Orthogonal polynomials and partial differential equations on the unit ball

Orthogonal polynomials of degree $n$ with respect to the weight function $W_\mu(x) = (1-\|x\|^2)^\mu$ 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.