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

SIGMA 14 (2018), 095, 28 pages      arXiv:1803.11230      https://doi.org/10.3842/SIGMA.2018.095
Contribution to the Special Issue on Painlevé Equations and Applications in Memory of Andrei Kapaev

### Tronquée Solutions of the Third and Fourth Painlevé Equations

Xiaoyue Xia
Department of Mathematics, The Ohio State University, 100 Math Tower, 231 West 18th Avenue, Columbus OH, 43210-1174, USA

Received April 04, 2018, in final form August 30, 2018; Published online September 08, 2018

Abstract
Recently in a paper by Lin, Dai and Tibboel, it was shown that the third and fourth Painlevé equations have tronquée and tritronquée solutions. We obtain global information about these tronquée and tritronquée solutions. We find their sectors of analyticity, their Borel summed representations in these sectors as well as the asymptotic position of the singularities near the boundaries of the analyticity sectors. We also correct slight errors in the paper mentioned.

Key words: the third and fourth Painlevé equations; asymptotic position of singularities; tronquée solutions; tritronquée solutions; Borel summed representation.

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