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International Journal of Mathematics and Mathematical Sciences
Volume 2010, Article ID 785949, 12 pages
http://dx.doi.org/10.1155/2010/785949
Research Article

On -Operators and Summation of Some -Series

Faculté des Sciences de Tunis, Tunis 1060, Tunisia

Received 14 October 2010; Accepted 11 December 2010

Academic Editor: Naseer Shahzad

Copyright © 2010 Sana Guesmi and Ahmed 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.

Abstract

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

1. Introduction

The operator 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 , we use the operators , which give These operators are intimately related to the Stirling numbers of second kind by the formula (see [1]) where is a suitable function. We note that the -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 in Quantum Calculus and some -transformation theorems that will be used to establish the sums of some -series.

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 -Stirling numbers, and presents some related applications. Section 4 attempts to give a new -analogue of formula (1.2) by studying the transformation theorem related to a -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 .

2.1. -Shifted Factorials

For , the -shifted factorials are defined by We also write We put For and , we adopt the following notation [5]: The -analogue of the Jordan factorial is given by and the -binomial coefficient is defined by

2.2. The Jackson's -Derivative

The -derivative of a function is defined by (see [4]) and provided exists. Note that when is differentiable, at , then tends to as tends to .

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

2.3. Elementary -Special Functions

Two -analogues of the exponential function are given by (see [4]) They satisfy the relations In 1910, F. H. Jackson defined a -analogue of the Gamma function by (see [4, 6]) It satisfies the following functional equations:

2.4. -Stirling Numbers of Noncentral Type

In [7], Charalambides introduced the so-called noncentral -Stirling numbers, which are -analogues of the Stirling numbers and classified into two kinds.

The noncentral -Stirling numbers of the first kind are defined by the following generating relation: and they are given by The noncentral -Stirling numbers of the second kind are defined by the following generating relation: and they are given by

Remark 2.1. Note that when , then and reduce to the -Stirling numbers, respectively, of the first and the second kind studied by Gould, Carlitz, and Kim (see [811]).

Properties
The noncentral -Stirling numbers satisfy the following properties. (i)For and , under the following conditions: (ii)For and , under the following conditions:

3. The Operator and Some Related Transformations Theorems

As in the classical case (see [1]), the iterate , , can be expanded in finite terms involving the -Stirling numbers. This is the purpose of the following result.

Lemma 3.1 (see [2, 3]). Letting be a differentiable function, then one has where

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

Example 3.2 (-binomial series). The -binomial theorem asserts that Using the fact that, for all , and the previous lemma, we deduce that On the other hand, the definition of -derivative (2.9) gives and by iteration we have Thus, So, taking , we obtain Remark that if tends to , we obtain the formula given in [1, page 366].

Example 3.3 (-Bessel function). We consider the function where is the first Jackson's -Bessel function of order (see [12, 13]).
By application of the operator to and the use of relation (3.4), we obtain Then, using Lemma 3.1 and the fact that we get

Example 3.4 (-polynomial exponential). Take . From relation (3.4) and Lemma 3.1, we obtain where which is called the -polynomial exponential. So,
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 -analogue context.

Theorem 3.5. Let and be two functions satisfying Then provided the series converges absolutely.

Proof. From Lemma 3.1 and the properties of the -Stirling numbers of the second kind (2.21), we obtain The result follows, then, from relations (3.4) and (3.19).

Corollary 3.6. Let . Then provided the series converges absolutely.

Proof. By taking , , in the previous theorem, and by application of relation (3.7), we obtain

Example 3.7. Let . Then, the fact that and Corollary 3.6 give Remark that when tends to , we obtain the formula given in [14, (6.4), page 3863].

Some others summation formulas are presented in the following statements.

Corollary 3.8. For , the transformation formulas lead to the following: (1); (2); (3); (4) provided the series converge absolutely.

Proof. The results are direct consequences of Theorem 3.5 by putting the following: (1) and remark that ;(2) and remark that ;(3); (4) and remark that .

Remark 3.9. The last formulas coincide with some of the ones given in [15] when tends to .

4. The Operator and Related Transformation Theorem

Lemma 4.1. For a suitable function , one has for

Proof. The formula can be obtained by induction with respect to . Indeed, for , we have Assuming that formula (4.1) is true for , then The result is easily deduced by formulas (2.20), and (2.21).

Theorem 4.2. Let and be two functions defined by If the series converges absolutely, then

Proof. From the previous lemma, we obtain for , So, the absolute convergence of the series (4.5) and the fact that achieve the proof.

Corollary 4.3. Let . Then

Proof. Put .
Using the representation (see [5]) relation (4.6) and the fact that give the desired result.

Remark 4.4. Note that recently Liu in his paper (see [16]) has obtained some interesting -identities in showing that the solutions of two difference equations involve some series of -operators of -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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