Orthogonal Polynomials on the Unit Ball and Fourth-Order Partial Differential Equations

TL;DR

This paper studies a special family of orthogonal polynomials on the unit ball, exploring their properties, connections to classical polynomials, and their relation to fourth-order partial differential equations.

Contribution

It introduces a new family of orthogonal polynomials on the unit ball with a sphere term and establishes their connection to classical polynomials and differential equations.

Findings

Connection formulas with classical multivariate orthogonal polynomials

Representation in terms of spherical harmonics

Differential properties related to fourth-order PDEs

Abstract

The purpose of this work is to analyse a family of mutually orthogonal polynomials on the unit ball with respect to an inner product which includes an additional term on the sphere. First, we will get connection formulas relating classical multivariate orthogonal polynomials on the ball with our family of orthogonal polynomials. Then, using the representation of these polynomials in terms of spherical harmonics, algebraic and differential properties will be deduced.