Orthogonal polynomials and partial differential equations on the unit ball

Abstract

Orthogonal polynomials of degree n with respect to the weight function Wμ (x) = (1-x2)μ on the unit ball in R are known to satisfy the partial differential equation [Δ-x,Δ2-(2μ + d) -1. The singular case of μ =-1,-2,. is studied in this paper. Explicit polynomial solutions are constructed and the equation for. =-2,-3,. is shown to have complete polynomial solutions if the dimension d is odd. The orthogonality of the solution is also discussed. © 2009 American Mathematical Society.