Bivariate version of the Hahn-Sonine theorem

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)\partial _{x} \overrightarrow {U}\!_{n}=\Lambda _{n}\overrightarrow {U}\!_{n}, \end{equation*} where $\deg \alpha \leq 2$, $\deg \beta \leq 1$, $\overrightarrow {U} _{n}=(U_{n0},U_{n-1,1},\cdots ,U_{0n})$ is a vector of polynomials in $x$ and $y$ for $n\geq 0$, and $\Lambda _{n}$ is an eigenvalue matrix of order $(n+1)\times (n+1)$ for $n\geq 0$. Also we obtain several characterizations for these polynomials. Finally, we point out that our results are able to cover more examples than Bertran’s.