Abstract and Applied Analysis
Volume 2012 (2012), Article ID 957350, 15 pages
http://dx.doi.org/10.1155/2012/957350
Research Article

Extended Laguerre Polynomials Associated with Hermite, Bernoulli, and Euler Numbers and Polynomials

Taekyun Kim1 and Dae San Kim2

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

Received 14 June 2012; Accepted 9 August 2012

Academic Editor: Pekka Koskela

Copyright © 2012 Taekyun Kim and Dae San 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 𝐏 𝑛 = { 𝑝 ( 𝑥 ) ∈ ℝ [ 𝑥 ] ∣ d e g 𝑝 ( 𝑥 ) ≤ 𝑛 } be an inner product space with the inner product ∫ ⟨ 𝑝 ( 𝑥 ) , 𝑞 ( 𝑥 ) ⟩ = ∞ 0 𝑥 𝛼 𝑒 − 𝑥 𝑝 ( 𝑥 ) 𝑞 ( 𝑥 ) 𝑑 𝑥 , where 𝑝 ( 𝑥 ) , 𝑞 ( 𝑥 ) ∈ 𝐏 𝑛 and 𝛼 ∈ ℝ with 𝛼 > − 1 . In this paper we study the properties of the extended Laguerre polynomials which are an orthogonal basis for 𝐏 𝑛 . From those properties, we derive some interesting relations and identities of the extended Laguerre polynomials associated with Hermite, Bernoulli, and Euler numbers and polynomials.