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Discrete Dynamics in Nature and Society
Volume 2012, Article ID 486158, 12 pages
http://dx.doi.org/10.1155/2012/486158
Research Article

Some Identities on Bernoulli and Euler Numbers

1Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea
2Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
3Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea

Received 15 November 2011; Accepted 23 December 2011

Academic Editor: Delfim F. M. Torres

Copyright © 2012 D. S. Kim 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

Recently, Kim introduced the fermionic p-adic integral on . By using the equations of the fermionic and bosonic p-adic integral on , we give some interesting identities on Bernoulli and Euler numbers.

1. Introduction/Preliminaries

Let be a fixed odd prime number. Throughout this paper, , , and will denote the ring of -adic integers, the field of -adic rational numbers, and the completion of algebraic closure of , respectively. Let be the set of natural numbers and . The -adic absolute value is normally defined by .

Let be the space of uniformly differentiable functions on and the space of continuous function on . For , the fermionic -adic integral on is defined by Kim as follows: The following fermionic -adic integral equation on is well known (see [13]): where .

From (1.1) and (1.2), we can derive the generating function of Euler polynomials as follows: where is the ordinary Euler polynomial (see [14]). In the special case, , is called the ordinary Euler number.

By (1.3), we get Witt’s formula for the Euler polynomial as follows: Thus, by (1.4), we have with the usual convention about replacing by (see [5, 6]). From (1.3), we note that where is the Kronecker symbol (see [3]). By (1.2) and (1.4), we get Thus, by (1.4) and (1.7), we have Equation (1.8) is equivalent to From (1.6), we can derive the following equation:

For , the bosonic -adic integral on is defined by From (1.11), we can easily derive the following -integral equation: where and .

It is well known that the Bernoulli polynomial can be represented by the bosonic -adic integral on as follows: where is called the Bernoulli polynomial (see [4, 713]). In the special case, , is called the Bernoulli number. By the definition of Bernoulli numbers and polynomials, we get Thus, by (1.13) and (1.14), we see that with the usual convention about replacing by (see [122]).

By (1.11), we easily get From (1.13), (1.14), and (1.16), we have By (1.15), we get Thus, by (1.17) and (1.18), we have

From (1.12) and (1.13), we get Thus, by (1.13) and (1.20), we have Equation (1.21) is equivalent to the following equation:

In this paper we derive some interesting and new identities for the Bernoulli and Euler numbers from the -adic integral equations on .

2. Some Identities on Bernoulli and Euler Numbers

From (1.1), we note that By (1.14) and (2.1), we get In the special case, , we have

Let us consider the following fermionic -adic integral on as follows: Therefore, by (1.4) and (2.4), we obtain the following theorem.

Theorem 2.1. For , one has

It is known that . If we take the fermionic -adic integral on both sides of (1.22), then we have From (2.2) and (2.6), we note that Therefore, by (1.4) and (2.7), we obtain the following theorem.

Theorem 2.2. For , one has

Corollary 2.3. For , one has

Let us take the bosonic -adic integral on both sides of (1.9) as follows: Thus, by (1.14) and (2.10), we obtain the following theorem.

Theorem 2.4. For , one has

On the other hand, by (2.2) and (2.10), we get where with . Therefore, by (2.12), we obtain the following theorem.

Theorem 2.5. For with , one has

By (1.9) and (1.22), we get Therefore, by (1.4), (1.14), and (2.14), we obtain the following theorem.

Theorem 2.6. For and , one has

It is easy to show that Therefore, by (2.16), we obtain the following corollay.

Corollary 2.7. For and , one has

For ,  -adic analogue of Bernstein operator of order for is given by where for is called the Bernstein polynomial of degree (see [8]). From the definition of , we note that .

Let us take the fermionic -adic integral on for the product of and as follows: From (2.18), we note that Therefore, by (2.19) and (2.20), we obtain the following theorem.

Theorem 2.8. For , one has In particular,

By (1.17) and the symmetric property of , we get From (1.4) and (2.2), we note that By (2.23) and (2.24), we see that From (2.20) and (2.25), we have Therefore, by (1.19) and (2.26), we obtain the following theorem.

Theorem 2.9. For with , one has In particular,

Acknowledgment

The first author was supported by National Research Foundation of Korea Grant funded by the Korean Government 2011-0002486.

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