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Orthogonal polynomials and partial differential equations on the unit ball

Authors: Miguel Piñar and Yuan Xu
Journal: Proc. Amer. Math. Soc. 137 (2009), 2979-2987
MSC (2000): Primary 33C50, 33E30, 42C05
Published electronically: April 14, 2009
MathSciNet review: 2506456
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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 $ \mathbb{R}$ are known to satisfy the partial differential equation

$\displaystyle \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.

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Additional Information

Miguel Piñar
Affiliation: Department of Applied Mathematics, University of Granada, Granada 18071, Spain

Yuan Xu
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222

Keywords: pde, orthogonal polynomials, several variables, unit ball
Received by editor(s): December 18, 2007
Published electronically: April 14, 2009
Additional Notes: Partially supported by Ministerio de Ciencia y Tecnología (MCYT) of Spain and by the European Regional Development Fund (ERDF) through the grant MTM 2005–08648–C02–02, and Junta de Andalucía, Grupo de Investigación FQM 0229. The work of the second author was supported in part by NSF Grant DMS-0604056
Communicated by: Peter A. Clarkson
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.