Orthogonal polynomials and partial differential equations on the unit ball

TL;DR

This paper investigates orthogonal polynomials on the unit ball related to a specific weight function, exploring their solutions to associated PDEs, especially in singular cases, and discusses their orthogonality properties.

Contribution

It extends the theory of orthogonal polynomials on the unit ball to singular weight cases, providing explicit solutions and conditions for polynomial solutions in odd dimensions.

Findings

Explicit polynomial solutions for singular cases are constructed.

Complete polynomial solutions exist for certain singular cases when the dimension is odd.

Orthogonality properties of these solutions are analyzed.

Abstract

Orthogonal polynomials of degree $n$ with respect to the weight function $W_\mu(x) = (1-\|x\|^2)^\mu$ on the unit ball in $\RR^d$ are known to satisfy the partial differential equation $$ [ \Delta - \la x, \nabla \ra^2 - (2 \mu +d) \la x, \nabla \ra \right ] P = -n(n+2 \mu+d) P $$ for $\mu > -1$. The singular case of $\mu = -1,-2, ...$ is studied in this paper. Explicit polynomial solutions are constructed and the equation for $\nu = -2,-3,...$ is shown to have complete polynomial solutions if the dimension $d$ is odd. The orthogonality of the solution is also discussed.