Abstract and Applied Analysis

Volume 2012 (2012), Article ID 298531, 11 pages

http://dx.doi.org/10.1155/2012/298531

## Explicit Formulas Involving -Euler Numbers and Polynomials

^{1}Department of Mathematics, Faculty of Science and Arts, University of Gaziantep, 27310 Gaziantep, Turkey^{2}Department of Applied Mathematics, Pukyong National University, Busan 608-737, Republic of Korea

Received 4 April 2012; Accepted 4 October 2012

Academic Editor: Gerd Teschke

Copyright © 2012 Serkan Araci et al. 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

We deal with -Euler numbers and -Bernoulli numbers. We derive some interesting relations for -Euler numbers and polynomials by using their generating function and derivative operator. Also, we derive relations between the -Euler numbers and -Bernoulli numbers via the -adic -integral in the -adic integer ring.

#### 1. Preliminaries

Imagine that is a fixed odd prime number. Throughout this paper we use the following notations, where denotes the ring of -adic rational integers, denotes the field of rational numbers, denotes the field of -adic rational numbers, and denotes the completion of algebraic closure of . Let be the set of natural numbers and .

The -adic absolute value is defined by

In this paper, we will assume that as an indeterminate.

is a -extension of , which is defined by We note that (see [1–12]).

We say that is a uniformly differentiable function at a point , if the difference quotient has a limit as and denote this by .

Let be the set of uniformly differentiable function on . For , let us start with the expression which represents -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

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 [9–12]).

In [6], the -Euler polynomials with weight are introduced as

From (1.7), we have where are called -Euler numbers with weight . Then, -Euler numbers are defined as where the usual convention about replacing by is used.

Similarly, the -Bernoulli polynomials and numbers with weight are defined, respectively, as (for more information, see [4]).

We, by using the Kim et al. method in [2], will investigate some interesting identities on the -Euler numbers and polynomials arising from their generating function and derivative operator. Consequently, we derive some properties on the -Euler numbers and polynomials and -Bernoulli numbers and polynomials by using -Volkenborn integral and fermionic -adic -integral on .

#### 2. On the -Euler Numbers and Polynomials

Let us consider Kim’s -Euler polynomials as follows:

Here is a fixed parameter. Thus, by expression of (2.1), we can readily see the following:

Last from equality, taking derivative operator as on the both sides of (2.2). Then, we easily see that where and is identity operator. By multiplying on both sides of (2.3), we get

Let us take derivative operator on both sides of (2.4). Then we get

Let (not ) be the constant term in a Laurent series of . Then, from (2.5), we get

By (2.1), we see

By expressions of (2.6) and (2.7), we see that

From (2.1), we note that

By (2.9), we easily see

Now, let us consider definition of integral from to in (2.8), then we have where is beta function which is defined by

As a result, we obtain the following theorem.

Theorem 2.1. *For , one has
*

Substituting into Theorem 2.1, we readily get

By (2.1), it follows that

Let in (2.1), we see that

Last from equality, we discover the following:

Here is Gauss’ symbol. Then, taking integral from to in both sides of last equality, we get

Consequently, we derive the following theorem.

Theorem 2.2. *The following identity
**
is true. *

In view of (2.1) and (2.17), we discover the following applications:

By expressions (2.17) and (2.20), we have the following theorem.

Theorem 2.3. *For , one has
*

#### 3. -adic Integral on Associated with Kim’s -Euler Polynomials

In this section, we consider Kim’s -Euler polynomials by means of -adic -integral on . Now we start with the following assertion.

Let . Then by (2.8),

On the other hand, in right hand side of (2.8),

Equating and , we get the following theorem.

Theorem 3.1. *For , one has
*

Let us take fermionic -adic -integral on in left hand side of (2.21), we get

In other words, we consider right hand side of (2.21) as follows:

Equating and , we get the following theorem.

Theorem 3.2. *For , one has
*

Now, we consider (2.8) and (2.1) by means of -Volkenborn integral. Then, by (2.8), we see

On the other hand,

Therefore, we get the following theorem.

Theorem 3.3. *For , one has
*

By using fermionic -adic -integral on in left hand side of (2.21), we get

Also, we consider right hand side of (2.21) as follows:

Equating and , we get the following corollary.

Corollary 3.4. *For , one gets
*

#### Acknowledgment

The authors would like to thank the referee for his/her valuable comments on this work.

#### References

- S. Araci, D. Erdal, and J.-J. Seo, “A study on the fermionic $p$-adic $q$-integral representation on ${\mathbb{Z}}_{p}$ associated with weighted $q$-Bernstein and $q$-Genocchi polynomials,”
*Abstract and Applied Analysis*, vol. 2011, Article ID 649248, 10 pages, 2011. View at Publisher · View at Google Scholar - T. Kim, B. Lee, S. H. Lee, and S. H. Rim, “Identities for the Bernoulli and Euler numbers and polynomials,”
*Ars Combinatoria*. In press. - D. Kim, T. Kim, S.-H. Lee, D.-V. Dolgy, and S.-H. Rim, “Some new identities on the Bernoulli numbers and polynomials,”
*Discrete Dynamics in Nature and Society*, vol. 2011, Article ID 856132, 11 pages, 2011. View at Publisher · View at Google Scholar - T. Kim, J. Choi, and Y. H. Kim, “Some identities on the $q$-Bernoulli numbers and polynomials with weight 0,”
*Abstract and Applied Analysis*, vol. 2011, Article ID 361484, 8 pages, 2011. View at Publisher · View at Google Scholar - T. Kim, “On a $q$-analogue of the $p$-adic log gamma functions and related integrals,”
*Journal of Number Theory*, vol. 76, no. 2, pp. 320–329, 1999. View at Publisher · View at Google Scholar - T. Kim and J. Choi, “On the $q$-Euler numbers and polynomials with weight 0,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 795304, 7 pages, 2012. View at Publisher · View at Google Scholar - T. Kim, “On the $q$-extension of Euler and Genocchi numbers,”
*Journal of Mathematical Analysis and Applications*, vol. 326, no. 2, pp. 1458–1465, 2007. View at Publisher · View at Google Scholar - T. Kim, “On the weighted $q$-Bernoulli numbers and polynomials,”
*Advanced Studies in Contemporary Mathematics*, vol. 21, no. 2, pp. 207–215, 2011. View at Google Scholar - T. Kim, “$q$-Volkenborn integration,”
*Russian Journal of Mathematical Physics*, vol. 9, no. 3, pp. 288–299, 2002. View at Google Scholar · View at Zentralblatt MATH - T. Kim, “$q$-Euler numbers and polynomials associated with $p$-adic $q$-integrals,”
*Journal of Nonlinear Mathematical Physics*, vol. 14, no. 1, pp. 15–27, 2007. View at Publisher · View at Google Scholar - T. Kim, “New approach to $q$-Euler polynomials of higher order,”
*Russian Journal of Mathematical Physics*, vol. 17, no. 2, pp. 218–225, 2010. View at Publisher · View at Google Scholar - T. Kim, “Some identities on the $q$-Euler polynomials of higher order and $q$-Stirling numbers by the fermionic $p$-adic integral on ${\mathbb{Z}}_{p}$,”
*Russian Journal of Mathematical Physics*, vol. 16, no. 4, pp. 484––491, 2009. View at Google Scholar