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Bivariate version of the Hahn-Sonine theorem

Authors:	Jeongkeun Lee

Institution:	Department of Mathematics, Sunmoon University, Asan, ChoongNam 336-840, Korea

Abstract:	We consider orthogonal polynomials in two variables whose derivatives with respect to 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 and 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.

Keywords:	Orthogonal polynomials in two variables Hahn-Sonine theorem

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