On Polar Legendre Polynomials

TL;DR

This paper introduces polar Legendre polynomials, a new class of orthogonal polynomials arising from an inverse Gauss problem, and explores their algebraic, differential, and asymptotic properties.

Contribution

It presents the definition and analysis of polar Legendre polynomials, including their orthogonality and properties, as a novel contribution to polynomial theory.

Findings

Polynomials are solutions to an inverse Gauss problem.

They are orthogonal with respect to a differential operator.

They exhibit specific algebraic and asymptotic behaviors.

Abstract

We introduce a new class of polynomials $\{P_{n}\}$, that we call polar Legendre polynomials, they appear as solutions of an inverse Gauss problem of equilibrium position of a field of forces with $n+1$ unit masses. We study algebraic, differential and asymptotic properties of this class of polynomials, that are simultaneously orthogonal with respect to a differential operator and a discrete-continuous Sobolev type inner product.