Orthogonal polynomials and partial differential equations on the unit ball

Orthogonal polynomials of degree $n$ with respect to the weight function $W_{\mu }(x)=(1-\mathrm{\parallel }x{\mathrm{\parallel }}^2{)}^{\mu }$ on the unit ball in ${\text{RR}}^d$ 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 $\mu =-1,-2,\mathrm{.}\mathrm{.}\mathrm{.}$ is studied in this paper. Explicit polynomial solutions are constructed and the equation for $\nu =-2,-3,\mathrm{.}\mathrm{.}\mathrm{.}$ is shown to have complete polynomial solutions if the dimension $d$ is odd. The orthogonality of the solution is also discussed.