Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2010, Article ID 785949, 12 pages
doi:10.1155/2010/785949
Research Article
On q-Operators and Summation of Some q-Series
Sana Guesmi and Ahmed Fitouhi
Faculté des Sciences de Tunis, Tunis 1060, Tunisia
Correspondence should be addressed to Ahmed Fitouhi, ahmed.fitouhi@fst.rnu.tn
Received 14 October 2010; Accepted 11 December 2010
Academic Editor: Naseer Shahzad
Copyright q 2010 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/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
∞ n k
n
k0 k x , we use the operators xd/dx , which give
1
d n
,
k x x
dx
1−x
k0
∞
n k
|x| < 1, n 0, 1, 2 . . . .
1.1
These operators are intimately related to the Stirling numbers of second kind { nk } by the
formula see 1
d
x
dx
n
fx n n
k1
k
xk
dk f
,
dxk
1.2
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 xd/dxn in Quantum Calculus
and some q-transformation theorems that will be used to establish the sums of some q-series.
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This paper is organized as follows. In Section 2, 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 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. q-Shifted Factorials
For a ∈ , the q-shifted factorials are defined by
a; q 0 1,
a; q
n
n−1 1 − aqk ,
k0
∞ a; q ∞ 1 − aqk .
2.1
k0
We also write
k a1 , . . . , ak ; q n aj ; q n ,
n 0, 1, . . . , ∞.
2.2
j1
We put
1 − qx
, x∈ ,
1−q
q; q n
nq ! n , n ∈ .
1−q
xq For a, x ∈
2.3
and n ∈ , we adopt the following notation 5:
x − anq ⎧
⎨1
if n 0,
⎩x − a x − aq · · · x − aqn−1
if n 1.
2.4
The q-analogue of the Jordan factorial is given by
xk,q xq x − 1q · · · x − k 1q
1 − qx 1 − qx−1 · · · 1 − qx−k1
,
k
1−q
2.5
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3
and the q-binomial coefficient is defined by
x
k
q
xk,q
kq !
2.6
.
2.2. The Jackson’s q-Derivative
The q-derivative Dq f of a function f is defined by see 4
f qx − fx
Dq fx q−1 x
if x /
0,
2.7
and Dq f0 f 0 provided f 0 exists. Note that when f is differentiable, at x, then
Dq fx tends to f x as q tends to 1− .
It is easy to see that for suitable functions f and g, we have
Dq fg x f qx Dq gx gxDq fx,
g qx Dq fx − f qx Dq gx
f
Dq
.
x g
gxg qx
2.8
2.9
2.3. Elementary q-Special Functions
Two q-analogues of the exponential function are given by see 4
eq z ∞
1
zn
,
!
n
1 − q z; q ∞
q
n0
−1
|z| < 1 − q ,
∞
zn
qnn−1/2
Eq z − 1 − q z; q ∞ ,
nq !
n0
2.10
z∈ .
They satisfy the relations
Dq eq z eq z,
Dq Eq z Eq qz ,
eq zEq −z Eq zeq −z 1,
Eq z e1/q z.
2.11
In 1910, F. H. Jackson defined a q-analogue of the Gamma function by see 4, 6
q; q
1−x
Γq x x ∞ 1 − q
,
q ;q ∞
x/
0, −1, −2, . . . .
2.12
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It satisfies the following functional equations:
Γq x 1 xq Γq x,
Γq 1 1,
Γq n 1 nq !,
n ∈ .
2.13
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 noncentral q-Stirling numbers of the first kind sq n, k; r are defined by the
following generating relation:
t − rn,q q− 2 −rn
n
n
sq n, k; rtkq ,
n 0, 1, . . . ,
2.14
k0
and they are given by
sq n, k; r 1
1−q
n−k
n
j−k
−1
q
n−j rn−j n
2
j
jk
j
k
q
.
2.15
The noncentral q-Stirling numbers of the second kind Sq n, k; r are defined by the following
generating relation:
tnq n
q
k −rk
2
Sq n, k; rt
− rk,q ,
n 0, 1, . . . ,
2.16
k0
and they are given by
k
j1 −rjk k n
1 k−j
Sq n, k; r r j q
−1 q 2
kq ! j0
j q
1
n
j−k
n−k −1 q
1−q
jk
j
n
rj−k
j
k
2.17
.
q
Remark 2.1. Note that when r 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 noncentral q-Stirling numbers satisfy the following properties.
i For n 1, 2, . . . and k 1, 2, . . . , n,
sq n, k; r sq n − 1, k − 1; r − n r − 1q sq n − 1, k; r,
2.18
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5
under the following conditions:
n
sq 0, 0; r 1,
sq n, 0; r q
sq 0, k; r 0,
k > 0,
2
rn
−rn,q ,
sq n, k; r 0,
n > 0,
2.19
k > n.
ii For n 1, 2, . . . and k 1, 2, . . . , n,
Sq n, k; r Sq n − 1, k − 1; r r kq Sq n − 1, k; r,
2.20
under the following conditions:
Sq 0, 0; r 1,
Sq 0, k; r 0,
Sq n, 0; r rnq ,
k > 0,
n > 0,
Sq n, k; r 0,
2.21
k > n.
3. The Operator xDq m and Some Related Transformations Theorems
As in the classical case see 1, the iterate xDq m , m ∈ , 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. Letting f be a differentiable function, then one has
m m
m
xDq fx xk Dqk fx,
k
q,1
k1
m 1, 2, . . . ,
3.1
where
k−1
j
k−1 m−1
1
m
j
kk−1/2
k−j q .
q
Sq m − 1, k − 1; 1 −1 q 2
k q,1
k − 1q ! j0
j
3.2
q
Now, let us give three applications of the previous lemma.
Example 3.2 q-binomial series. The q-binomial theorem asserts that
a
∞ qa ; q
q x; q ∞
n n
x ,
1 Φ0 q ; −; q, x x; q ∞
n0 q; q n
a
|x| < 1.
3.3
Using the fact that, for all m ∈ ,
xDq
m
n
xn nm
q x
3.4
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International Journal of Mathematics and Mathematical Sciences
and the previous lemma, we deduce that
a
∞ qa ; q
m q x; q ∞
m
n
m n
k k
nq x x Dq
,
k q,1
x; q ∞
n0 q; q n
k1
|x| < 1.
3.5
On the other hand, the definition of q-derivative 2.9 gives
a
a1
q x; q ∞
q x; q ∞
aq ,
Dq
x; q ∞
x; q ∞
3.6
a
q x; q ∞
Γq a k qak x; q ∞
.
Γq a
x; q ∞
x; q ∞
3.7
and by iteration we have
Dqk
Thus,
∞ qa ; q
m 1
m
n
n nm
x
xk Γq a k qak x; q .
q
∞
Γq a x; q ∞ k1 k q,1
n0 q; q n
3.8
So, taking a 1, we obtain
∞
n
nm
q x
n0
m kq !
1 m
xk .
1 − x k1 k q,1
xq; q k
3.9
Remark that if q tends to 1− , we obtain the formula given in 13, page 366.
Example 3.3 q-Bessel function. We consider the function
Cp x x −p
√
−1k xk
1 1 − q 2x, q ,
Jp
2
k0 kq ! k p q !
∞
3.10
1
where Jp ·, 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
xDq
m
C0 x ∞
km
q
k
−1k 2 x .
k0
kq !
3.11
Then, using Lemma 3.1 and the fact that
Dq Cp x −Cp1 x,
3.12
International Journal of Mathematics and Mathematical Sciences
7
we get
∞
m km
q
m
k
x
−1k −1k xk Ck x.
2
k
q,1
k0
k1
kq !
3.13
Example 3.4 q-polynomial exponential. Take fx eq x. From relation 3.4 and
Lemma 3.1, we obtain
m
∞ n
m
q n
x ,
xDq eq x eq xΦm,q x n
q!
n1
3.14
where
Φm,q x m m
k
k1
xk ,
3.15
q,1
which is called the q-polynomial exponential. So,
m m
k1
k
xk Eq −x
q,1
∞ nm
q
nq !
n1
xn ∞ n
−1k q
n1 k1
k
2
n − km
q
kq !n − kq !
xn .
3.16
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. Let fx and gx be two functions satisfying
fx ∞
an xnq ,
gx n0
∞
cn xn .
3.17
n0
Then
∞
∞
m m
n
cn fnx am
xk Dq k gx
k
q,1
n0
m1
k1
3.18
provided the series
∞
cn fnxn
n0
converges absolutely.
3.19
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International Journal of Mathematics and Mathematical Sciences
Proof. From Lemma 3.1 and the properties of the q-Stirling numbers of the second kind 2.21,
we obtain
∞
am
m1
m m
k1
k
xk Dqk gx q,1
∞
m
am xDq gx
m1
∞
∞
m n
am xDq
cn x .
3.20
n0
m1
The result follows, then, from relations 3.4 and 3.19.
Corollary 3.6. Let fx ∞
n0
an xnq . Then
m ∞
∞
kq !
m
n
fnx am
xk k q,1
x; q k1
n0
m1
k1
provided the series
∞
n0
3.21
fnxn converges absolutely.
Proof. By taking gx 1/1 − x application of relation 3.7, we obtain
∞
n0
xn , |x| < 1, in the previous theorem, and by
m ∞
∞
1
m
fnxn am
xk Dqk
k q,1
1−x
n0
m1
k1
∞
am
Γq k 1 qk1 x; q ∞
x
k q,1
Γq 1
x; q ∞
m m
m1
k1
∞
m m
am
m1
k1
k
3.22
kq !
xk .
k q,1
x; q k1
Example 3.7. Let fx xn,q . Then, the fact that
xn,q q− 2 n
n
sq n, k; 0xkq ,
n 0, 1, . . .
3.23
k0
and Corollary 3.6 give
n
∞
n
m lq !
q− 2 m
sq n, m; 0
xl kn,q xk .
l q,1
1 − x m1
qx; q l
k0
l1
Remark that when q tends to 1− , we obtain the formula given in 1, 6.4, page 3863.
Some others summation formulas are presented in the following statements.
3.24
International Journal of Mathematics and Mathematical Sciences
Corollary 3.8. For fx 1
2
3
4
∞
n0
∞
n0
n0 1
n0 fn/nq !x
n0
∞
n1
an
n
eq x
∞
n1
qnn−1 fn/nq !xn n
n1
an
∞
n1
α−k
k1
∞
− qα nq /1 − qnq fnxn ∞
∞
an xnq , the transformation formulas lead to the following:
qnn−1/2 αn q fnxn ∞
9
qkk−1/2 { nk }q,1 kq ! αk q xk 1 qk xq ;
an
n
an
k1
n
k1
{ nk }q,1
αk−1 k−1
{ nk }q,1 xk eq x
n
k1
q
xk /1 − xαk
q ;
∞
n1
an Φn,q x;
{ nk }q,1 qkk−1/2 xk /−1 − qqk x; q∞ provided the series converge absolutely.
Proof. The results are direct consequences of Theorem 3.5 by putting the following:
1 gx
q
kk−1/2
1 xαq
∞
n0
qnn−1/2 αn q xn and remark that Dqk 1 xαq
αq α − 1q · · · α − k 1q 1 q
2 gx 1/1 − xαq ∞
n0 1
k
α−k
xq ;
− qα nq /1 − qnq xn and remark that Dqk 1/1 − xαq αq α 1q · · · α k − 1q /1 − xαk
q ;
3 gx eq x;
4 gx Eq x and remark that Dqk Eq x qkk−1/2 Eq qk x.
Remark 3.9. The last formulas coincide with some of the ones given in 15 when q tends to 1− .
4. The Operator x; q1 Dq m and Related Transformation Theorem
Lemma 4.1. For a suitable function f, one has for m 1, 2, . . .
m
m
x; q1 Dq fx −1m−k Sq m − 1, k − 1; 1 x; q k Dqk fx.
4.1
k1
Proof. The formula can be obtained by induction with respect to m. Indeed, for m 1, we
have
x; q 1 Dq fx x; q 1 Dq fx Sq 0, 0; 1 x; q 1 Dq fx.
4.2
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International Journal of Mathematics and Mathematical Sciences
Assuming that formula 4.1 is true for m, then
m
k
m1
m−k
x; q 1 Dq
fx x; q 1 Dq
−1 Sq m − 1, k − 1; 1 x; q k Dq fx
k1
m
x; q 1 −1m−k Sq m − 1, k − 1; 1 qx; q k Dqk1 fx
k1
m
− x; q 1 −1m−k Sq m − 1, k − 1; 1kq qx; q k−1 Dqk fx
k1
m1
−1m−k1 Sq m − 1, k − 2; 1 x; q k Dqk fx
k2
−
m
−1m−k kq Sq m − 1, k − 1; 1 x; q k Dqk fx
k1
m
−1m−k1 Sq m − 1, k − 2; 1 − kq Sq m − 1, k − 1; 1 x; q k Dqk fx
k2
Sq m − 1, m − 1; 1 x; q m1 Dqm1 fx
− −1m−1 Sq m − 1, 0; 1 x; q 1 Dq fx.
4.3
The result is easily deduced by formulas 2.20, and 2.21.
Theorem 4.2. Let fx and gx be two functions defined by
fx ∞
−1n αn xnq ,
gx n0
∞ cn x; q n .
4.4
n0
If the series
∞
cn fn x; q n
4.5
∞
m
cn fn x; q n αm −1m−k Sq m − 1, k − 1; 1 x; q k Dqk gx.
4.6
n0
converges absolutely, then
∞
n0
m1
k1
Proof. From the previous lemma, we obtain for m 1, 2, . . .,
∞
m
∞
m
αm x; q1 Dq gx αm −1m−k Sq m − 1, k − 1; 1 x; q k Dqk gx.
m0
m1
k1
4.7
International Journal of Mathematics and Mathematical Sciences
11
So, the absolute convergence of the series 4.5 and the fact that
m x; q n −1m nm
x; q1 Dq
q x; q n ,
m∈
4.8
achieve the proof.
∞
m
m
m0 −1 αm xq .
Corollary 4.3. Let fx ∞
αm
m1
Then
n
m
k
−q sq k, 0, −nfk x; q k .
−1m−k Sq m − 1, k − 1; 1 x; q k nk,q xn−k k1
k0
4.9
Proof. Put gx xn .
Using the representation see 5
x n
n
k
−1
k0
n
k
q−n−1k q
q
k 2
x; q
k
n
k
−q sq k, 0, −n x; q k ,
4.10
k0
relation 4.6 and the fact that
Dqk xn nk,q xn−k
4.11
give the desired result.
Remark 4.4. Note that recently Liu in his paper see 16 has obtained some interesting qidentities in showing that the solutions of two difference equations involve some series of
q-operators Dqn 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
see 3, 16.
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