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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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Інститут математики НАН України
2012
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| Series: | Symmetry, Integrability and Geometry: Methods and Applications |
| Online Access: | http://dspace.nbuv.gov.ua/handle/123456789/149188 |
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| Summary: | 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. |
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