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

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 nth Yablonskii-Vorob'ev polynomial equals [(n+1)/2]. We prove thi...

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Bibliographic Details
Main Author:	Pieter Roffelsen
Format:	Article
Language:	English
Published:	National Academy of Science of Ukraine 2012-12-01
Series:	Symmetry, Integrability and Geometry: Methods and Applications
Subjects:	second Painlevé equation rational solutions real roots interlacing of roots Yablonskii-Vorob'ev polynomials
Online Access:	http://dx.doi.org/10.3842/SIGMA.2012.099

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spelling	doaj-4f622ffc13cf44dda8b59f06c797eefb2020-11-24T22:54:19ZengNational Academy of Science of UkraineSymmetry, Integrability and Geometry: Methods and Applications1815-06592012-12-018099On the Number of Real Roots of the Yablonskii-Vorob'ev PolynomialsPieter RoffelsenWe 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 nth Yablonskii-Vorob'ev polynomial equals [(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 nth Yablonskii-Vorob'ev polynomial.http://dx.doi.org/10.3842/SIGMA.2012.099second Painlevé equationrational solutionsreal rootsinterlacing of rootsYablonskii-Vorob'ev polynomials
collection	DOAJ
language	English
format	Article
sources	DOAJ
author	Pieter Roffelsen
spellingShingle	Pieter Roffelsen On the Number of Real Roots of the Yablonskii-Vorob'ev Polynomials Symmetry, Integrability and Geometry: Methods and Applications second Painlevé equation rational solutions real roots interlacing of roots Yablonskii-Vorob'ev polynomials
author_facet	Pieter Roffelsen
author_sort	Pieter Roffelsen
title	On the Number of Real Roots of the Yablonskii-Vorob'ev Polynomials
title_short	On the Number of Real Roots of the Yablonskii-Vorob'ev Polynomials
title_full	On the Number of Real Roots of the Yablonskii-Vorob'ev Polynomials
title_fullStr	On the Number of Real Roots of the Yablonskii-Vorob'ev Polynomials
title_full_unstemmed	On the Number of Real Roots of the Yablonskii-Vorob'ev Polynomials
title_sort	on the number of real roots of the yablonskii-vorob'ev polynomials
publisher	National Academy of Science of Ukraine
series	Symmetry, Integrability and Geometry: Methods and Applications
issn	1815-0659
publishDate	2012-12-01
description	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 nth Yablonskii-Vorob'ev polynomial equals [(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 nth Yablonskii-Vorob'ev polynomial.
topic	second Painlevé equation rational solutions real roots interlacing of roots Yablonskii-Vorob'ev polynomials
url	http://dx.doi.org/10.3842/SIGMA.2012.099
work_keys_str_mv	AT pieterroffelsen onthenumberofrealrootsoftheyablonskiivorobevpolynomials
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On the Number of Real Roots of the Yablonskii-Vorob'ev Polynomials

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