Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 002, 49 pages      arXiv:1707.05222      https://doi.org/10.3842/SIGMA.2018.002
Contribution to the Special Issue on Painlevé Equations and Applications in Memory of Andrei Kapaev

Poles of Painlevé IV Rationals and their Distribution

Davide Masoero a and Pieter Roffelsen b
a) Grupo de Física Matemática e Departamento de Matemática da Universidade de Lisboa, Campo Grande Edifício C6, 1749-016 Lisboa, Portugal
b) School of Mathematics and Statistics F07, The University of Sydney, NSW 2006, Australia

Received July 20, 2017, in final form December 18, 2017; Published online January 06, 2018

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.

Key words: Painlevé fourth equation; singularities of Painlevé transcendents; isomonodromic deformations; generalised Hermite polynomials; generalised Okamoto polynomials.

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