Two-dimensional diffusion orthogonal polynomials ordered by a weighted degree

Abstract

We study the following problem: describe the triplets $(\Omega ,g,\mu )$, $\mu =\rho dx$, where $g=(g^{ij}(x))$ is the (co)metric associated with the symmetric second order differential operator $L(f)=\frac{1}{\rho }{\sum }_{ij}{\partial }_i(g^{ij}\rho {\partial }_{j}f)$ defined on a domain $\Omega$ of ${\mathbb{R}}^d$ and such that there exists an orthonormal basis of ${\mathcal{L}}^2(\mu )$ made of polynomials which are eigenvectors of $L$, where the polynomials are ranked according to some weighted degree. In a joint paper with D. Bakry and M. Zani this problem was solved in dimension 2 for the usual degree. In the present paper we solve it still in dimension 2, but for a weighted degree with arbitrary positive weights.

30 pages, 8 figures