Exceptional Orthogonal Polynomials and Rational Solutions to Painlevé Equations

DOI10.1007/978-3-030-36744-2_15zbMath1443.33014arXiv1912.07597OpenAlexW3012164624MaRDI QIDQ3298081

Robert Milson, Daivd Gómez-Ullate

Publication date: 21 July 2020

Published in: Orthogonal Polynomials (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1912.07597

zbMATH Keywords

Sturm-Liouville problemsrational solutionsDarboux transformationsPainlevé equationsexceptional polynomialsclassical polynomialsMaya diagramsWronskian determinantsDarboux dressing chains

Mathematics Subject Classification ID

Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) (33C45) Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies (34M55)

Related Items (6)

Classical and quantum walks on paths associated with exceptional Krawtchouk polynomials ⋮ A family of fourth-order superintegrable systems with rational potentials related to Painlevé VI ⋮ DEK-type orthogonal polynomials and a modification of the Christoffel formula ⋮ Exceptional Hahn and Jacobi polynomials with an arbitrary number of continuous parameters ⋮ Exceptional Jacobi polynomials which are deformations of Jacobi polynomials ⋮ About families of orthogonal polynomials satisfying Heun's differential equation

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Complex SUSY transformations and the Painlevé IV equation
• A conjecture on exceptional orthogonal polynomials
• Admissibility condition for exceptional Laguerre polynomials
• Oscillation theorems for the Wronskian of an arbitrary sequence of eigenfunctions of Schrödinger's equation
• Zeros of exceptional Hermite polynomials
• Solvable rational extensions of the isotonic oscillator
• Two-step Darboux transformations and exceptional Laguerre polynomials
• Multistep DBT and regular rational extensions of the isotonic oscillator
• Exceptional Hahn and Jacobi orthogonal polynomials
• An extension of Bochner's problem: exceptional invariant subspaces
• Studies on the Painlevé equations. III: Second and fourth Painlevé equations, \(P_{II}\) and \(P_{IV}\)
• Dressing chains and the spectral theory of the Schrödinger operator
• Nonlinear chains and Painlevé equations
• Spectral curve, Darboux coordinates and Hamiltonian structure of periodic dressing chains
• Poles of Painlevé IV rationals and their distribution
• Painlevé differential equations in the complex plane
• Painlevé equations -- nonlinear special functions.
• A modification of Crum's method
• Exceptional Meixner and Laguerre orthogonal polynomials
• An extended class of orthogonal polynomials defined by a Sturm-Liouville problem
• A Bochner type characterization theorem for exceptional orthogonal polynomials
• Exceptional Charlier and Hermite orthogonal polynomials
• Universal characters, integrable chains and the Painlevé equations
• Connection between quantum systems involving the fourth Painlevé transcendent and k-step rational extensions of the harmonic oscillator related to Hermite exceptional orthogonal polynomial
• Extensions of solvable potentials with finitely many discrete eigenstates
• Krein–Adler transformations for shape-invariant potentials and pseudo virtual states
• Rational solutions of the Noumi and Yamada system of type $A_4^{(1)}$A4(1)
• Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry
• The Symmetric Fourth Painlevé Hierarchy and Associated Special Polynomials
• Determinant structure of the rational solutions for the Painlevé IV equation
• Symmetries in the fourth Painlevé equation and Okamoto polynomials
• A determinant formula for a class of rational solutions of Painlevé V equation
• The Darboux transformation and algebraic deformations of shape-invariant potentials
• Equidistant spectra of anharmonic oscillators
• Orthogonal Polynomials and Painlevé Equations
• Exceptional Laguerre Polynomials
• Durfee Rectangles and Pseudo‐Wronskian Equivalences for Hermite Polynomials
• Supersymmetry and algebraic Darboux transformations
• Shape invariance and equivalence relations for pseudo-Wronskians of Laguerre and Jacobi polynomials
• The fourth Painlevé equation and associated special polynomials
• Two-step rational extensions of the harmonic oscillator: exceptional orthogonal polynomials and ladder operators
• Cyclic Maya diagrams and rational solutions of higher order Painlevé systems
• Higher order recurrence relation for exceptional Charlier, Meixner, Hermite and Laguerre orthogonal polynomials
• Determinant structure of the rational solutions for the Painlevé II equation
• Rational extensions of the quantum harmonic oscillator and exceptional Hermite polynomials
• New ladder operators for a rational extension of the harmonic oscillator and superintegrability of some two-dimensional systems
• Third-order differential ladder operators and supersymmetric quantum mechanics
• Darboux transformations and the symmetric fourth Painlevé equation
• Monodromy-free Schrödinger operators with quadratically increasing potentials
• Application of the \(\tau\)-function theory of Painlevé equations to random matrices: PIV, PII and the GUE.

This page was built for publication: Exceptional Orthogonal Polynomials and Rational Solutions to Painlevé Equations