Event Details
- Date: Thursday, September 24.
- Time: 17:00h - 19:00h.
- Place: Gauss Seminar, IMAG, UGR.
- Area: GOYA Seminar.
- Speakers: Misael E. Marriaga (Universidad Rey Juan Carlos, Madrid, Spain).
- Title: Generalized symmetric classical orthogonal polynomials in two variables.
- Abstract: In this talk I will discuss bivariate symmetric orthogonal polynomials governed by reflection-type interaction weights. In several variables, the geometry of the domain and the symmetry of the weight determine not only the orthogonal basis, but also the natural differential operators acting on it. I will focus on weights with singular interaction factors along reflection walls and on the corresponding orthogonality problems on fundamental chambers. The main theme is an operator-theoretic construction of these systems. For product domains associated with the classical Hermite, Laguerre and Jacobi weights, for the unit disk through generalized Zernike polynomials, and for the unit triangle with Jacobi-type weights, the symmetric orthogonal families arise as eigenfunctions of self-adjoint second-order differential operators. I will explain how lowering and raising operators shift degrees and parameters, and how their compositions produce additional commuting operators. The passage to elementary symmetric variables plays a central role: apparent rational singularities disappear and the operators acquire polynomial coefficients. This leads to explicit descriptions of the differential-operator algebras for which the transformed polynomials form a joint eigenbasis, revealing a common Koornwinder-type structure behind these bivariate models with interaction terms. Join work with Miguel A. Piñar and Gema Alhama (Universidad de Granada, Granada, Spain).
