International Journal of Mathematics and Mathematical Sciences
Volume 2012 (2012), Article ID 343981, 15 pages
http://dx.doi.org/10.1155/2012/343981
Research Article

Euler Basis, Identities, and Their Applications

D. S. Kim1 and T. Kim2

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

Received 11 June 2012; Accepted 9 August 2012

Academic Editor: Yilmaz Simsek

Copyright © 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 𝑉 𝑛 = { 𝑝 ( 𝑥 ) ∈ ℚ [ 𝑥 ] | d e g 𝑝 ( 𝑥 ) ≤ 𝑛 } be the ( 𝑛 + 1 ) -dimensional vector space over ℚ . We show that { 𝐸 0 ( 𝑥 ) , 𝐸 1 ( 𝑥 ) , … , 𝐸 𝑛 ( 𝑥 ) } is a good basis for the space 𝑉 𝑛 , for our purpose of arithmetical and combinatorial applications. Thus, if 𝑝 ( 𝑥 ) ∈ ℚ [ 𝑥 ] is of degree 𝑛 , then ∑ 𝑝 ( 𝑥 ) = 𝑛 𝑙 = 0 𝑏 𝑙 𝐸 𝑙 ( 𝑥 ) for some uniquely determined 𝑏 𝑙 ∈ ℚ . In this paper we develop method for computing 𝑏 𝑙 from the information of 𝑝 ( 𝑥 ) .