Diffusion orthogonal polynomials in 3-dimensional domains bounded by developable surfaces

The following problem is studied: describe the triplets $(\mathrm{\Omega },g,\mu )$, $\mu =\rho dx$, where $g=(g^{ij}(x))$ is the (co)metric associated with the symmetric second order differential operator $L(f)=\frac{1}{\rho }{\sum }_{ij}{\mathrm{\partial }}_i(g^{ij}\rho {\mathrm{\partial }}_{j}f)$ defined on a domain $\mathrm{\Omega }$ of ${\mathbb{R}}^n$ and such that there exists an orthonormal basis of ${\mathcal{L}}^2(\mu )$ made of polynomials which are eigenvectors of $L$, and the basis is compatible with the filtration of the space of polynomials with respect 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 author’s subsequent paper this problem was solved in dimension 2 for any weighted degree. In the present paper this problem is solved in dimension 3 for the usual degree under the condition that $\mathrm{\partial }\mathrm{\Omega }$ contains a piece of a tangent developable surface. The proof is based on Plücker-like formulas in the form given by Ragni Piene. All the found solutions are generalized for any dimension.