Poles of Painlevé IV Rationals and their Distribution

Abstract

We study the distribution of singularities (poles and zeros) of rational solutions of the Painlevé IV equation by means of the isomonodromic deformation method. Singularities are expressed in terms of the roots of generalised Hermite $H_{m,n}$ and generalised Okamoto $Q_{m,n}$ polynomials. We show that roots of generalised Hermite and Okamoto polynomials are described by an inverse monodromy problem for an anharmonic oscillator of degree two. As a consequence they turn out to be classified by the monodromy representation of a class of meromorphic functions with a finite number of singularities introduced by Nevanlinna. We compute the asymptotic distribution of roots of the generalised Hermite polynomials in the asymptotic regime when $m$ is large and $n$ fixed.

References in corpus (7)

• Universality for mathematical and physical systems
• Poles of Integrale Tritronquee and Anharmonic Oscillators. Asymptotic localization from WKB analysis
• Movable algebraic singularities of second-order ordinary differential equations
• Y-System and Deformed Thermodynamic Bethe Ansatz
• On the Number of Real Roots of the Yablonskii-Vorob'ev Polynomials
• Irrationality of the Roots of the Yablonskii-Vorob'ev Polynomials and Relations between Them
• Large-degree asymptotics of rational Painleve-IV functions associated to generalized Hermite polynomials

Cited by in corpus (7)

• Quantum spectral problems and isomonodromic deformations
• Coefficients of Wronskian Hermite polynomials
• Asymptotic behavior of Wronskian polynomials that are factorized via $p$-cores and $p$-quotients
• On symmetric solutions of the fourth $q$-Painlevé equation
• Lectures on exceptional orthogonal polynomials and rational solutions to Painlevé equations
• Complete classification of rational solutions of $A_{2n}$-Painlevé systems
• Isomonodromy sets of accessory parameters for Heun class equations