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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Bivariate version of the Hahn-Sonine theorem


Author: Jeongkeun Lee
Journal: Proc. Amer. Math. Soc. 128 (2000), 2381-2391
MSC (1991): Primary 33C50, 35P99
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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Additional Information

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

DOI: http://dx.doi.org/10.1090/S0002-9939-00-05648-3
PII: S 0002-9939(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