
SIGMA 6 (2010), 095, 11 pages arXiv:1012.2933
http://dx.doi.org/10.3842/SIGMA.2010.095
Irrationality of the Roots of the YablonskiiVorob'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 YablonskiiVorob'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 YablonskiiVorob'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; YablonskiiVorob'ev polynomials.
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