Discrete Dynamics in Nature and Society

Volume 2012 (2012), Article ID 269847, 15 pages

doi:10.1155/2012/269847

Research Article

Integral Formulae of Bernoulli Polynomials

Dae San Kim,¹ Dmitry V. Dolgy,² Hyun-Mee Kim,³ Sang-Hun Lee,³ and Taekyun Kim⁴

¹Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea

²Hanrimwon, Kwangwoon University, Seoul 139-701, Republic of Korea

³Division of General Education, Kwangwoon University, Seoul 139-701, Republic of Korea

⁴Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea

Received 24 February 2012; Accepted 10 May 2012

Academic Editor: Lee Chae Jang

Copyright © 2012 Dae San 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, some interesting and new identities are introduced in (Hwang et al., Communicated). From these identities, we derive some new and interesting integral formulae for the Bernoulli polynomials.

1. Introduction

As is well known, the Bernoulli polynomials are defined by generating functions as follows:

(see [1–11]). In the special case, are called the th Bernoulli numbers. The Euler polynomials are also defined by

with the usual convention about replacing by (see [1–11]). From (1.1) and (1.2), we can easily derive the following equation:

By (1.1) and (1.3), we get

From (1.1), we have

Thus, by (1.5), we get

It is known that are called the th Euler numbers (see [7]). The Euler polynomials are also given by

(see [6]). From (1.7), we can derive the following equation:

By the definition of Bernoulli and Euler numbers, we get the following recurrence formulae:

where is the kronecker symbol (see [5]). From (1.6), (1.8), and (1.9), we note that

where . The following identity is known in [5]:

From the identities of Bernoulli polynomials, we derive some new and interesting integral formulae of an arithmetical nature on the Bernoulli polynomials.

2. Integral Formulae of Bernoulli Polynomials

From (1.1) and (1.2), we note that

Therefore, by (1.2) and (2.1), we obtain the following theorem.

Theorem 2.1. For , one has

Let us take the definite integral from to on both sides of (1.4): for ,

By (2.3), we get

Therefore, by (2.4), we obtain the following theorem.

Theorem 2.2. For , with , one has

Let us take , , and in (1.11). Then we have

It is easy to show that

Let us consider the integral from to in (2.6):

By (2.6) and (2.8), we get

Therefore, by (2.9), we obtain the following theorem.

Theorem 2.3. For , one has

Lemma 2.4. Let . For , one has

(see [5]).

Let us take , , in Lemma 2.4. Then we have

Taking integral from to in (2.12), we get

It is easy to show that

Thus, by (2.13) and (2.14), we get

Therefore, by (2.2) and (2.15), we obtain the following theorem.

Theorem 2.5. For , one has

3. -Adic Integral on Associated with Bernoulli and Euler Numbers

Let be a fixed odd prime number. Throughout this section, , , 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 normalized exponential valuation of with . Let be the space of uniformly differentiable functions on . For , the bosonic -adic integral on is defined by

(see [8]). Thus, by (3.1), we get

where , and . Let us take . Then we have

From (3.3), we have

From (1.2), we can derive the following integral equation:

Thus, from (3.4) and (3.5), we get

From (3.6), we have

The fermionic -adic integral on is defined by Kim as follows [6, 7]:

Let . Then we have

Continuing this process, we obtain the following equation:

Thus, by (3.10), we have

Let us take . By (3.9), we get

From (3.2), we have the Witt's formula for the th Euler polynomials and numbers as follows:

By (3.11) and (3.13), we get

Let us consider the following -adic integral on :

From (1.4) and (3.15), we have

Therefore, by (3.15) and (3.16), we obtain the following theorem.

Theorem 3.1. For , one has

Now, we set

By (1.4), we get

Therefore, by (3.18) and (3.19), we obtain the following theorem.

Theorem 3.2. For , one has

Let us consider the following integral on :

From (2.2), we have

Therefore, by (3.21) and (3.22), we obtain the following theorem.

Theorem 3.3. For , one has

Now, we set

By (2.2), we get

Therefore, by (3.24) and (3.25), we obtain the following corollary.

Corollary 3.4. For , we have

Let us assume that . From Lemma 2.4 and (3.13), we note that

By (3.27), we get

Thus, by (3.28) and (3.13), we obtain the following lemma (see [5]).

Lemma 3.5. Let . For , one has

Let us consider the formula in Lemma 3.5 with . Then we have

Taking on both sides of (3.30),

By the same method, we get

Therefore, by (3.31) and (3.32), we obtain the following proposition.

Proposition 3.6. Let . Then one has

Replacing by , we have

From (3.4) and (3.7), we derive some identity for the first term of the LHS of (3.34).

The first term of the LHS of  (3.34)

where

The second term of the LHS of (3.34)

Therefore, by (3.34), (3.35), and (3.37), we obtain the following theorem.

Theorem 3.7. Let with . Then one has

where

Remark 3.8. Here, we note that

Acknowledgment

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

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