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 convergentq-series.
1. Introduction
The operatorxd/dxnhas 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 ∞
k0knxk, we use the operatorsxd/dx
n, which give
∞
k0 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 {nk} by the formulasee1
x d
dx n
fx n
k1
n k
xkd kf
dxk, 1.2
wherefis a suitable function. We note that theq-analogue of formula1.2has 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 theq-Stirling numbers, and presents some related applications. Section4attempts to give a new q-analogue of formula1.2by studying the transformation theorem related to aq-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 book4, for the notations and for a deep study in this way. Throughout this paper, we fixq∈0,1.
2.1.
q
-Shifted Factorials
Fora∈ , theq-shifted factorials are defined by
a;q 01, a;q n n−1
k0
1−aqk, a;q ∞
∞
k0
1−aqk. 2.1
We also write
a1, . . . , ak;q n k
j1
aj;q n, n0,1, . . . ,∞. 2.2
We put
xq 1−q x
1−q , x∈ ,
nq!
q;q n
1−q n, n∈ .
2.3
Fora, x∈ andn∈, we adopt the following notation5:
x−anq ⎧ ⎨ ⎩
1 if n0,
x−ax−aq · · ·x−aqn−1 if n 1. 2.4
Theq-analogue of the Jordan factorial is given by
xk,q xqx−1q· · ·x−k1q
1−qx 1−qx−1 · · ·1−qx−k1
1−q k ,
and theq-binomial coefficient is defined by
x k
q
xk,q
kq!. 2.6
2.2. The Jackson’s
q
-Derivative
Theq-derivativeDqfof a functionfis defined bysee4
Dqfx
fqx −fx
q−1 x ifx /0, 2.7
and Dqf0 f0 provided f0exists. Note that when f is differentiable, atx, then Dqfxtends tofxasqtends to 1−.
It is easy to see that for suitable functionsfandg, we have
Dqfg x fqx Dqgx gxDqfx, 2.8
Dq f
g
x g
qx Dqfx−fqx Dqgx
gxgqx . 2.9
2.3. Elementary
q
-Special Functions
Twoq-analogues of the exponential function are given bysee4
eqz
∞
n0 zn nq!
1
1−q z;q ∞, |z|<
1−q −1,
Eqz
∞
n0
qnn−1/2 z n
nq!
−1−q z;q ∞, z∈ .
2.10
They satisfy the relations
Dqeqz eqz, DqEqz Eqqz ,
eqzEq−z Eqzeq−z 1, Eqz e1/qz.
2.11
In 1910, F. H. Jackson defined aq-analogue of the Gamma function bysee4,6
Γqx
q;q ∞
qx;q ∞
It satisfies the following functional equations:
Γqx1 xqΓqx, Γq1 1, Γqn1 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 sqn, k;r are defined by the following generating relation:
t−rn,qq−n2−rn n
k0
sqn, k;rtqk, n0,1, . . . , 2.14
and they are given by
sqn, k;r 1 1−q n−k
n
jk
−1j−kq n−j
2
rn−j
n j
q
j k
. 2.15
The noncentralq-Stirling numbers of the second kindSqn, k;rare defined by the following generating relation:
tnq n
k0 q
k 2
−rk
Sqn, k;rt−rk,q, n0,1, . . . , 2.16
and they are given by
Sqn, k;r 1 kq!
k
j0
−1k−jq j1
2
−rjk
k j
q
rjnq
1
1−q n−k n
jk
−1j−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 theq-Stirling numbers, respectively, of the first and the second kind studied by Gould, Carlitz, and Kim see8–11.
Properties
The noncentralq-Stirling numbers satisfy the following properties. iForn1,2, . . .andk1,2, . . . , n,
under the following conditions:
sq0,0;r 1, sqn,0;r q n
2
rn
−rn,q, n >0,
sq0, k;r 0, k >0, sqn, k;r 0, k > n.
2.19
iiForn1,2, . . .andk1,2, . . . , n,
Sqn, k;r Sqn−1, k−1;r rkqSqn−1, k;r, 2.20
under the following conditions:
Sq0,0;r 1, Sqn,0;r rnq, n >0,
Sq0, k;r 0, k >0, Sqn, k;r 0, k > n.
2.21
3. The Operator
xDq
mand Some Related Transformations Theorems
As in the classical casesee1, the iteratexDqm,m∈ , can be expanded in finite terms involving theq-Stirling numbers. This is the purpose of the following result.
Lemma 3.1see2,3. Lettingfbe a differentiable function, then one has
xDq mfx m
k1
m k
q,1
xkDkqfx, m1,2, . . . , 3.1
where
m
k
q,1
qkk−1/2Sqm−1, k−1; 1
1 k−1q!
k−1
j0
−1jq j
2 k−1
j
q
k−jmq−1. 3.2
Now, let us give three applications of the previous lemma.
Example 3.2q-binomial series. Theq-binomial theorem asserts that
1Φ0
qa;−;q, x
∞
n0
qa;q n
q;q n x n
qax;q
∞
x;q ∞ , |x|<1. 3.3
Using the fact that, for allm∈,
and the previous lemma, we deduce that
∞
n0
qa;q n
q;q n n m qxn
m
k1
m k
q,1 xkDqk
qax;q
∞
x;q ∞
, |x|<1. 3.5
On the other hand, the definition ofq-derivative2.9gives
Dq
qax;q ∞
x;q ∞
aq
qa1x;q ∞
x;q ∞ , 3.6
and by iteration we have
Dk q
qax;q
∞
x;q ∞
ΓqΓak qa
qakx;q
∞
x;q ∞ . 3.7
Thus,
∞
n0
qa;q n
q;q n n m qxn
1 Γqa
x;q ∞
m
k1
m k
q,1
xkΓqak
qakx;q
∞. 3.8
So, takinga1, we obtain
∞
n0
nmqxn 1 1−x
m
k1
m k
q,1
xkkq!
xq;q k. 3.9
Remark that ifqtends to 1−, we obtain the formula given in13, page 366.
Example 3.3q-Bessel function. We consider the function
Cpx
∞
k0
−1kxk kq!kpq!
x 2
−p Jp1
1−q √2x, q, 3.10
whereJp1·, qis the first Jackson’sq-Bessel function of orderpsee12,13.
By application of the operatorxDqmtoC0xand the use of relation3.4, we obtain
xDq
m C0x
∞
k0
−1k k m q
kq!2
xk. 3.11
Then, using Lemma3.1and the fact that
we get
∞
k0
−1k k m q
kq!2 xk
m
k1
m k
q,1
−1kxkCkx. 3.13
Example 3.4 q-polynomial exponential. Take fx eqx. From relation 3.4 and Lemma3.1, we obtain
xDq meqx eqxΦm,qx
∞
n1
nmq nq!x
n, 3.14
where
Φm,qx m
k1
m k
q,1
xk, 3.15
which is called theq-polynomial exponential. So,
m
k1
m k
q,1
xkEq−x
∞
n1
nmq nq!x
n∞ n1
n
k1
−1kq
k 2
n−km q kq!n−kq!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 aq-analogue context.
Theorem 3.5. Letfxandgxbe two functions satisfying
fx
∞
n0
anxnq, gx
∞
n0
cnxn. 3.17
Then
∞
n0
cnfnxn
∞
m1 am
m
k1
m k
q,1
xkDqkgx 3.18
provided the series
∞
n0
cnfnxn 3.19
Proof. From Lemma3.1and the properties of theq-Stirling numbers of the second kind2.21, we obtain
∞
m1 am
m
k1
m k
q,1
xkDkqgx
∞
m1 am
xDq mgx
∞ m1
amxDq m ∞
n0 cnxn
.
3.20
The result follows, then, from relations3.4and3.19.
Corollary 3.6. Letfx n∞0anxnq. Then
∞
n0
fnxn
∞
m1 am
m
k1
m k
q,1
xkkq!
x;q k1 3.21
provided the seriesn∞0fnxnconverges absolutely.
Proof. By taking gx 1/1 −x n∞0xn, |x| < 1, in the previous theorem, and by application of relation3.7, we obtain
∞
n0
fnxn ∞ m1
am m
k1
m k
q,1 xkDk
q
1 1−x
∞ m1
am m
k1
m k
q,1
xkΓqk1 Γq1
qk1x;q
∞
x;q ∞
∞ m1
am m
k1
m k
q,1
xkkq! x;q k1.
3.22
Example 3.7. Letfx xn,q. Then, the fact that
xn,qq−n2 n
k0
sqn, k; 0xqk, n0,1, . . . 3.23
and Corollary3.6give
∞
k0
kn,qxk q −n2
1−x n
m1
sqn, m; 0 m
l1
m l
q,1
xllq!
qx;q l. 3.24
Remark that whenqtends to 1−, we obtain the formula given in1,6.4, page 3863.
Corollary 3.8. Forfx n∞0anxnq, the transformation formulas lead to the following:
1n∞0qnn−1/2α
nqfnxn ∞
n1an n
k1qkk−1/2{nk}q,1kq!αkqxk1qkx α−k q ;
2n∞01−qαn
q/1−q n
qfnxn ∞
n1an n
k1{nk}q,1 αk−1
k−1
qxk/1−x αk q ;
3n∞0fn/nq!xneqxn∞1annk1{nk}q,1xkeqx ∞
n1anΦn,qx;
4n∞0qnn−1fn/n
q!xn ∞
n1an n
k1{nk}q,1qkk−1/2xk/−1−qqkx;q∞
provided the series converge absolutely.
Proof. The results are direct consequences of Theorem3.5by putting the following:
1gx 1xαq n∞0qnn−1/2α
nqxn and remark that Dqk1xαq qkk−1/2α
qα−1q· · ·α−k1q1qkx α−k q ;
2gx 1/1−xαq n∞01−qαn
q/1−q n
qxnand remark thatDqk1/1−xαq αqα1q· · ·αk−1q/1−xαqk;
3gx eqx;
4gx Eqxand remark thatDkqEqx qkk−1/2Eqqkx.
Remark 3.9. The last formulas coincide with some of the ones given in15whenqtends to 1−.
4. The Operator
x
;
q
1Dq
mand Related Transformation Theorem
Lemma 4.1. For a suitable functionf, one has form1,2, . . .
x;q1Dqmfx m
k1
−1m−kSqm−1, k−1; 1x;q kDkqfx. 4.1
Proof. The formula can be obtained by induction with respect tom. Indeed, for m 1, we have
Assuming that formula4.1is true form, then
x;q 1Dqm1fx x;q 1Dq
m
k1
−1m−kSqm−1, k−1; 1x;q kDkqfx
x;q 1 m
k1
−1m−kSqm−1, k−1; 1
qx;q kDk1 q fx
−x;q 1 m
k1
−1m−kSqm−1, k−1; 1kqqx;q k−1Dqkfx
m1 k2
−1m−k1Sqm−1, k−2; 1x;q kDkqfx
−m
k1
−1m−kkqSqm−1, k−1; 1
x;q kDk qfx
m k2
−1m−k1Sqm−1, k−2; 1−kqSqm−1, k−1; 1x;q kDqkfx
Sqm−1, m−1; 1x;q m1Dmq1fx
−−1m−1Sqm−1,0; 1
x;q 1Dqfx.
4.3
The result is easily deduced by formulas2.20, and2.21.
Theorem 4.2. Letfxandgxbe two functions defined by
fx
∞
n0
−1nαnxnq, gx
∞
n0 cn
x;q n. 4.4
If the series
∞
n0 cnfn
x;q n 4.5
converges absolutely, then
∞
n0 cnfn
x;q n
∞
m1 αm
m
k1
−1m−kSqm−1, k−1; 1
x;q kDqkgx. 4.6
Proof. From the previous lemma, we obtain form1,2, . . .,
∞
m0
αmx;q1Dqmgx
∞
m1 αm
m
k1
So, the absolute convergence of the series4.5and the fact that
x;q1Dqmx;q n −1mnmqx;q n, m∈ 4.8
achieve the proof.
Corollary 4.3. Letfx m∞0−1mαmxmq. Then
∞
m1 αm
m
k1
−1m−kSqm−1, k−1; 1x;q knk,qxn−k n
k0
−q ksqk,0,−nfkx;q k.
4.9
Proof. Putgx xn.
Using the representationsee5
xn n
k0
−1k
n k
q
q−n−1kq k
2
x;q k n
k0
−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 papersee16has obtained some interesting q-identities in showing that the solutions of two difference equations involve some series of q-operatorsDn
q ofq-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.
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