International Journal of Mathematics and Mathematical Sciences

Volume 2012 (2012), Article ID 689797, 10 pages

doi:10.1155/2012/689797

Research Article

Arithmetic Identities Involving Bernoulli and Euler Numbers

H.-M. Kim¹ and D. S. Kim²

¹Department of Mathematics, Kookmin University, Seoul 136-702, Republic of Korea

²Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea

Received 12 June 2012; Accepted 23 October 2012

Academic Editor: A. Bayad

Copyright © 2012 H.-M. Kim and D. S. Kim. 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 give some arithmatic identities for the Bernoulli and Euler numbers. These identities are derived from the several -adic integral equations on .

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 , respectively. The -adic norm is normalized so that . Let be the set of natural numbers and .

Let be the space of uniformly differentiable functions on . For , the bosonic -adic integral on is defined by

and the fermionic -adic integral on is defined by Kim as follows (see [1–8]):

The Euler polynomials, , are defined by the generating function as follows (see [1–16]):

In the special case, , is called the th Euler number.

By (1.3) and the definition of Euler numbers, we easily see that

with the usual convention about replacing by (see [10]). Thus, by (1.3) and (1.4), we have

where is the Kronecker symbol (see [9, 10, 17–19]).

From (1.2), we can also derive the following integral equation for the fermionic -adic integral on as follows:

see [1, 2]. By (1.3) and (1.6), we get

Thus, by (1.7), we have

see [1–8, 13–16].

The Bernoulli polynomials, , are defined by the generating function as follows:

see [18]. In the special case, , is called the th Bernoulli number. From (1.9) and the definition of Bernoulli numbers, we note that

see [1–19], with the usual convention about replacing by . By (1.9) and (1.10), we easily see that

see [13].

From (1.1), we can derive the following integral equation on :

where and .

By (1.12), we have

Thus, by (1.13), we can derive the following Witt’s formula for the Bernoulli polynomials:

In [19], it is known that for ,

where if or .

The purpose of this paper is to give some arithmetic identities involving Bernoulli and Euler numbers. To derive our identities, we use the properties of -adic integral equations on .

2. Arithmetic Identities for Bernoulli and Euler Numbers

Let us take the bosonic -adic integral on in (1.15) as follows:

On the other hand, we get

By (2.1) and (2.2), we get

Therefore, by (2.3), we obtain the following theorem.

Theorem 2.1. For , one has

Now we consider the fermionic -adic integral on in (1.15) as follows:

On the other hand, we get

By (2.5) and (2.6), we get

Therefore, by (2.7), we obtain the following theorem.

Theorem 2.2. For , one has

Replacing by in (1.15), we have the identity:

Let us take the bosonic -adic integral on in (2.9) as follows:

On the other hand, we see that

By (2.10) and (2.11), we get

Therefore, by (2.12), we obtain the following theorem.

Theorem 2.3. For , one has

We consider the fermionic -adic integral on in (2.9) as follows:

On the other hand, we get

By (2.14) and (2.15), we obtain the following theorem.

Theorem 2.4. For , one has

Acknowledgment

This Research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (2012R1A1A2003786).

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