Geodesic
Two-dimensional diffusion orthogonal polynomials ordered by a weighted degree
Funkcionalʹnyj analiz i ego priloženiâ, Tome 57 (2023) no. 3, pp. 39-73.

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We study the problem of describing the triples $(\Omega,g,\mu)$, $\mu=\rho\,dx$, where $g= (g^{ij}(x))$ is the (co)metric associated with a symmetric second-order differential operator $\mathbf{L}(f) = \frac{1}{\rho}\sum_{ij} \partial_i (g^{ij} \rho\,\partial_j f)$ defined on a domain $\Omega$ of $\mathbb{R}^d$ and such that there exists an orthonormal basis of $\mathcal{L}^2(\mu)$ consisting of polynomials which are eigenvectors of $\mathbf{L}$ and this basis is compatible with the filtration of the space of polynomials by some weighted degree. In a joint paper of D. Bakry, M. Zani, and the author 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.
Mots-clés : orthogonal polynomials
Keywords: diffusion operator. 
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S. Yu. Orevkov. Two-dimensional diffusion orthogonal polynomials ordered by a weighted degree. Funkcionalʹnyj analiz i ego priloženiâ, Tome 57 (2023) no. 3, pp. 39-73. http://geodesic.mathdoc.fr/item/FAA_2023_57_3_a2/

[1] D. Bakry, X. Bressaud, “Diffusions with polynomial eigenvectors via finite subgroups of $O(3)$”, Ann. Fac. Sci. Toulouse Math. (6), 25:2–3 (2016), 683–721 | DOI | MR | Zbl

[2] D. Bakry, O. Zribi, “Curvature dimension bounds on the deltoid model”, Ann. Fac. Sci. Toulouse Math. (6), 25:1 (2016), 65–90 | DOI | MR | Zbl

[3] D. Bakry, S. Orevkov, M. Zani, “Orthogonal polynomials and diffusion operators”, Ann. Fac. Sci. Toulouse Math. (6), 30:5 (2021), 985–1073 | DOI | MR

[4] V. S. Kulikov, “A remark on classical Pluecker's formulae”, Ann. Fac. Sci. Toulouse Math. (6), 25:5 (2016), 959–967 | DOI | MR | Zbl

[5] L. Soukhanov, On the phenomena of constant curvature in the diffusion-orthogonal polynomials, arXiv: 1409.5332

[6] L. Soukhanov, “Diffusion-orthogonal polynomial systems of maximal weighted degree”, Ann. Fac. Sci. Toulouse Math. (6), 26:2 (2017), 511–518 | DOI | MR | Zbl