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Bivariate version of the Hahn-Sonine theorem


Author: Jeongkeun Lee
Journal: Proc. Amer. Math. Soc. 128 (2000), 2381-2391
MSC (1991): Primary 33C50, 35P99
DOI: https://doi.org/10.1090/S0002-9939-00-05648-3
Published electronically: February 21, 2000
MathSciNet review: 1709757
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Abstract:

We consider orthogonal polynomials in two variables whose derivatives with respect to $x$ are orthogonal. We show that they satisfy a system of partial differential equations of the form \begin{equation*}\alpha(x,y)\partial_{x}^{2}\overrightarrow{U}_{n}+\beta(x,y)\pa... ...l _{x} \overrightarrow{U}_{n}=\Lambda_{n}\overrightarrow{U}_{n}, \end{equation*}where $\deg\alpha\leq2$, $\deg\beta\leq1$, $\overrightarrow{U} _{n}=(U_{n0},U_{n-1,1},\cdots,U_{0n})$ is a vector of polynomials in $x$ and $y$ for $n\geq0$, and $\Lambda_{n}$ is an eigenvalue matrix of order $ (n+1)\times(n+1)$ for $n\geq0$. Also we obtain several characterizations for these polynomials. Finally, we point out that our results are able to cover more examples than Bertran's.


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  • 1. W. A. Al-Salam and T. S. Chihara, Another characterization of the classical orthogonal polynomials, SIAM J. Math. Anal. 3 (1972), 65-70. MR 47:5320
  • 2. M. Bertran, Notes on orthogonal polynomials in $ \upsilon$-variables, SIAM J. Math. Anal. 6(2) (1975), 250-257. MR 51:959
  • 3. S. Bochner, Über Sturm-Liouvillesche Polynomsysteme, Math. Z. 29 (1929), 65-72.
  • 4. W. C. Connett and A. L. Schwartz, Continuous $2$-variable polynomial hypergroups. In (O. Gebuhrer W. C. Connett and A. L. Schwartz, editors), Applications of hypergroups and related measure algebras, pp. 89-109, Providence, R.I., 1995, American Mathematical Society. Contemporary Mathematics, 183. MR 96g:43006
  • 5. W. Hahn, Über die Jacobischen Polynom und zwei verwandte Polynomklassen, Math. Z. 39 (1935), 634-638.
  • 6. 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
  • 7. Y. J. Kim, K. H. Kwon and J. K. Lee, Partial differential equations having orthogonal polynomial solutions, J. Comp. Appl. Math. 99 (1998), 239-253. MR 99j:35007
  • 8. M. A. Kowalski, Orthogonality and recursion formulas for polynomials in $n$ variables, SIAM J. Math. Anal. 13 (1982), 316-323. MR 83j:42022b
  • 9. M. A. Kowalski, Algebraic characterizations of orthogonality in the space of polynomials, Lecture Notes in Math. 1171 (1985), 101-110. MR 87j:42066
  • 10. H. L. Krall, On derivatives of orthogonal polynomials II, Bull. Amer. Math. Soc. 47 (1941), 261-264. MR 2:282
  • 11. H. L. Krall and I. M. Sheffer, Orthogonal polynomials in two variables, Ann. Mat. pura Appl. 4 (1967), 325-376. MR 37:4499
  • 12. K. H. Kwon, J. K. Lee and L. L. Littlejohn, Orthogonal polynomial eigenfunctions of second order partial differential equations, preprint.
  • 13. K. H. Kwon, J. K. Lee and B. H. Yoo, Characterizations of classical orthogonal polynomials, Results in Math. 24 (1993), 119-128. MR 94i:33011
  • 14. K. H. Kwon and L. L. Littlejohn, Classification of classical orthogonal polynomials, J. Korean Math. Soc. 34 No. 4 (1997), 973-1008. MR 99k:33028
  • 15. L. L. Littlejohn, Orthogonal polynomial solutions to ordinary and partial differential equations, Proc. 2nd Intern. Symp. Orthogonal polynomials and their applications, M. Alfaro et al. ed., Segovia (Spain), 1986, Lecture Notes in Math. Vol. 1329, Springer Verlag, Berlin, 1988, 98-124. MR 89j:33013
  • 16. A. L. Schwartz, Partial differential equations and bivariate orthogonal polynomials, preprint.
  • 17. N. Ja Sonine, Über die angenäherte Berechnung der bestimmten Integrale und über die dabei vorkommenden ganzen Functionen, Warsaw Univ. Izv 18 (1887), 1-76.
  • 18. Y. Xu, On multivariate orthogonal polynomials, SIAM J. Math. Anal. 24 (3) (1993), 783-794. MR 94i:42031

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Additional Information

Jeongkeun Lee
Affiliation: Department of Mathematics, Sunmoon University, Asan, ChoongNam 336-840, Korea
Email: jklee@omega.sunmoon.ac.kr

DOI: https://doi.org/10.1090/S0002-9939-00-05648-3
Keywords: Orthogonal polynomials in two variables, Hahn-Sonine theorem
Received by editor(s): September 19, 1998
Published electronically: February 21, 2000
Communicated by: Hal L. Smith
Article copyright: © Copyright 2000 American Mathematical Society

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