Orthogonal polynomials and partial differential equations on the unit ball

Abstract

Orthogonal polynomials of degree $n$ with respect to the weight function $W_{\mu }(x)=(1-\Vert x{\Vert }^2{)}^{\mu }$ on the unit ball in $\RR^d$ are known to satisfy the partial differential equation $$ [ Δ- \la x, \nabla \ra^2 - (2 μ+d) \la x, \nabla \ra \right ] P = -n(n+2 μ+d) P $$ 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.

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