Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 099, 9 pages      arXiv:1208.2337

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

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.

Key words: second Painlevé equation; rational solutions; real roots; interlacing of roots; Yablonskii-Vorob'ev polynomials.

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  1. Clarkson P.A., Special polynomials associated with rational solutions of the Painlevé equations and applications to soliton equations, Comput. Methods Funct. Theory 6 (2006), 329-401.
  2. Clarkson P.A., Mansfield E.L., The second Painlevé equation, its hierarchy and associated special polynomials, Nonlinearity 16 (2003), R1-R26.
  3. Fukutani S., Okamoto K., Umemura H., Special polynomials and the Hirota bilinear relations of the second and the fourth Painlevé equations, Nagoya Math. J. 159 (2000), 179-200.
  4. Kaneko M., Ochiai H., On coefficients of Yablonskii-Vorob'ev polynomials, J. Math. Soc. Japan 55 (2003), 985-993, math.QA/0205178.
  5. Roffelsen P., Irrationality of the roots of the Yablonskii-Vorob'ev polynomials and relations between them, SIGMA 6 (2010), 095, 11 pages, arXiv:1012.2933.
  6. Taneda M., Remarks on the Yablonskii-Vorob'ev polynomials, Nagoya Math. J. 159 (2000), 87-111.
  7. Vorob'ev A.P., On the rational solutions of the second Painlevé equation, Differ. Uravn. 1 (1965), 79-81.
  8. Yablonskii A.I., On rational solutions of the second Painlevé equation, Vesti AN BSSR, Ser. Fiz.-Tech. Nauk (1959), no. 3, 30-35.

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