Orthogonal polynomials and partial differential equations on the unit ball

Orthogonal polynomials of degree $n$ with respect to the weight function $W_\mu(x) = (1-\Vert x\Vert^2)^\mu$ on the unit ball in $\mathbb{R}$ are known to satisfy the partial differential equation

$$\left[ \Delta - \langle x, \nabla \rangle^2 - (2 \mu +d) \langle x, \nabla \rangle \right ] P = -n(n+2 \mu+d) P$$

for $\mu > -1$. The singular case of $\mu = -1,-2, \ldots$ is studied in this paper. Explicit polynomial solutions are constructed and the equation for $\nu = -2,-3,\ldots$ is shown to have complete polynomial solutions if the dimension $d$ is odd. The orthogonality of the solution is also discussed.