Journal of Function Spaces and Applications

VolumeΒ 2012, Article IDΒ 214961, 13 pages

http://dx.doi.org/10.1155/2012/214961

## Novel Identities for -Genocchi Numbers and Polynomials

Department of Mathematics, Faculty of Science and Arts, University of Gaziantep, 27310 Gaziantep, Turkey

Received 26 February 2012; Revised 25 April 2012; Accepted 9 May 2012

Academic Editor: GesturΒ Γlafsson

Copyright Β© 2012 Serkan Araci.

#### Abstract

The essential aim of this paper is to introduce novel identities for *q*-Genocchi numbers and polynomials by using the method by T. Kim et al. (article in press). We show that these polynomials are related to *p*-adic analogue of Bernstein polynomials. Also, we derive relations between *q*-Genocchi and *q*-Bernoulli numbers.

#### 1. Preliminaries

Imagine that be a fixed odd prime number. We now start with definition of the following notations. Let be the field -adic rational numbers and let be the completion of algebraic closure of .

Thus,

Then is integral domain, which is defined by or

We assume that with as an indeterminate. The -adic absolute value , is normally defined by where with , and .

is a -extension of , which is defined by we note that (see [1β17]).

Throughout this paper, we use notation of , where denotes set of Natural numbers.

We say that is a uniformly differentiable function at a point , if the difference quotient, has a limit as , and we denote this by . Then, for , we can start with the following expression: which represents a -adic -analogue of Riemann sums for . The integral of on will be defined as the limit of these sums, when it exists. The -adic -integral of function is defined by Kim in [7, 12] as

The bosonic integral is considered as a bosonic limit , . Similarly, the fermionic -adic integral on is introduced by Kim as follows: (for more details, see [13β16]).

From (1.9), it is well-known equality that where (for details, see [2, 3, 8, 9, 12, 13, 15β17]).

The -Genocchi polynomials with wegiht are introduced as

From (1.11), we have where are called -Genocchi numbers with weight . Then, -Genocchi numbers are defined as with the usual convention about replacing by is used (for details, see [3]).

Let be the space of continuous functions on . For , -adic analogue of Bernstein operator for is defined by where . Here, is called -adic Bernstein polynomials, which are defined by (for details, see [1, 4, 5, 7]).

The -Bernoulli polynomials and numbers with weight are defined by Kim et al., respectively, (for more information, see [10]).

The author, by using derivative operator, will investigate some interesting identities on the -Genocchi numbers and polynomials arising from their generating function. Also, the author derives some relations between -Genocchi numbers and -Bernoulli numbers by using Kimβs -Volkenborn integral and fermionic -adic -integral on .

#### 2. Novel Properties of -Genocchi Numbers and Polynomials with Weight 0

Let . Then, by using (1.10), we easily procure the following:

From the last equality, by (1.11), we get Araci, Acikgoz, and Qiβs -Genocchi polynomials with weight in [3] as follows:

Here, we assume that is a fixed parameter. Let Thus, by expression of (2.3), we can readily see the following:

Last from equality, taking derivative operator as on the both sides of (2.4), then, we easily see that where and is identity operator. By multiplying on both sides of (2.5), we get

Let us take derivative operator on the both sides of (2.6). Then, we get

Let (not ) be the constant term in a Laurent series of in (2.3). Then, we get

By (2.3), we easily see

We see that the members of (2.11) are proportional to the Bernstein polynomials with the following theorem.

Theorem 2.1. *For , one has
*

* Proof. *By expressions of (2.8) and (2.9), we see that

By applying basic operations to above equality, we can easily reach to the desired result.

As a special case, we derive the following.

Corollary 2.2. *For , one has
*

*Proof. *When into (2.10), we derive the following identity:
Here, is greatest integer . Then, we complete the proof of Theorem.

From (2.2), we note that

By (2.14) and (1.11), we easily see that

Now, let us consider definition of integral from to in (2.11), then we have where is beta function which is defined by

As a result, we obtain the following theorem.

Theorem 2.3. *For , one has
*

* Proof. *By taking integral from to in (2.11), we easily reach to desired result.

Substituting into Theorem 2.1, we readily get

By differentiating both sides of (2.11) with respect to , we have the following:

We now give interesting theorem for -Genocchi numbers with weight as follows.

Theorem 2.4. *For , one has
*

* Proof. *It is proved by using definition of integral on the both sides in the following equality, that is,
Last from equality, we discover the following:
Then, taking integral from to both sides of last equality, we get

Thus, we complete the proof of the theorem.

Theorem 2.5. *For , one has
*

* Proof. *In view of (2.2) and (2.23), we discover the following applications:

Thus, we give evidence of the theorem.

As into Theorem 2.5, it leads to the following interesting property.

Corollary 2.6. *For , one has
**
where is ordinary Genocchi polynomials, which is defined by the means of the following generating function [9]:
*

#### 3. Some Identities -Genocchi Numbers and -Bernoulli Numbers by Using Kimβs -Adic -Integrals on

In this section, we consider -Genocchi numbers and -Bernoulli numbers by means of -adic -integral on . Now, we start with the following theorem.

Theorem 3.1. *For ,, one has
*

*Proof. *For , then by (2.11),
On the other hand, the right hand side of (2.11),
Combining and , we arrive to the proof of the theorem.

Theorem 3.2. *For , one has
**
here .*

* Proof. *Let us take fermionic -adic -inetgral on left-hand side of Theorem 2.5, we get
In other word, we consider the right-hand side of Theorem 2.5 as follows:

Equating and , we complete the proof of the theorem.

As in the above theorem, we reach interesting property in Analytic Numbers Theory concerning ordinary Genocchi polynomials.

Corollary 3.3. *For , one has
*

Theorem 3.4. *For , one has
*

*Proof. *We consider (2.11) and (2.2) by means of -Volkenborn integral. Then, by (2.11), we see
On the other hand,
Therefore, we get the proof of theorem.

Corollary 3.5. *For , one gets
**
where *

*Proof. *By using -adic -integral on left-hand side of Theorem 2.5, we get
Also, we compute the right-hand side of Theorem 2.5 as follows:

Equating and , we get the proof of Corollary.

As in the above theorem, we easily derive the following corollary.

Corollary 3.6. *For , one has
*

#### Acknowledgment

The author would like to express his sincere gratitude to the referee for his/her valuable comments and suggestions which have improved the presentation of the paper.

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