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Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 269847, 15 pages
Integral Formulae of Bernoulli Polynomials
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.
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.
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 ). The Euler polynomials are also given by (see ). 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 ). From (1.6), (1.8), and (1.9), we note that where . The following identity is known in : 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
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
Theorem 2.3. For , one has
Lemma 2.4. Let . For , one has (see ).
It is easy to show that
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 ). 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 :
Theorem 3.1. For , one has
Now, we set
Theorem 3.2. For , one has
Let us consider the following integral on :
Theorem 3.3. For , one has
Now, we set
Corollary 3.4. For , we have
By (3.27), we get
Lemma 3.5. Let . For , one has
Proposition 3.6. Let . Then one has
Replacing by , we have
Theorem 3.7. Let with . Then one has where
Remark 3.8. Here, we note that
The first author was supported by National Research Foundation of Korea Grant funded by the Korean Government 2011-0002486.
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