/*2 *z*
*n*

*n _{q}*!

−1−*q* *z;q* _{∞}*,* *z*∈ *.*

2.10

They satisfy the relations

*Dqeqz* *eqz,* *DqEqz* *Eqqz* *,*

*eqzEq*−*z* *Eqzeq*−*z* 1, *Eqz* *e*1*/qz.*

2.11

In 1910, F. H. Jackson defined a*q-analogue of the Gamma function by*see4,6

Γ*qx*

*q;q* _{∞}

*qx*_{;}* _{q}*
∞

It satisfies the following functional equations:

Γ*qx*1 *xq*Γ*qx,* Γ*q*1 1, Γ*qn*1 *nq*!, *n*∈*.* 2.13

**2.4.**

**2.4.**

*q*

**-Stirling Numbers of Noncentral Type**

**-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 noncentral *q-Stirling numbers of the first kind* *sqn, k;r* are defined by the
following generating relation:

*t*−*r _{n,q}q*−

*n*2−

*rn*

*n*

*k*0

*sqn, k;rt _{q}k,*

*n*0,1, . . . , 2.14

and they are given by

*sqn, k;r* _{} 1
1−*q* *n*−*k*

*n*

*jk*

−1*j*−*kq*
_{n}_{−}_{j}

2

*rn*−*j*

*n*
*j*

*q*

*j*
*k*

*.* 2.15

The noncentral*q-Stirling numbers of the second kindSqn, k;r*are defined by the following
generating relation:

*tn _{q}*

*n*

*k*0
*q*

*k*
2

−*rk*

*Sqn, k;rt*−*rk,q,* *n*0,1, . . . , 2.16

and they are given by

*Sqn, k;r* 1
*k _{q}*!

*k*

*j*0

−1*k*−*jq*
_{j}_{}_{1}

2

−*rjk*

*k*
*j*

*q*

*rjn _{q}*

_{} 1

1−*q* *n*−*k*
*n*

*jk*

−1*j*−*kqrj*−*k*

*n*
*j*

*j*
*k*

*q*
*.*

2.17

*Remark 2.1. Note that when* *r* 0, then *sqn, k;r* and *Sqn, k;r* reduce to the*q-Stirling*
numbers, respectively, of the first and the second kind studied by Gould, Carlitz, and Kim
see8–11.

*Properties*

The noncentral*q-Stirling numbers satisfy the following properties.*
iFor*n*1,2, . . .and*k*1,2, . . . , n,

under the following conditions:

*sq*0,0;*r* 1, *sqn,*0;*r* *q*
_{n}

2

*rn*

−*r _{n,q},*

*n >*0,

*sq*0, k;*r* 0, *k >*0, *sqn, k;r* 0, *k > n.*

2.19

iiFor*n*1,2, . . .and*k*1,2, . . . , n,

*Sqn, k;r* *Sqn*−1, k−1;*r* *rkqSqn*−1, k;*r,* 2.20

under the following conditions:

*Sq*0,0;*r* 1, *Sqn,*0;*r* *rn _{q},*

*n >*0,

*Sq*0, k;*r* 0, *k >*0, *Sqn, k;r* 0, *k > n.*

2.21

**3. The Operator**

*xDq*

*m*

**and Some Related Transformations Theorems**

As in the classical casesee1, the iterate*xDqm*,*m*∈ , can be expanded in finite terms
involving the*q-Stirling numbers. This is the purpose of the following result.*

**Lemma 3.1**see2,3*. Lettingfbe a diﬀerentiable function, then one has*

*xDq* *mfx*
*m*

*k*1

*m*
*k*

*q,*1

*xkDk _{q}fx,*

*m*1,2, . . . , 3.1

*where*

*m*

*k*

*q,*1

*qkk*−1*/*2*Sqm*−1, k−1; 1

1
*k*−1* _{q}*!

*k*−1

*j*0

−1*jq*
_{j}

2
_{k}_{−}_{1}

*j*

*q*

*k*−*jm _{q}*−1

*.*

_{}3.2

Now, let us give three applications of the previous lemma.

*Example 3.2q-binomial series*. The*q-binomial theorem asserts that*

1Φ0

*qa*;−;*q, x*

∞

*n*0

*qa*_{;}_{q}*n*

*q;q* _{n}*x*
*n*_{}

*qa _{x;}_{q}*

∞

*x;q* _{∞} *,* |*x*|*<*1. 3.3

Using the fact that, for all*m*∈,

and the previous lemma, we deduce that

∞

*n*0

*qa*;*q* _{n}

*q;q* _{n}*n*
*m*
*qxn*

*m*

*k*1

*m*
*k*

*q,*1
*xkD _{q}k*

*qa _{x;}_{q}*

∞

*x;q* _{∞}

*,* |*x*|*<*1. 3.5

On the other hand, the definition of*q-derivative*2.9gives

*Dq*

*qa _{x;}_{q}*
∞

*x;q* _{∞}

*a _{q}*

*qa*1* _{x;}_{q}*
∞

*x;q* _{∞} *,* 3.6

and by iteration we have

*Dk*
*q*

*qa _{x;}_{q}*

∞

*x;q* _{∞}

Γ*q*_{Γ}*ak*
*qa*

*qak _{x;}_{q}*

∞

*x;q* _{∞} *.* 3.7

Thus,

∞

*n*0

*qa*_{;}_{q}*n*

*q;q* _{n}*n*
*m*
*qxn*

1
Γ*qa*

*x;q* _{∞}

*m*

*k*1

*m*
*k*

*q,*1

*xk*Γ*qak*

*qakx;q*

∞*.* 3.8

So, taking*a*1, we obtain

∞

*n*0

*nm _{q}xn* 1
1−

*x*

*m*

*k*1

*m*
*k*

*q,*1

*xkkq*!

*xq;q* * _{k}.* 3.9

Remark that if*q*tends to 1−, we obtain the formula given in13, page 366.

*Example 3.3q-Bessel function*. We consider the function

*Cpx*

∞

*k*0

−1*kxk*
*k _{q}*!

*kp*!

_{q}* _{x}*
2

−*p*
*Jp*1

1−*q* √2x, q*,* 3.10

where*Jp*1·*, q*is the first Jackson’s*q-Bessel function of orderp*see12,13.

By application of the operator*xDqm*to*C*0*x*and the use of relation3.4, we obtain

*xDq*

m
*C*0*x*

∞

*k*0

−1*k* *k*
*m*
*q*

*k _{q}*!2

*xk.* _{}_{3.11}_{}

Then, using Lemma3.1and the fact that

we get

∞

*k*0

−1*k* *k*
*m*
*q*

*k _{q}*!2

*xk*

*m*

*k*1

*m*
*k*

*q,*1

−1*kxkCkx.* _{}_{3.13}_{}

*Example 3.4* *q-polynomial exponential*. Take *fx* *eqx*. From relation 3.4 and
Lemma3.1, we obtain

*xDq* *meqx* *eqx*Φ*m,qx*

∞

*n*1

*nm _{q}*

*n*!

_{q}*x*

*n _{,}*

_{}

_{3.14}

_{}

where

Φ*m,qx*
*m*

*k*1

*m*
*k*

*q,*1

*xk,* 3.15

which is called the*q-polynomial exponential. So,*

*m*

*k*1

*m*
*k*

*q,*1

*xkEq*−*x*

∞

*n*1

*nm _{q}*

*n*!

_{q}*x*

*n*_{}∞
*n*1

*n*

*k*1

−1*kq*

*k*
2

_{}_{n}_{−}_{k}_{}*m*
*q*
*k _{q}*!

*n*−

*k*!

_{q}*x*

*n _{.}*

_{}

_{3.16}

_{}

In many mathematical fields there are some transformation theorems using the Stirling
numbers leading one to compute certain sumssee14. The purpose of the following result
is to give a*q-analogue context.*

**Theorem 3.5. Let**fxandgxbe two functions satisfying

*fx*

∞

*n*0

*anxn _{q},*

*gx*

∞

*n*0

*cnxn.* 3.17

*Then*

∞

*n*0

*cnfnxn*

∞

*m*1
*am*

*m*

*k*1

*m*
*k*

*q,*1

*xkDqkgx* 3.18

*provided the series*

∞

*n*0

*cnfnxn* 3.19

*Proof. From Lemma*3.1and the properties of the*q-Stirling numbers of the second kind*2.21,
we obtain

∞

*m*1
*am*

*m*

*k*1

*m*
*k*

*q,*1

*xkDk _{q}gx*

∞

*m*1
*am*

*xDq* *mgx*

∞
*m*1

*amxDq* *m*
_{}_{}_{∞}

*n*0
*cnxn*

*.*

3.20

The result follows, then, from relations3.4and3.19.

**Corollary 3.6. Let**fx_{n}_{}∞_{0}*anxnq. Then*

∞

*n*0

*fnxn*

∞

*m*1
*am*

*m*

*k*1

*m*
*k*

*q,*1

*xkkq*!

*x;q* _{k}_{}_{1} 3.21

*provided the series _{n}*∞

_{}

_{0}

*fnxn*

_{converges absolutely.}*Proof. By taking* *gx* 1/1 −*x* _{n}_{}∞_{0}*xn*_{,} _{|}_{x}_{|} _{<}_{1, in the previous theorem, and by}
application of relation3.7, we obtain

∞

*n*0

*fnxn* _{}∞
*m*1

*am*
*m*

*k*1

*m*
*k*

*q,*1
*xk _{D}k*

*q*

1
1−*x*

∞
*m*1

*am*
*m*

*k*1

*m*
*k*

*q,*1

*xk*Γ*qk*1
Γ*q*1

*qk*1_{x;}_{q}

∞

*x;q* _{∞}

∞
*m*1

*am*
*m*

*k*1

*m*
*k*

*q,*1

*xk*_{}*kq*!
*x;q* _{k}_{}_{1}*.*

3.22

*Example 3.7. Letfx* *x _{n,q}*. Then, the fact that

*x _{n,q}q*−

*n*2

*n*

*k*0

*sqn, k; 0xqk,* *n*0,1, . . . 3.23

and Corollary3.6give

∞

*k*0

*k _{n,q}xk*

*q*−

*n*

_{2}

1−*x*
*n*

*m*1

*sqn, m; 0*
*m*

*l*1

*m*
*l*

*q,*1

*xllq*!

*qx;q* * _{l}.* 3.24

Remark that when*q*tends to 1−, we obtain the formula given in1,6.4, page 3863.

**Corollary 3.8. For**fx_{n}_{}∞_{0}*anxn _{q}, the transformation formulas lead to the following:*

1* _{n}*∞

_{}

_{0}

*qnn*−1

*/*2

_{}

*α*

*nqfnxn*
_{}_{∞}

*n*1*an*
_{n}

*k*1*qkk*−1*/*2{*nk*}*q,*1*kq*!*αkqxk*1*qkx*
*α*−*k*
*q* *;*

2* _{n}*∞

_{}

_{0}1−

*qα*

_{}

*n*

*q/*1−*q*
*n*

*qfnxn*
_{}_{∞}

*n*1*an*
*n*

*k*1{*nk*}*q,*1
_{α}_{}_{k}_{−1}

*k*−1

*qxk/*1−*x*
*αk*
*q* *;*

3* _{n}*∞

_{}

_{0}

*fn/n*!

_{q}*xneqx*∞

_{n}_{}

_{1}

*ann*

_{k}_{}

_{1}{

*nk*}

*q,*1

*xkeqx*

_{}

_{∞}

*n*1*an*Φ*n,qx;*

4* _{n}*∞

_{}

_{0}

*qnn*−1

_{}

_{f}_{}

_{n}_{}

_{/}_{}

_{n}_{}

*q*!*xn*
_{}_{∞}

*n*1*an*
*n*

*k*1{*nk*}*q,*1*qkk*−1*/*2*xk/*−1−*qqkx;q*∞

*provided the series converge absolutely.*

*Proof. The results are direct consequences of Theorem*3.5by putting the following:

1*gx* 1*xα _{q}*

*∞*

_{n}_{}

_{0}

*qnn*−1

*/*2

_{}

*α*

*nqxn* and remark that *Dqk*1*xαq*
*qkk*−1*/*2_{}_{α}_{}

*qα*−1*q*· · ·*α*−*k*1*q*1*qkx*
*α*−*k*
*q* ;

2*gx* 1/1−*xα _{q}*

*∞*

_{n}_{}

_{0}1−

*qα*

_{}

*n*

*q/*1−*q*
*n*

*qxn*and remark that*Dqk*1/1−*xαq*
*α _{q}α*1

*· · ·*

_{q}*αk*−1

*1−*

_{q}/*xα*;

_{q}k3*gx* *eqx*;

4*gx* *Eqx*and remark that*DkqEqx* *qkk*−1*/*2*Eqqkx*.

*Remark 3.9. The last formulas coincide with some of the ones given in*15when*q*tends to 1−.

**4. The Operator**

*x*

### ;

*q*

_{1}

*Dq*

*m*

**and Related Transformation Theorem**

* Lemma 4.1. For a suitable functionf, one has form*1,2, . . .

*x;q*_{1}*Dqmfx*
*m*

*k*1

−1*m*−*kSqm*−1, k−1; 1*x;q* * _{k}Dk_{q}fx.* 4.1

*Proof. The formula can be obtained by induction with respect tom. Indeed, for* *m* 1, we
have

Assuming that formula4.1is true for*m, then*

*x;q* _{1}*Dqm*1*fx* *x;q* _{1}*Dq*

*m*

*k*1

−1*m*−*kSqm*−1, k−1; 1*x;q* _{k}Dk_{q}fx

*x;q* _{1}
*m*

*k*1

−1*m*−*kSqm*−1, k−1; 1

*qx;q* * _{k}Dk*1

*q*

*fx*

−*x;q* _{1}
*m*

*k*1

−1*m*−*kSqm*−1, k−1; 1*k _{q}qx;q*

_{k}_{−1}

*D*

_{q}kfx*m*1
*k*2

−1*m*−*k*1*Sqm*−1, k−2; 1*x;q* _{k}Dk_{q}fx

−*m*

*k*1

−1*m*−*kk _{q}Sqm*−1, k−1; 1

*x;q* _{k}Dk*qfx*

*m*
*k*2

−1*m*−*k*1*Sqm*−1, k−2; 1−*k _{q}Sqm*−1, k−1; 1

*x;q*

_{k}D_{q}kfx*Sqm*−1, m−1; 1*x;q* _{m}_{}_{1}*Dm _{q}*1

*fx*

−−1*m*−1*Sqm*−1,0; 1

*x;q* _{1}*Dqfx.*

4.3

The result is easily deduced by formulas2.20, and2.21.

**Theorem 4.2. Let**fxandgxbe two functions defined by

*fx*

∞

*n*0

−1*nαnxnq,* *gx*

∞

*n*0
*cn*

*x;q* * _{n}.* 4.4

*If the series*

∞

*n*0
*cnfn*

*x;q* * _{n}* 4.5

*converges absolutely, then*

∞

*n*0
*cnfn*

*x;q* _{n}

∞

*m*1
*αm*

*m*

*k*1

−1*m*−*kSqm*−1, k−1; 1

*x;q* * _{k}D_{q}kgx.* 4.6

*Proof. From the previous lemma, we obtain form*1,2, . . .,

∞

*m*0

*αmx;q*_{1}*Dqmgx*

∞

*m*1
*αm*

*m*

*k*1

So, the absolute convergence of the series4.5and the fact that

*x;q*_{1}*Dqmx;q* * _{n}* −1

*mnm*

_{q}x;q

_{n},*m*∈ 4.8

achieve the proof.

**Corollary 4.3. Let**fx* _{m}*∞

_{}

_{0}−1

*mαmxm*

_{q}. Then∞

*m*1
*αm*

*m*

*k*1

−1*m*−*kSqm*−1, k−1; 1*x;q* * _{k}n_{k,q}xn*−

*k*

*n*

*k*0

−*q* *ksqk,*0,−*nfkx;q* _{k}.

4.9

*Proof. Putgx* *xn*.

Using the representationsee5

*xn*
*n*

*k*0

−1*k*

*n*
*k*

*q*

*q*−*n*−1*kq*
_{k}

2
_{}

*x;q* _{k}*n*

*k*0

−*q* *ksqk,*0,−*nx;q* * _{k},* 4.10

relation4.6and the fact that

*Dkqxn* *nk,qxn*−*k* 4.11

give the desired result.

*Remark 4.4. Note that recently Liu in his paper*see16has obtained some interesting
q-identities in showing that the solutions of two diﬀerence equations involve some series of
*q-operatorsDn*

*q* of*q-Cauchy type.*

**Acknowledgments**

The authors would like to thank the Board of editors for their helpful comments. They are thankful to the anonymous reviewer for his remarks, who pointed the references out to them see3,16.

**References**

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