Quadrature rules for $L^1$-weighted norms of orthogonal polynomials

TL;DR

This paper derives quadrature rules for calculating $L^1$-weighted norms of classical orthogonal polynomials, enabling direct computation of certain positive integrals with practical examples.

Contribution

It introduces new quadrature formulas for $L^1$-weighted norms of Hermite, Laguerre, and Jacobi polynomials based on their zeros, facilitating direct integral calculations.

Findings

Derived explicit quadrature rules for orthogonal polynomials' $L^1$-weighted norms

Provided practical examples demonstrating the formulas' utility

Enabled direct computation of positive integrals using these formulas

Abstract

In this paper we obtain $L^1$-weighted norms of classical orthogonal polynomials (Hermite, Laguerre and Jacobi polynomials) in terms of the zeros of these orthogonal polynomials; these expressions are usually known as quadrature rules. In particular these new formulae are useful to calculate directly some positive defined integrals as several examples show.