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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

In this paper, we show that for several second-order partial differential equations

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.

**1.**P. Appell and J. K. de Fériet,*Fonctions Hypergéometriques et Hypersphériques; Polynomes d'Hermite,*Paris, Gauthier-Villars et Cie (1926).**2.**V. K. Jain,*On the orthogonal polynomials of two variables,*Indian J. Pure Appl. Math.**11**(4) (1980), 492-501. MR**81c:42030****3.**Y. J. Kim, K. H. Kwon, and J. K. Lee,*Orthogonal polynomials in two variables and second order partial differential equations,*J. Comp. Appl. Math.**82**(1997), 239-260. MR**98m:33031****4.**Y. J. Kim, K. H. Kwon, and J. K. Lee,*Partial differential equations having orthogonal polynomial solutions,*J. Comp. Appl. Math.**99**(1998) no. 1-2, 239-253. MR**99j:35007****5.**T. H. Koornwinder,*Two-variable analogues of the classical orthogonal polynomials,*Theory and Applications of Special Functions, R. Askey Ed., Academic Press (1975), 435-495. MR**53:5967****6.**H. L. Krall and O. Frink,*A new class of orthogonal polynomials: The Bessel polynomials,*Trans. Amer. Math. Soc.**65**(1949), 100-115. MR**10:453a****7.**H. L. Krall and I. M. Sheffer,*Orthogonal polynomials in two variables,*Ann. Mat. Pura Appl. (4)**76**(1967), 325- 376. MR**37:4499****8.**K. H. Kwon, S. S. Kim, and S. S. Han,*Orthogonality of Tchebychev sets of polynomials,*Bull. London Math. Soc.**24**(1992), 361-367.MR**93g:33007****9.**K. H. Kwon, J. K. Lee, and B. H. Yoo,*Characterizations of classical orthogonal polynomials,*Results in Math.**24**(1993), 119-128. MR**98i:33010****10.**K. H. Kwon and L. L. Littlejohn,*Classification of classical orthogonal polynomials,*J. Korean Math. Soc.,**34**(1997), 973-1008. MR**99k:35007****11.**L. L. Littlejohn,*Orthogonal polynomial solutions to ordinary and partial differential equations,*Proc. 2nd Intern. Symp. Orthogonal Polynomial and their Applications, M. Alfaro et al. Eds, Segovia (Spain), 1986, Lect. Notes Math. Vol.**1329**, Springer-Verlag, Berlin, 1988, 98-124. MR**89j:33013****12.**F. Marcellán, A. Branquinho, and J. Petronilho,*Classical orthogonal polynomials: A functional approach,*Acta Appl. Math.**34**(1994), 283-303. MR**95b:33024****13.**P. Maroni,*An integral representation for the Bessel form,*J. Comp. Appl. Math.**57**(1995), 251-260. MR**96a:42038****14.**P. K. Suetin,*Orthogonal polynomials in two variables*, Nauka, Moscow, 1988 (in Russian). MR**90g:42055****15.**Y. Xu,*A class of bivariate orthogonal polynomials and cubature formula,*Numer. Math.**69**(1994), 233-241. MR**96b:65027**

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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