International Journal of Mathematics and Mathematical Sciences

Volume 2012 (2012), Article ID 184649, 9 pages

doi:10.1155/2012/184649

Research Article

Identities on the Bernoulli and Genocchi Numbers and Polynomials

Seog-Hoon Rim,¹ Joohee Jeong,¹ and Sun-Jung Lee²

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

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

Received 9 June 2012; Accepted 9 August 2012

Academic Editor: Yilmaz Simsek

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.

We give some interesting identities on the Bernoulli numbers and polynomials, on the Genocchi numbers and polynomials by using symmetric properties of the Bernoulli and Genocchi polynomials.

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 the algebraic closure of . Let be the set of natural numbers and . The -adic norm on is normalized so that . Let be the space of continuous functions on . For , the fermionic -adic integral on is defined by Kim as follows:

(see [1–16]). From (1.1), we have

(see [1–16]), where .

Let us take . Then, by (1.2), we get

where are the th ordinary Genocchi numbers (see [8, 15]).

From the same method of (1.3), we can also derive the following equation:

where are called the th Genocchi polynomials (see [14, 15]).

By (1.3), we easily see that

(see [15]). By (1.3) and (1.4), we get Witt's formula for the th Genocchi numbers and polynomials as follows:

From (1.2), we have

where the symbol is the Kronecker symbol (see [4, 5]).

Thus, by (1.5) and (1.7), we get

(see [15]). From (1.4), we can derive the following equation:

By (1.6) and (1.9), we see that

Thus, by (1.10), we get .

From (1.5) and (1.8), we have

The Bernoulli polynomials are defined by

(see [6, 9, 12]) with the usual convention about replacing by .

In the special case, , is called the -th Bernoulli number. By (1.12), we easily see that

(see [6]). Thus, by (1.12) and (1.13), we get reflection symmetric formula for the Bernoulli polynomials as follows:

(see [6, 9, 12]). From (1.14) and (1.15), we can also derive the following identity:

In this paper, we investigate some properties of the fermionic -adic integrals on . By using these properties, we give some new identities on the Bernoulli and the Euler numbers which are useful in studying combinatorics.

2. Identities on the Bernoulli and Genocchi Numbers and Polynomials

Let us consider the following fermionic -adic integral on as follows:

On the other hand, by (1.14) and (1.15), we get

Equating (2.1) and (2.2), we obtain the following theorem.

Theorem 2.1. For , one has

By using the reflection symmetric property for the Euler polynomials, we can also obtain some interesting identities on the Euler numbers.

Now, we consider the fermionic -adic integral on for the polynomials as follows:

On the other hand, by (1.8), (1.10), and (1.11), we get

Equating (2.4) and (2.5), we obtain the following theorem.

Theorem 2.2. For , one has

Let us consider the fermionic -adic integral on for the product of and as follows:

On the other hand, by (1.10) and (1.14), we get

By (2.7) and (2.8), we easily see that

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

Theorem 2.3. For , one has

Corollary 2.4. For , one has

Let us consider the fermionic -adic integral on for the product of the Bernoulli polynomials and the Bernstein polynomials. For , with , are called the Bernstein polynomials of degree , see [11]. It is easy to show that ,

On the other hand, by (1.14) and (2.12), we get

Equating (2.12) and (2.13), we see that

Thus, from (2.14), we obtain the following theorem.

Theorem 2.5. For , one has

Finally, we consider the fermionic -adic integral on for the product of the Euler polynomials and the Bernstein polynomials as follows:

On the other hand, by (1.10) and (2.12), we get

Equating (2.16) and (2.17), we obtain

Therefore, by (2.18), we obtain the following theorem.

Theorem 2.6. For , one has

Acknowledgment

This paper was supported by Kynugpook National University Research Fund, 2012.

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