349 citations · 2.2k across the 57 of their papers we have counts for
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Complete classification of rational solutions of -Painlevé systems
David Gómez-Ullate, Yves Grandati, Robert Milson
We provide a complete classification and an explicit representation of rational solutions to the fourth Painlevé equation PIV and its higher order generalizations known as the $A_{…
Constructing Coherent States for the Rational Extensions of the Harmonic Oscillator Potential
Zoé McIntyre, Robert Milson
Using the formalism of Maya diagrams and ladder operators, we describe the algebra of annihilating operators for the class of rational extensions of the harmonic oscillators. This…
Rational solutions of Painleve systems
David Gomez-Ullate, Yves Grandati, Robert Milson
Although the solutions of Painlevé equations are transcendental in the sense that they cannot be expressed in terms of known elementary functions, there do exist rational solutions…
Lectures on exceptional orthogonal polynomials and rational solutions to Painlevé equations
David Gómez-Ullate, Robert Milson
These are the lecture notes for a course on exceptional polynomials taught at the \textit{AIMS-Volkswagen Stiftung Workshop on Introduction to Orthogonal Polynomials and Applicatio…
Ladder operators and rational extensions
David Gomez-Ullate, Yves Grandati, Zoe McIntyre +1
This note presents the classification of ladder operators corresponding to the class of rational extensions of the harmonic oscillator. We show that it is natural to endow the clas…
Rational solutions of dressing chains and higher order Painleve equations
D. Gomez-Ullate, Y. Grandati, S. Lombardo +1
We present a new approach to determine the rational solutions of the higher order Painleve equations associated to periodic dressing chain systems. We obtain new sets of solutions,…
Cyclic Maya diagrams and rational solutions of higher order Painlevé systems
Peter A. Clarkson, David Gómez-Ullate, Yves Grandati +1
This paper focuses on the construction of rational solutions for the Painlevé system, also called the Noumi-Yamada system, which are considered the higher order generaliza…
Rational extensions of the quantum harmonic oscillator and exceptional Hermite polynomials
David Gomez-Ullate, Yves Grandati, Robert Milson
We prove that every rational extension of the quantum harmonic oscillator that is exactly solvable by polynomials is monodromy free, and therefore can be obtained by applying a fin…
A conjecture on Exceptional Orthogonal Polynomials
David Gomez-Ullate, Niky Kamran, Robert Milson
Exceptional orthogonal polynomial systems (X-OPS) arise as eigenfunctions of Sturm-Liouville problems and generalize in this sense the classical families of Hermite, Laguerre and J…
Two-step Darboux transformations and exceptional Laguerre polynomials
David Gomez-Ullate, Niky Kamran, Robert Milson
It has been recently discovered that exceptional families of Sturm-Liouville orthogonal polynomials exist, that generalize in some sense the classical polynomials of Hermite, Lague…
On orthogonal polynomials spanning a non-standard flag
David Gomez-Ullate, Niky Kamran, Robert Milson
We survey some recent developments in the theory of orthogonal polynomials defined by differential equations. The key finding is that there exist orthogonal polynomials defined by…
Exceptional orthogonal polynomials and the Darboux transformation
David Gomez-Ullate, Niky Kamran, Robert Milson
We adapt the notion of the Darboux transformation to the context of polynomial Sturm-Liouville problems. As an application, we characterize the recently described Laguerre po…
An extended class of orthogonal polynomials defined by a Sturm-Liouville problem
David Gomez-Ullate, Niky Kamran, Robert Milson
We present two infinite sequences of polynomial eigenfunctions of a Sturm-Liouville problem. As opposed to the classical orthogonal polynomial systems, these sequences start with a…
An Extension of Bochner's Problem: Exceptional Invariant Subspaces
David Gomez-Ullate, Niky Kamran, Robert Milson
A classical result due to Bochner characterizes the classical orthogonal polynomial systems as solutions of a second-order eigenvalue equation. We extend Bochner's result by droppi…
Spectral residues of second-order differential equations: towards a new class of summation identities and inversion formulas
Robert Milson
The present article deals with differential equations with spectral parameter from the point of view of formal power series. The treatment does not make use of the notion of eigenv…