Orthogonal polynomials and partial differential equations on the unit ball

Abstract

Orthogonal polynomials of degree n n with respect to the weight function W μ ( x ) = ( 1 − ‖ x ‖ 2 ) μ W_\mu (x) = (1-\|x\|^2)^\mu on the unit ball in R \mathbb {R} are known to satisfy the partial differential equation \[ [ Δ − ⟨ x , ∇ ⟩ 2 − ( 2 μ + d ) ⟨ x , ∇ ⟩ ] P = − n ( n + 2 μ + d ) P \left [ \Delta - \langle x, abla \rangle ^2 - (2 \mu +d) \langle x, abla \rangle \right ] P = -n(n+2 \mu +d) P \] for μ > − 1 \mu > -1 . The singular case of μ = − 1 , − 2 , … \mu = -1,-2, \ldots is studied in this paper. Explicit polynomial solutions are constructed and the equation for ν = − 2 , − 3 , … u = -2,-3,\ldots is shown to have complete polynomial solutions if the dimension d d is odd. The orthogonality of the solution is also discussed.