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 SIGMA, 2012, Volume 8, 099, 9 pages (Mi sigma776)

On the Number of Real Roots of the Yablonskii–Vorob'ev Polynomials

Pieter Roffelsen

Radboud University Nijmegen, IMAPP, FNWI, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands

Abstract: We study the real roots of the Yablonskii–Vorob'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 $n$th Yablonskii–Vorob'ev polynomial equals $[\frac{n+1}{2}]$. We prove this conjecture using an interlacing property between the roots of the Yablonskii–Vorob'ev polynomials. Furthermore we determine precisely the number of negative and the number of positive real roots of the $n$th Yablonskii–Vorob'ev polynomial.

Keywords: second Painlevé equation; rational solutions; real roots; interlacing of roots; Yablonskii–Vorob'ev polynomials

DOI: https://doi.org/10.3842/SIGMA.2012.099

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ArXiv: 1208.2337
MSC: 34M55
Received: August 14, 2012; in final form December 7, 2012; Published online December 14, 2012
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Citation: Pieter Roffelsen, “On the Number of Real Roots of the Yablonskii–Vorob'ev Polynomials”, SIGMA, 8 (2012), 099, 9 pp.

Citation in format AMSBIB
\Bibitem{Rof12} \by Pieter~Roffelsen \paper On the Number of Real Roots of the Yablonskii--Vorob'ev Polynomials \jour SIGMA \yr 2012 \vol 8 \papernumber 099 \totalpages 9 \mathnet{http://mi.mathnet.ru/sigma776} \crossref{https://doi.org/10.3842/SIGMA.2012.099} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3007260} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000312436900001} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84871477585} 

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This publication is cited in the following articles:
1. R. J. Buckingham, P. D. Miller, “Large-degree asymptotics of rational Painlevé-II functions: noncritical behaviour”, Nonlinearity, 27:10 (2014), 2489–2577
2. Peter D. Miller, Yue Sheng, “Rational Solutions of the Painlevé-II Equation Revisited”, SIGMA, 13 (2017), 065, 29 pp.
3. Davide Masoero, Pieter Roffelsen, “Poles of Painlevé IV Rationals and their Distribution”, SIGMA, 14 (2018), 002, 49 pp.
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