Orthogonal polynomials and partial differential equations on the unit ball

Abstract:

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