Generalizations of an integral for Legendre polynomials by Persson and Strang

Details Generalizations of an integral for Legendre polynomials by Persson and Strang Generalizations of an integral for Legendre polynomials by Persson and Strang

2010-05-13

arxiv.org/abs/1005.2285v2

J. Math. Anal. Appl. 388 (2012), 125-135

Mathematics

Classical Analysis and ODEs

14 pages; minor corrections and reference added; J. Math. Anal. Appl. (2011)

Scientific paper

10.1016/j.jmaa.2011.12.001

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.

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