Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2004; 41(6): 1049-1070

Printed November 1, 2004

Copyright © The Korean Mathematical Society.

Orthogonal polynomials satisfying partial differential equations belonging to the basic class

J. K. Lee, L. L. Littlejohn, and B. H. Yoo

SunMoon University, Utah State University, Andong University

Abstract

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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