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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| Format: | Article |
| Language: | English |
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National Academy of Science of Ukraine
2012-12-01
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| Series: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Online Access: | http://dx.doi.org/10.3842/SIGMA.2012.099 |
| _version_ | 1819138827336286208 |
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| author | Pieter Roffelsen |
| author_facet | Pieter Roffelsen |
| author_sort | Pieter Roffelsen |
| collection | DOAJ |
| 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. |
| first_indexed | 2024-12-22T11:12:57Z |
| format | Article |
| id | doaj.art-4f622ffc13cf44dda8b59f06c797eefb |
| institution | Directory Open Access Journal |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2024-12-22T11:12:57Z |
| publishDate | 2012-12-01 |
| publisher | National Academy of Science of Ukraine |
| record_format | Article |
| series | Symmetry, Integrability and Geometry: Methods and Applications |
| spelling | doaj.art-4f622ffc13cf44dda8b59f06c797eefb2022-12-21T18:28:07ZengNational 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 |
| 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 |
| title | 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_short | 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 |
| 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 |