Journal of Inequalities and Applications
Volume 2011 (2011), Article ID 372874, 25 pages
doi:10.1155/2011/372874
Research Article

Some Properties of Orthogonal Polynomials for Laguerre-Type Weights

HeeSun Jung1 and Ryozi Sakai2

1Department of Mathematics Education, Sungkyunkwan University, Seoul 110-745, Republic of Korea
2Department of Mathematics, Meijo University, Nagoya 468-8502, Japan

Received 3 November 2010; Accepted 21 February 2011

Academic Editor: P. Y. H. Pang

Copyright © 2011 HeeSun Jung and Ryozi Sakai. 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 ℝ + = [ 0 , ∞ ) , let 𝑅 ∶ ℝ + → ℝ + be a continuous, nonnegative, and increasing function, and let 𝑝 𝑛 , 𝜌 ( 𝑥 ) be the orthonormal polynomials with the weight 𝑤 𝜌 ( 𝑥 ) = 𝑥 𝜌 𝑒 − 𝑅 ( 𝑥 ) , 𝜌 > − 1 / 2 . For the zeros { 𝑥 𝑘 , 𝑛 , 𝜌 } 𝑛 𝑘 = 1 of 𝑝 𝑛 , 𝜌 ( 𝑥 ) = 𝑝 𝑛 ( 𝑤 2 𝜌 ; 𝑥 ) , we estimate 𝑝 ( 𝑗 ) 𝑛 , 𝜌 ( 𝑥 𝑘 , 𝑛 , 𝜌 ) , where 𝑗 is a positive integer. Moreover, we investigate the various weighted 𝐿 𝑝 -norms ( 0 < 𝑝 ⩽ ∞ ) of 𝑝 𝑛 , 𝜌 ( 𝑥 ) .