An Orthogonality Property of the Legendre Polynomials

TL;DR

This paper reveals a new orthogonality property of Legendre polynomials involving the arcsine measure and Christoffel function, expanding their classical orthogonality framework.

Contribution

It introduces a novel orthogonality relation for Legendre polynomials under a specific weighted measure, enhancing understanding of their mathematical properties.

Findings

Legendre polynomials are orthogonal under the arcsine measure with Christoffel weight.

The new orthogonality extends classical properties of Legendre polynomials.

Provides insights into polynomial orthogonality under non-standard measures.

Abstract

We give a remarkable additional orthogonality property of the classical Legendre polynomials on the real interval $[-1,1]$: polynomials up to degree $n$ from this family are mutually orthogonal under the arcsine measure weighted by the degree-$n$ normalized Christoffel function.