Generalizations of an integral for Legendre polynomials by Persson and Strang

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• 作者：Enno Diekema^a ; ^{e.diekema@gmail.com} ; Tom H. Koornwinder^b ; ^{T.H.Koornwinder@uva.nl}
• 关键词：Orthogonal polynomials ; Gegenbauer polynomials ; Jacobi polynomials ; Hahn polynomials ; Evaluation of definite integrals ; Christoffel&ndash ; Darboux formula
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2012
• 出版时间：1 April 2012
• 年：2012
• 卷：388
• 期：1
• 页码：125-135
• 全文大小：184 K

文摘

Persson and Strang (2003) evaluated the integral over of a squared odd degree Legendre polynomial divided by 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.