Generalizations of an integral for Legendre polynomials by Persson and Strang

TL;DR

This paper generalizes an integral involving Legendre polynomials to a broader class of orthogonal polynomials, providing explicit evaluations and extending results to Hahn polynomials.

Contribution

It introduces a unified approach to evaluate integrals of squared orthogonal polynomials across various families, including Gegenbauer, Hermite, Jacobi, Laguerre, and Hahn polynomials.

Findings

Explicit integral evaluations for Gegenbauer and Hermite polynomials.

Extension to nonsymmetric cases with Jacobi and Laguerre polynomials.

Generalized results for Hahn polynomials with necessary adaptations.

Abstract

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.