Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 974632, 13 pages
http://dx.doi.org/10.1155/2012/974632
Research Article

Hermite Polynomials and their Applications Associated with Bernoulli and Euler Numbers

Dae San Kim,1 Taekyun Kim,2 Seog-Hoon Rim,3 and Sang Hun Lee4

1Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea
2Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
3Department of Mathematics Education, Kyungpook National University, Taegu 702-701, Republic of Korea
4Division of General Education, Kwangwoon University, Seoul 139-701, Republic of Korea

Received 7 May 2012; Accepted 15 May 2012

Academic Editor: Garyfalos Papaschinopoulos

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.

Abstract

We derive some interesting identities and arithmetic properties of Bernoulli and Euler polynomials from the orthogonality of Hermite polynomials. Let 𝐏 𝑛 = { 𝑝 ( 𝑥 ) ∈ ℚ [ 𝑥 ] ∣ d e g 𝑝 ( 𝑥 ) ≤ 𝑛 } be the ( 𝑛 + 1 ) -dimensional vector space over ℚ . Then we show that { 𝐻 0 ( 𝑥 ) , 𝐻 1 ( 𝑥 ) , … , 𝐻 𝑛 ( 𝑥 ) } is a good basis for the space 𝐏 𝑛 for our purpose of arithmetical and combinatorial applications.