Orthogonal polynomials and partial differential equations on the unit ball

Details Orthogonal polynomials and partial differential equations on the unit ball Orthogonal polynomials and partial differential equations on the unit ball

2007-12-19

arxiv.org/abs/0712.3091v1

Mathematics

Classical Analysis and ODEs

9

Scientific paper

Orthogonal polynomials of degree $n$ with respect to the weight function $W_\mu(x) = (1-\|x\|^2)^\mu$ on the unit ball in $\RR^d$ 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 $\mu = -1,-2, ...$ 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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