Orthogonal polynomial eigenfunctions of second-order partial differerential equations
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- by K. H. Kwon, J. K. Lee and L. L. Littlejohn PDF
- Trans. Amer. Math. Soc. 353 (2001), 3629-3647 Request permission
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.References
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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
- 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.
- © Copyright 2001 American Mathematical Society
- 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
- MathSciNet review: 1837252