J. Korean Math. Soc. 2004; 41(6): 1049-1070
Printed November 1, 2004
Copyright © The Korean Mathematical Society.
J. K. Lee, L. L. Littlejohn, and B. H. Yoo
SunMoon University, Utah State University, Andong University
We classify all partial differential equations with polynomial coefficients in $x$ and $y$ of the form \[ A(x)u_{xx}+2B(x,y)u_{xy}+C(y)u_{yy}+D(x)u_{x}+E(y)u_{y}=\lambda_{n}u, \] which has weak orthogonal polynomials as solutions and show that partial derivatives of all orders are orthogonal. Also, we construct orthogonal polynomials in $d$-variables satisfying second order partial differential equations in $d$-variables.
Keywords: orthogonal polynomials in two variables, partial differential equation in the basic class
MSC numbers: 33C50, 35P99
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