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International Journal of Mathematics and Mathematical Sciences

VolumeΒ 2012Β (2012), Article IDΒ 343981, 15 pages

http://dx.doi.org/10.1155/2012/343981

## Euler Basis, Identities, and Their Applications

^{1}Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea^{2}Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea

Received 11 June 2012; Accepted 9 August 2012

Academic Editor: YilmazΒ Simsek

Copyright Β© 2012 D. S. Kim and T. 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

Let be the -dimensional vector space over . We show that is a good basis for the space , for our purpose of arithmetical and combinatorial applications. Thus, if is of degree , then for some uniquely determined . In this paper we develop method for computing from the information of .

#### 1. Introduction

The Euler polynomials, , are given by (see [1β20]) with the usual convention about replacing by . In the special case, are called the th Euler numbers. The Bernoulli numbers are also defined by (see [1β20]) with the usual convention about replacing by . As is well known, the Bernoulli polynomials are given by (see [9β15]) From (1.1), (1.2), and (1.3), we note that where is the kronecker symbol.

Let with . The formula is proved in [4β6]. Let be the -dimensional vector space over . Probably, is the most natural basis for this space. But is also a good basis for the space , for our purpose of arithmetical and combinatorial applications. Thus, if is of degree , then for some uniquely determined . Further, where . In this paper we develop methods for computing from the information of . Apply these results to arithmetically and combinatorially interesting identities involving . Finally, we give some applications of those obtained identities.

#### 2. Euler Basis, Identities, and Their Applications

Let us take the polynomial of degree as follows: From (2.1), we have By (1.7) and (2.2), we get Thus, we have By (1.6), (2.1), (2.3), and (2.4), we get Let us consider the following triple identities: where the sum runs over all with . Thus, by (2.7), we get From (1.7) and (2.8), we have Therefore, by (2.7) and (2.9), we obtain the following theorem.

Theorem 2.1. *For , and with , one has
*

Let us take the polynomial as follows: Then, by (2.11), we get

From (1.6), (1.7), and (2.12), we have Note that Therefore, we obtain the following theorem.

Theorem 2.2. *For with , one has
*

*Remark 2.3. *By the same method, we obtain the following identities.

(I)

(II)
Let us consider the polynomial as follows:
Thus, by (2.18), we get
From (1.6), (1.7), (2.18), and (2.19), we have
Here we note that
It is easy to show that
Therefore, by (1.6), (2.18), (2.20), and (2.22), we obtain the following theorem.

Theorem 2.4. *For with , one has
*

*Remark 2.5. *By the same method, we can derive the following identities.

(I)

(II)

Now we generalize the above consideration to the completely arbitrary case. Let where the sum runs over all nonnegative integers satisfying . From (2.26), we note that

By (1.6), (1.7), (2.18), and (2.27), we get Note that Therefore, by (1.6), (2.28), and (2.29), we obtain the following theorem.

Theorem 2.6. *For with , one has
*

Let us consider the polynomial of degree as Then, from (2.31), we have By (1.7) and (2.32), we get From (2.33), we can derive the following equation: Observe now that Therefore, by (1.6), (2.31), (2.34), (2.35), and (2.36), we obtain the following theorem.

Theorem 2.7. *For with , one has
*

Let us consider the following polynomial of degree . Thus, by (2.38), we get From (1.7), we have Thus, by (2.40), we get Now, we note that Therefore, by (1.6), (2.38), (2.41), and (2.42), we obtain the following theorem.

Theorem 2.8. *For with , one has
*

By the same method, we can obtain the following identity:

#### Acknowledgments

This paper 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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