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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.

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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:

xanq ⎧ ⎨ ⎩

1 if n0,

xaxaq · · ·xaqn−1 if n 1. 2.4

Theq-analogue of the Jordan factorial is given by

xk,q xqx−1q· · ·xk1q

1−qx 1qx−1 · · ·1qxk1

1−q k ,

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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

fqxfx

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 Dqfxfqx 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 ,

eqzEqz Eqzeqz 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

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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:

trn,qqn2−rn n

k0

sqn, k;rtqk, n0,1, . . . , 2.14

and they are given by

sqn, k;r 1 1−q nk

n

jk

−1jkq nj

2

rnj

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;rtrk,q, n0,1, . . . , 2.16

and they are given by

Sqn, k;r 1 kq!

k

j0

−1kjq j1

2

rjk

k j

q

rjnq

1

1−q nk n

jk

−1jkqrjk

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,

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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

m

and 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 k1

j

q

kjmq−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∈,

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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

whereJp, 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

(7)

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

xkEqx

n1

nmq nq!x

nn1

n

k1

−1kq

k 2

nkm q kq!nkq!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

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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 n0anxnq. Then

n0

fnxn

m1 am

m

k1

m k

q,1

xkkq!

x;q k1 3.21

provided the seriesn0fnxnconverges absolutely.

Proof. By taking gx 1/1 −x n0xn, |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,qqn2 n

k0

sqn, k; 0xqk, n0,1, . . . 3.23

and Corollary3.6give

k0

kn,qxk qn2

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.

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Corollary 3.8. Forfx n0anxnq, the transformation formulas lead to the following:

1n0qnn−1/2α

nqfnxn

n1an n

k1qkk−1/2{nk}q,1kq!αkqxk1qkx αk q ;

2n01−n

q/1−q n

qfnxn

n1an n

k1{nk}q,1 αk−1

k−1

qxk/1−x αk q ;

3n0fn/nq!xneqxn1annk1{nk}q,1xkeqx

n1anΦn,qx;

4n0qnn−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 1q n0qnn−1/2α

nqxn and remark that Dqk1xαq qkk−1/2α

−1q· · ·αk1q1qkx αk q ;

2gx 1/1−q n01−n

q/1−q n

qxnand remark thatDqk1/1−xαq αqα1q· · ·αk−1q/1−qk;

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

1

Dq

m

and Related Transformation Theorem

Lemma 4.1. For a suitable functionf, one has form1,2, . . .

x;q1Dqmfx m

k1

−1mkSqm−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

(10)

Assuming that formula4.1is true form, then

x;q 1Dqm1fx x;q 1Dq

m

k1

−1mkSqm−1, k−1; 1x;q kDkqfx

x;q 1 m

k1

−1mkSqm−1, k−1; 1

qx;q kDk1 q fx

x;q 1 m

k1

−1mkSqm−1, k−1; 1kqqx;q k−1Dqkfx

m1 k2

−1mk1Sqm−1, k−2; 1x;q kDkqfx

m

k1

−1mkkqSqm−1, k−1; 1

x;q kDk qfx

m k2

−1mk1Sqm−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

−1mkSqm−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

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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 m0−1mαmxmq. Then

m1 αm

m

k1

−1mkSqm−1, k−1; 1x;q knk,qxnk n

k0

q ksqk,0,−nfkx;q k.

4.9

Proof. Putgx xn.

Using the representationsee5

xn n

k0

−1k

n k

q

qn−1kq k

2

x;q k n

k0

q ksqk,0,−nx;q k, 4.10

relation4.6and the fact that

Dkqxn nk,qxnk 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.

References

1 P. M. Knopf, “The operatorxd/dxnand its applications to series,” Mathematics Magazine, vol. 76, no. 5, pp. 364–371, 2003.

2 B. A. Kupershmidt, “q-probability. I. Basic discrete distributions,” Journal of Nonlinear Mathematical

Physics, vol. 7, no. 1, pp. 73–93, 2000.

3 L. J. Rogers, “On a three-fold symmety in the elements of Heine’s series,” Proceedings of the London

Mathematical Society, vol. 24, pp. 171–179, 1893.

4 G. Gasper and M. Rahman, Basic Hypergeometric Series, vol. 35 of Encyclopedia of Mathematics and Its

Applications, Cambridge University Press, Cambridge, UK, 1990.

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6 G. E. Andrews, R. Askey, and R. Roy, Special Functions, vol. 71 of Encyclopedia of Mathematics and Its

Applications, Cambridge University Press, Cambridge, UK, 1999.

7 Ch. A. Charalambides, “Non-central generalized q-factorial coefficients and q-Stirling numbers,”

Discrete Mathematics, vol. 275, no. 1–3, pp. 67–85, 2004.

8 L. Carlitz, “On abelian fields,” Transactions of the American Mathematical Society, vol. 35, no. 1, pp. 122–136, 1933.

9 H. W. Gould, “The operatoraxΔnand Stirling numbers of the first kind,” The American Mathematical Monthly, vol. 71, no. 8, pp. 850–858, 1964.

10 H. W. Gould, “Theq-Stirling numbers of first and second kinds,” Duke Mathematical Journal, vol. 28, pp. 281–289, 1961.

11 T. Kim, “q-Bernoulli numbers associated withq-Stirling numbers,” Advances in Difference Equations,

vol. 2008, Article ID 743295, 10 pages, 2008.

12 M. E. H. Ismail, “The zeros of basic Bessel functions, the functions axx, and associated

orthogonal polynomials,” Journal of Mathematical Analysis and Applications, vol. 86, no. 1, pp. 1–19, 1982.

13 M. E. H. Ismail, “On Jacksons third—Bessel function andq-exponentials,” preprint, 2001.

14 K. N. Boyadzhiev, “A series transformation formula and related polynomials,” International Journal of

Mathematics and Mathematical Sciences, no. 23, pp. 3849–3866, 2005.

15 T.-X. He, L. C. Hsu, and P. J.-S. Shiue, “A symbolic operator approach to several summation formulas for power series. II,” Discrete Mathematics, vol. 308, no. 16, pp. 3427–3440, 2008.

References

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