Generalizations of an integral for Legendre polynomials by Persson and Strang

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:388

Article

Koornwinder, Tom H.¹ 

[1] Univ Amsterdam, Korteweg de Vries Inst, NL-1090 GE Amsterdam, Netherlands

关键词: Orthogonal polynomials;    Gegenbauer polynomials;    Jacobi polynomials;    Hahn polynomials;    Evaluation of definite integrals;    Christoffel-Darboux formula;   

DOI  :  10.1016/j.jmaa.2011.12.001

来源: Elsevier

【 摘 要 】

Persson and Strang (2003) evaluated the integral over [-1,1] of a squared odd degree Legendre polynomial divided by x(2) as being equal to 2. We consider a similar integral for orthogonal polynomials with respect to a general even orthogonality measure, with Gegenbauer and Hermite polynomials as explicit special cases. Next, after a quadratic transformation, we are led to the general nonsymmetric case, with Jacobi and Laguerre polynomials as explicit special cases. Examples of indefinite summation also occur in this context. The paper concludes with a generalization of the earlier results for Hahn polynomials. There some adaptations have to be made in order to arrive at relatively nice explicit evaluations. (C) 2011 Elsevier Inc. All rights reserved.

【 授权许可】

Free