
SIGMA 8 (2012), 099, 9 pages arXiv:1208.2337
http://dx.doi.org/10.3842/SIGMA.2012.099
On the Number of Real Roots of the YablonskiiVorob'ev Polynomials
Pieter Roffelsen
Radboud Universiteit Nijmegen, IMAPP, FNWI, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands
Received August 14, 2012, in final form December 07, 2012; Published online December 14, 2012
Abstract
We study the real roots of the YablonskiiVorob'ev polynomials, which are special polynomials used to represent rational solutions of the second Painlevé equation.
It has been conjectured that the number of real roots of the nth YablonskiiVorob'ev polynomial equals [(n+1)/2].
We prove this conjecture using an interlacing property between the roots of the YablonskiiVorob'ev polynomials.
Furthermore we determine precisely the number of negative and the number of positive real roots of the nth YablonskiiVorob'ev polynomial.
Key words:
second Painlevé equation; rational solutions; real roots; interlacing of roots; YablonskiiVorob'ev polynomials.
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