Event Details

  • Date: Thursday, September 24.
  • Time: 17:30h - 19:30h.
  • Place: Aula A-10, Facultad de Ciencias, UGR.
  • Area: GOYA Seminar.
  • Language: English.

First talk

  • Time: 17:30h.
  • Speaker: 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.

    Joint work with Miguel A. Piñar and Gema Alhama (Universidad de Granada, Granada, Spain).

Second talk

  • Time: 18:30h.
  • Speaker: Cristina Rodríguez Perales (Universidad de Almería, Spain).
  • Title: Nevai-Sobolev orthogonal polynomials: zeros, local asymptotics and differential properties.
  • Abstract:

    In this talk, we consider the orthogonal polynomials \(\{q_n\}_{n\geq 0}\) with respect to a Sobolev inner product involving the Freud weight \(\exp(-x^4)\). This weight is a particular case of the Freud weights and, due to its historical relevance, we will refer to it as Nevai weight. So that, we consider the inner product

    \[ (p,q)_S = \int_{-\infty}^\infty p(x)q(x)\,\exp(-x^4)\,dx + \lambda\int_{-\infty}^\infty p'(x)q'(x)\exp(-x^4)\,dx, \quad \lambda>0. \]

    Our first objective is to study the local asymptotics known as Mehler--Heine asymptotics for this family of polynomials. Moreover, this result together with the well-known Hurwitz's Theorem allows us to establish the asymptotic behavior of the zeros of these polynomials. Motivated by this consequence, we also address the problem of computing the zeros of the polynomials \(\{q_n\}_{n\geq 0}\). With that purpose, we construct a generalized eigenvalue problem from the five-term recurrence relation satisfied by these polynomials, so that the computation of the zeros is equivalent to computing the generalized eigenvalues. We also present numerical experiments illustrating the computation of the zeros for different values of \(\lambda\). Finally, if we have time we will deduce the fourth-order differential equation satisfied by these polynomials.

    Joint work with Juan F. Mañas-Mañas and Juan J. Moreno-Balcázar.

  • Keywords: Sobolev orthogonal polynomials; Mehler--Heine asymptotics; Zeros; Computation.
  • References:
    1. A. Cachafeiro, F. Marcellán, J. J. Moreno--Balcázar (2003). On asymptotic properties of Freud--Sobolev orthogonal polynomials, J. Approx. Theory 125, 26--41.
    2. J. F. Mañas-Mañas, J. J. Moreno-Balcázar, C. Rodríguez-Perales (2026). Nevai--Sobolev orthogonal polynomials, Mediterr. J. Math. 23, 21.
    3. J. F. Mañas-Mañas, J. J. Moreno-Balcázar, C. Rodríguez-Perales (2026). Zeros of Nevai--Sobolev orthogonal polynomials, Notebook Archive. https://notebookarchive.org/2026-07-0gvs3pi
    4. P. Nevai (1983). Orthogonal polynomials associated with \(\exp(-x^4)\), Proc. Canad. Math. Soc. 3, 263--285.
  • Funding:

    Work financed by the PPIT-UAL grant, by MICIU/AEI/10.13039/501100011033 grant PID2025-170285NB-I00; and Research Group FQM--0229 of the University of Almería.