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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Orthogonal polynomial eigenfunctions of second-order partial differerential equations


Authors: K. H. Kwon, J. K. Lee and L. L. Littlejohn
Journal: Trans. Amer. Math. Soc. 353 (2001), 3629-3647
MSC (1991): Primary 33C50, 35P99
DOI: https://doi.org/10.1090/S0002-9947-01-02784-2
Published electronically: April 18, 2001
MathSciNet review: 1837252
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Abstract:

In this paper, we show that for several second-order partial differential equations
\begin{align*}L[u]&=A(x,y)u_{xx}+2B(x,y)u_{xy}+C(x,y)u_{yy}+D(x,y)u_{x}+E(x,y)u_{y} &=\lambda_{n}u \end{align*}
which have orthogonal polynomial eigenfunctions, these polynomials can be expressed as a product of two classical orthogonal polynomials in one variable. This is important since, otherwise, it is very difficult to explicitly find formulas for these polynomial solutions. From this observation and characterization, we are able to produce additional examples of such orthogonal polynomials together with their orthogonality that widens the class found by H. L. Krall and Sheffer in their seminal work in 1967. Moreover, from our approach, we can answer some open questions raised by Krall and Sheffer.


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Additional Information

K. H. Kwon
Affiliation: Department of Mathematics, KAIST, Taejon 305-701, Korea
Email: khkwon@jacobi.kaist.ac.kr

J. K. Lee
Affiliation: Department of Mathematics, SunMoon University, Asan, Choongnam, Korea
Email: jklee@omega.sunmoon.ac.kr

L. L. Littlejohn
Affiliation: Department of Mathematics and Statistics, Utah State University, Logan, Utah 84322-3900
Email: lance@math.usu.edu

DOI: https://doi.org/10.1090/S0002-9947-01-02784-2
Keywords: Orthogonal polynomials in two variables, second order partial differential equations
Received by editor(s): June 14, 1999
Published electronically: April 18, 2001
Additional Notes: The first author (KHK) acknowledges partial financial support from GARC at Seoul National University, the Korea Ministry of Education (BSRI 98-1420) and KOSEF (98-0701-03-01-5). The second author (JKL) thanks KOSEF for financial support and the third author (LLL) acknowledges partial financial support (DMS-9970478) from the National Science Foundation.
Article copyright: © Copyright 2001 American Mathematical Society