Bivariate Version of the Hahn-sonine Theorem

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 α(x, y)∂ x −→ Un + β(x, y)∂x −→ Un = Λn −→ Un, where degα ≤ 2, deg β ≤ 1, −→U n = (Un0, Un−1,1, · · · , U0n) is a vector of polynomials in x and y for n ≥ 0, and Λn is an eigenvalue matrix of order (n+ 1)× (n+ 1) for n ≥ 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.