Constructive Approximation

, Volume 39, Issue 1, pp 85–99 | Cite as

Distributions of Poles to Painlevé Transcendents via Padé Approximations

  • V. Y. NovokshenovEmail author


A version of the Fair–Luke algorithm has been used to find the Padé approximate solutions to the Painlevé I, II, and IV equations. The distributions of poles in the complex plane are studied to check the dynamics of movable poles and the emergence of rational and truncated solutions, as well as various patterns formed by the poles. The high-order approximations allow us to check asymptotic expansions at infinity and estimate the range of asymptotic domains. The Coulomb gas interpretation of the pole ensembles is discussed in view of the patterns arising in Painlevé IV transcendents.


Painlevé equations Meromorphic solutions Distribution of poles Padé approximations Continued fractions Riemann–Hilbert problem Stieltjes relations Coulomb gas 

Mathematics Subject Classification (2000)

30D35 30E10 33E17 34M55 34M60 41A21 


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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Institute of Mathematics, Ufa Scientific CenterRussian Academy of SciencesUfaRussia

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