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

SIGMA 6 (2010), 095, 11 pages      arXiv:1012.2933     https://doi.org/10.3842/SIGMA.2010.095

### Irrationality of the Roots of the Yablonskii-Vorob'ev Polynomials and Relations between Them

Pieter Roffelsen
Radboud Universiteit Nijmegen, IMAPP, FNWI, Heyendaalseweg 135, 6525 AJ Nijmegen, the Netherlands

Received November 13, 2010, in final form December 08, 2010; Published online December 14, 2010

Abstract
We study the Yablonskii-Vorob'ev polynomials, which are special polynomials used to represent rational solutions of the second Painlevé equation. Divisibility properties of the coefficients of these polynomials, concerning powers of 4, are obtained and we prove that the nonzero roots of the Yablonskii-Vorob'ev polynomials are irrational. Furthermore, relations between the roots of these polynomials for consecutive degree are found by considering power series expansions of rational solutions of the second Painlevé equation.

Key words: second Painlevé equation; rational solutions; power series expansion; irrational roots; Yablonskii-Vorob'ev polynomials.

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