International Journal of Mathematics and Mathematical Sciences

Volume 2012 (2012), Article ID 472010, 8 pages

doi:10.1155/2012/472010

Research Article

Integral Formulae of Bernoulli and Genocchi Polynomials

Seog-Hoon Rim,¹ Joung-Hee Jin,² and Joohee Jeong¹

¹Department of Mathematics Education, Kyungpook National University, Tagegu 702-701, Republic of Korea

²Department of Mathematics, Kyungpook National University, Tagegu 702-701, Republic of Korea

Received 19 June 2012; Accepted 19 July 2012

Academic Editor: Taekyun Kim

Copyright © 2012 Seog-Hoon Rim 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 the work of Kim et al. (2012). From these identities, we derive some new and interesting integral formulae for Bernoulli and Genocchi polynomials.

1. Introduction

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

(see [1–5]) with the usual convention about replacing by . In the special case, , are called the th Bernoulli numbers.

The Genocchi polynomials are also defined by

(see [1, 6–10]) with the usual convention about replacing by . In the special case, , are called the th Genocchi numbers.

From (1.1), we note that

(see [1–5]). Thus, by (1.3), we get

(see [2]). From (1.2), we note that

From (1.5), we can derive the following equation:

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

where is the Kronecker symbol (see [2]). From (1.4), (1.6), and (1.7), we note that

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

2. Integral Formula of Bernoulli and Genocchi Polynomials

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

By comparing the coefficients on the both sides of (2.1), we obtain the following theorem.

Theorem 2.1. For , one has

From (1.1) and (1.2), also notes that

By comparing the coefficients on the both sides of (2.3), we obtain the following theorem.

Theorem 2.2. For , one has

Let one take the definite integral from 0 to 1 on both sides of Theorem 2.1. For ,

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

Theorem 2.3. For with , one has

3. -Adic Integral on Associated with Bernoulli and Genocchi 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 [2, 5, 11]). From (3.1), we can derive the following integral equation:

where and (see [2]). Let us take . Then we have

(see [2, 5]). From (3.3), we have

(see [2, 5]). Thus, by (3.2) and (3.4), we get

(see [2]). From (3.5), we have

(see [2]). The fermionic -adic integral on is defined by Kim as follows [2, 8, 9]:

From (3.7), we obtain the following integral equation:

(see [2]), where . Thus, by (3.8), we have

(see [2]). Let us take . Then we have

From (3.10), we have

Thus, by (3.9) and (3.11), we get

Let us consider the following -adic integral on :

From Theorem 2.1 and (3.13), one has

Therefore, by (3.13) and (3.14), we obtain the following theorem.

Theorem 3.1. For , one has

Now, one sets

By Theorem 2.1, one gets

Therefore, by (3.16) and (3.17), we obtain the following theorem.

Theorem 3.2. For , one has

Let us consider the following -adic integral on :

From Theorem 2.2, one has

Therefore, by (3.19) and (3.20), we obtain the following theorem.

Theorem 3.3. For , one has

Now, one sets

By Theorem 2.2, one gets

Therefore, by (3.22) and (3.23), we obtain the following corollary.

Corollary 3.4. For , one has

Acknowledgement

This research was supported by Kyungpook National University research Fund, 2012.

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