Abstract and Applied Analysis

Volume 2012 (2012), Article ID 795304, 7 pages

http://dx.doi.org/10.1155/2012/795304

## On the -Euler Numbers and Polynomials with Weight

Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea

Received 13 October 2011; Accepted 29 November 2011

Academic Editor: Ibrahim Sadek

Copyright © 2012 T. Kim and J. Choi. 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

The purpose of this paper is to investigate some properties of -Euler numbers and polynomials with weight 0. From those -Euler numbers with weight 0, we derive some identities on the -Euler numbers and polynomials with weight 0.

#### 1. Introduction

Let be a fixed odd prime number. Throughout this paper , , and will denote the ring of -adic rational integers, the field of -adic rational numbers, and the completion of algebraic closure of . The -adic absolute value is defined by where for with and . In this paper, we assume that and with . As well-known definition, the Euler polynomials are defined by with the usual convention about replacing by (see [1–15]).

In this special case, , are called the th Euler numbers (see [1]). Recently, the -Euler numbers with weight are defined by with the usual convention about replacing by (see [3, 12]). The -number of is defined by (see [1–15]). Note that . Let us define the notation of -Euler numbers with weight 0 as . The purpose of this paper is to investigate some interesting identities on the -Euler numbers with weight 0.

#### 2. On the Extended -Euler Numbers of Higher-Order with Weight 0

Let be the space of continuous functions on . For , the fermionic -adic -integral on is defined by Kim as follows: (see [1–12]). By (2.1), we get where and (see [4, 5]).

By (1.2), (2.1), and (2.2), we see that

In the special case, , we get where are the th Frobenius-Euler numbers. From (2.4), we note that the -Euler numbers with weight 0 are given by

Therefore, by (2.5), we obtain the following theorem.

Theorem 2.1. *For , one has
**
where are called the th Frobenius-Euler numbers.*

Let us define the generating function of the -Euler numbers with weight 0 as follows:

Then, by (2.3) and (2.7), we get

Now we define the -Euler polynomials with weight 0 as follows:

From (2.10), we have where are called the th Frobenius-Euler polynomials (see [9]).

Therefore, by (2.11), we obtain the following theorem.

Theorem 2.2. *For , one has
**
where are called the th Frobenius-Euler polynomials.*

From (2.2) and Theorem 2.2, we note that where with (mod 2).

Therefore, by (2.13), we obtain the following corollary.

Corollary 2.3. *For , with (mod 2) and , one has
*

In particular, , we get , where and are called the th Euler numbers and polynomials which are defined by

By (2.2), we easily see that

Thus, by (2.16), we get

Therefore, by (2.16), we obtain the following theorem.

Theorem 2.4. *For , one has
*

where are called the th Frobenius-Euler polynomials and are called the th Frobenius-Euler numbers. In particular, , we have where are called the th Euler numbers.

From (2.5) and Theorem 2.2, we note that where the usual convention about replacing by . By Theorems 2.2 and 2.4, we get

From (2.20) and (2.21), we have

For , by (2.20) and (2.22), we have

Therefore, by (2.23), we obtain the following theorem.

Theorem 2.5. *For , one has
*

For , we have

Therefore, by (2.25), we obtain the following theorem.

Theorem 2.6. *For , one has
*

From (2.20), we have

By Theorem 2.6 and (2.27), we get

Therefore, by (2.28), we obtain the following theorem.

Theorem 2.7. *For , one has
*

Let be the space of continuous functions on . For , -adic analogue of Bernstein operator of order for is given by where (see [1, 6, 7]).

For , -adic Bernstein polynomial of degree is defined by (see [1, 6, 7]).

Let us take the fermionic -adic -integral on for one Bernstein polynomials in (2.31) as follows:

By simple calculation, we easily get

Therefore, by (2.32) and (2.33), we obtain the following theorem.

Theorem 2.8. *For with , one has
*

In particular, , we get

#### References

- S. Araci, D. Erdal, and J. J. Seo, “A Study on the fermionic
*p*-adic*q*integral 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 - D. Erdal, J. J. Seo, and S. Araci, “New construction weighted (
*h, g*)-Genocchi numbers and polynomials related to Zeta type function,”*Discrete Dynamics in Nature and Society*, vol. 2011, Article ID 487490, 7 pages, 2011. View at Publisher · View at Google Scholar - T. Kim, B. Lee, J. Choi, Y. H. Kim, and S. H. Rim, “On the q-Euler numbers and weighted q-Bernstein polynomials,”
*Advanced Studies in Contemporary Mathematics*, vol. 21, pp. 13–18, 2011. 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 Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - T. Kim, J. Y. Choi, and J. Y. Sug, “Extended q-euler numbers and polynomials associated with fermionic p-adic q-integral on ${\mathbb{Z}}_{p}$,”
*Russian Journal of Mathematical Physics*, vol. 14, no. 2, pp. 160–163, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - T. Kim, “A note on q-Bernstein polynomials,”
*Russian Journal of Mathematical Physic*, vol. 18, no. 4, pp. 73–82, 2011. View at Google Scholar - L. C. Jang, W.-J. Kim, and Y. Simsek, “A study on the p-adic integral representation
on ${\mathbb{Z}}_{p}$ associated with Bernstein and Bernoulli polynomials,”
*Advances in Difference Equations*, vol. 2010, Article ID 163217, 6 pages, 2010. View at Publisher · View at Google Scholar - L. C. Jang, K.-W. Hwang, and Y.-H. Kim, “A note on (
*h, q*)-Genocchi polynomials and numbers of higher order,”*Advances in Difference Equations*, vol. 2010, Article ID 309480, 6 pages, 2010. View at Publisher · View at Google Scholar - M. Can, M. Cenkci, V. Kurt, and Y. Simsek, “Twisted Dedekind type sums associated with Barnes' type multiple Frobenius-Euler l-functions,”
*Advanced Studies in Contemporary Mathematics*, vol. 18, no. 2, pp. 135–160, 2009. View at Google Scholar · View at Scopus - Y. Simsek, “Special functions related to Dedekind-type DC-sums and their applications,”
*Russian Journal of Mathematical Physics*, vol. 17, no. 4, pp. 495–508, 2010. View at Publisher · View at Google Scholar · View at Scopus - Y. Simsek, O. Yurekli, and V. Kurt, “On interpolation functions of the twisted generalized Frobenius-Euler numbers,”
*Advanced Studies in Contemporary Mathematics*, vol. 15, pp. 187–194, 2007. View at Google Scholar - C. S. Ryoo, “A note on the weighted q-Euler numbers and polynomials,”
*Advanced Studies in Contemporary Mathematics*, vol. 21, pp. 47–54, 2011. View at Google Scholar - S. H. Rim, S. J. Lee, E. J. Moon, and J. H. Jin, “On the q-Genocchi numbers and polynomials associated with Q-zeta function,”
*Proceedings of the Jangjeon Mathematical Society*, vol. 12, no. 3, pp. 261–267, 2009. View at Google Scholar · View at Scopus - C. S. Ryoo, “On the generalized Barnes type multiple q-Euler polynomials twisted by ramified roots of unity,”
*Proceedings of the Jangjeon Mathematical Society*, vol. 13, no. 2, pp. 255–263, 2010. View at Google Scholar · View at Scopus - A. Bayad, “Modular properties of elliptic Bernoulli and Euler functions,”
*Advanced Studies in Contemporary Mathematics*, vol. 20, no. 3, pp. 389–401, 2010. View at Google Scholar · View at Scopus