Filter integrals for orthogonal polynomials

TL;DR

This paper explores integral identities involving orthogonal polynomials such as Legendre, Hermite, Chebyshev, Laguerre, and Gegenbauer, extending known results and providing new formulas for these polynomial families.

Contribution

It generalizes a known integral identity for Legendre polynomials to other orthogonal polynomial families, offering new analytical expressions.

Findings

Derived integral formulas for Hermite, Chebyshev, Laguerre, and Gegenbauer polynomials.

Extended the specific Legendre polynomial identity to polynomials with even indices.

Provided a unified approach to integral identities across multiple orthogonal polynomial families.

Abstract

Motivated by an expression by Persson and Strang on an integral involving Legendre polynomials, stating that the square of $P_{2n+1}(x)/x$ integrated over $[-1,1]$ is always $2$, we present analog results for Hermite, Chebyshev, Laguerre and Gegenbauer polynomials as well as the original Legendre polynomial with even index.