Person:
Gómez Ullate, David

Profile Picture
Email Address
Birth Date
Research Projects
Organizational Units
Job Title
First Name
David
Last Name
Gómez Ullate
Affiliation
IE University
School
IE School of Science & Technology
Department
Applied Mathematics
Identifiers
Name

Filters

2018 - 20203
Reset filters

Settings

Author: Grandati, Yves ×

Search Results

Now showing 1 - 3 of 3
  • Thumbnail Image
    Publication
    Corrigendum on the proof of completeness for exceptional Hermite polynomials
    (Science Direct, 2020-05) Gómez Ullate, David; Grandati, Yves; Milson, Robert; Ministry of Economy, Industry and Competitiveness; Federación Española de Enfermedades Raras; https://ror.org/02jjdwm75
    Exceptional orthogonal polynomials are complete families of orthogonal polynomials that arise as eigenfunctions of a Sturm Liouville problem. Antonio Durán discovered a gap in the original proof of completeness for exceptional Hermite polynomials, that has propagated to analogous results for other exceptional families. In this paper we provide an alternative proof that follows essentially the same arguments, but provides a direct proof of the key lemma on which the completeness proof is based. This direct proof makes use of the theory of trivial monodromy potentials developed by Duistermaat and Grünbaum and Oblomkov
  • Thumbnail Image
    Publication
    Cyclic Maya diagrams and rational solutions of higher order Painlevé systems
    (Wiley, 2020-01-22) Gómez Ullate, David; Clarkson, Peter; Grandati, Yves; Milson, Robert; https://ror.org/02jjdwm75
    This paper focuses on the construction of rational solutions for the A2n-Painlev´e system, also called the Noumi-Yamada system, which are considered the higher order generalizations of PIV. In this even case, we introduce a method to construct the rational solutions based on cyclic dressing chains of Schr¨odinger operators with potentials in the class of rational extensions of the harmonic oscillator. Each potential in the chain can be indexed by a single Maya diagram and expressed in terms of a Wronskian determinant whose entries are Hermite polynomials. We introduce the notion of cyclic Maya diagrams and we characterize them for any possible period, using the concepts of genus and interlacing. The resulting classes of solutions can be expressed in terms of special polynomials that generalize the families of generalized Hermite, generalized Okamoto and Umemura polynomials, showing that they are particular cases of a larger family.
  • Thumbnail Image
    Publication
    Shape invariance and equivalence relations for pseudo-Wronskians of Laguerre and Jacobi polynomials
    (Cornell University, 2018-02-15) Gómez Ullate, David; Grandati, Yves; Milson, Robert; https://ror.org/02jjdwm75
    In a previous paper we derived equivalence relations for pseudo-Wronskian determinants of Hermite polynomials. In this paper we obtain the analogous result for Laguerre and Jacobi polynomials. The equivalence formulas are richer in this case since rational Darboux transformations can be defined for four families of seed functions, as opposed to only two families in the Hermite case. The pseudo-Wronskian determinants of Laguerre and Jacobi type will thus depend on two Maya diagrams, while Hermite pseudo-Wronskians depend on just one Maya diagram. We show that these equivalence relations can be interpreted as the general transcription of shape invariance and specific discrete symmetries acting on the parameters of the isotonic oscillator and Darboux-P¨oschl-Teller potential.