The Relationship Between Semiclassical Laguerre Polynomials and the Fourth Painlevé Equation

Clarkson, Peter A.; Jordaan, Kerstin

doi:10.1007/s00365-013-9220-4

Cited by 53 publications

(74 citation statements)

References 67 publications

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“…where ( ) is the corresponding pseudo-Wronskian (28)- (30), and ∈ ℤ is the index of . Up to an additive constant, every rational extension of the harmonic oscillator takes the form (33).…”

Section: Definitionmentioning

confidence: 99%

“…The relation between dressing chains of Darboux transformations for the class of operators (33) and flip operations on Maya diagrams is made explicit by the following proposition.…”

Section: Definitionmentioning

confidence: 99%

“…">1.the Yablonskii‐Vorob'ev polynomials arise in the transition behavior for the semiclassical sine‐Gordon equation, in boundary value problems, in moving boundary problems, and in symmetry reductions of a Korteweg capillarity system and cold plasma physics; 2.the generalized Hermite polynomials arise as multiple integrals in random matrix theory, in supersymmetric quantum mechanics, in the description of vortex dynamics with quadrupole background flow, and as coefficients of recurrence relations for orthogonal polynomials; 3.the generalized Okamoto polynomials arise in supersymmetric quantum mechanics and generate rational‐oscillatory solutions of the de‐focusing nonlinear Schrödinger equation; and …”

Section: Introductionmentioning

confidence: 99%

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Cyclic Maya diagrams and rational solutions of higher order Painlevé systems

Clarkson

Gómez‐Ullate

2020

Stud Appl Math

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This paper focuses on the construction of rational solutions for the A2n‐Painlevé 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 Schrödinger 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 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.

show abstract

Section: Definitionmentioning

confidence: 99%

“…The relation between dressing chains of Darboux transformations for the class of operators (33) and flip operations on Maya diagrams is made explicit by the following proposition.…”

Section: Definitionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Cyclic Maya diagrams and rational solutions of higher order Painlevé systems

Clarkson

Gómez‐Ullate

2020

Stud Appl Math

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“…31. We also mention the recent paper 32 where a discussion on the relationship between the recurrence coefficients of orthogonal polynomials with respect to a semiclassical Laguerre weight and classical solutions of Painlevé IV can be found. Krein 33,34 used orthogonal polynomials with matrix coefficients on the real line, and thereafter, they were studied sporadically until the last decade of the 20th century; some relevant papers are Refs.…”

Section: Introductionmentioning

confidence: 97%

“…. We also mention the recent paper where a discussion on the relationship between the recurrence coefficients of orthogonal polynomials with respect to a semiclassical Laguerre weight and classical solutions of Painlevé IV can be found.…”

Section: Introductionmentioning

confidence: 97%

Riemann‐Hilbert problem and matrix discrete Painlevé II systems

Cassatella‐Contra

Mañas

2019

Stud Appl Math

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Matrix Szegő biorthogonal polynomials for quasi‐definite matrices of Hölder continuous weights are studied. A Riemann‐Hilbert problem is uniquely solved in terms of the matrix Szegő polynomials and its Cauchy transforms. The Riemann‐Hilbert problem is given as an appropriate framework for the discussion of the Szegő matrix and the associated Szegő recursion relations for the matrix orthogonal polynomials and its Cauchy transforms. Pearson‐type differential systems characterizing the matrix of weights are studied. These are linear systems of ordinary differential equations that are required to have trivial monodromy. Linear ordinary differential equations for the matrix Szegő polynomials and its Cauchy transforms are derived. It is shown how these Pearson systems lead to nonlinear difference equations for the Verblunsky matrices and two examples, of Fuchsian and non‐Fuchsian type, are considered. For both cases, a new matrix version of the discrete Painlevé II equation for the Verblunsky matrices is found. Reductions of these matrix discrete Painlevé II systems presenting locality are discussed.

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Classification of Laguerre–Hahn orthogonal polynomials of class one

Filipuk

Rebocho

2019

Mathematische Nachrichten

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We study orthogonal polynomials related to Stieltjes functions satisfying Riccati type differential equations with polynomial coefficients,We derive recurrences for the three-term recurrence relation coefficients of the orthogonal polynomials, including connections with some forms of discrete Painlevé equations. K E Y W O R D SLaguerre-Freud equations, orthogonal polynomials, Painlevé equations, Stieltjes function M S C ( 2 0 1 0 ) 33C45, 33C47, 42C05 MOTIVATIONOrthogonal polynomials { ( ) = + lower degree terms} ≥0 on the real line may be fully characterized by the orthogonality relation1) or, equivalently, by a three-term recurrence relation [27] +1 ( ) = ( − ) ( ) − −1 ( ), = 1, 2, … , (1.2) with 0 ( ) = 1, 1 ( ) = − 0 and ≠ 0, ≥ 1. Here, , is the Kronecker delta, and is a positive Borel measure supported on a finite or infinite interval of the real line, , with finite moments 244

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The Relationship Between Semiclassical Laguerre Polynomials and the Fourth Painlevé Equation

Cited by 53 publications

References 67 publications

Cyclic Maya diagrams and rational solutions of higher order Painlevé systems

Cyclic Maya diagrams and rational solutions of higher order Painlevé systems

Riemann‐Hilbert problem and matrix discrete Painlevé II systems

Classification of Laguerre–Hahn orthogonal polynomials of class one

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