Orthogonal polynomials and partial differential equations on the unit ball

2007/12/19 by Miguel A. Piñar, Yuan Xu, Pinar, Miguel +1
Engineering · Mathematics · #33C50 #33E30 #42C05 #Aerospace Engineering and Control Systems #Algebraic and Geometric Analysis #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Numerical methods for differential equations

paper · pdf · doi:10.48550/arxiv.0712.3091

openalex publication_date 2007/12/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Orthogonal polynomials of degree n with respect to the weight function W_μ(x) = (1-‖x‖²)^μ on the unit ball in \RR^d are known to satisfy the partial differential equation [ Δ- \la x, ∇ \ra² - (2 μ+d) \la x, ∇ \ra ] P = -n(n+2 μ+d) P for μ> -1. The singular case of μ= -1,-2, ... is studied in this paper. Explicit polynomial solutions are constructed and the equation for ν= -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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