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Poles of Painlevé IV Rationals and their Distribution
Symmetry, integrability and geometry: methods and applications, Tome 14 (2018) Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: Painlevé fourth equation; singularities of Painlevé transcendents; isomonodromic deformations; generalised Hermite polynomials; generalised Okamoto polynomials. 
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     author = {Davide Masoero and Pieter Roffelsen},
     title = {Poles of {Painlev\'e} {IV} {Rationals} and their {Distribution}},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2018_14_a1/}
}
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Davide Masoero; Pieter Roffelsen. Poles of Painlevé IV Rationals and their Distribution. Symmetry, integrability and geometry: methods and applications, Tome 14 (2018). http://geodesic.mathdoc.fr/item/SIGMA_2018_14_a1/

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