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

SIGMA 6 (2010), 095, 11 pages      arXiv:1012.2933     https://doi.org/10.3842/SIGMA.2010.095

Irrationality of the Roots of the Yablonskii-Vorob'ev Polynomials and Relations between Them

Pieter Roffelsen
Radboud Universiteit Nijmegen, IMAPP, FNWI, Heyendaalseweg 135, 6525 AJ Nijmegen, the Netherlands

Received November 13, 2010, in final form December 08, 2010; Published online December 14, 2010

Abstract
We study the Yablonskii-Vorob'ev polynomials, which are special polynomials used to represent rational solutions of the second Painlevé equation. Divisibility properties of the coefficients of these polynomials, concerning powers of 4, are obtained and we prove that the nonzero roots of the Yablonskii-Vorob'ev polynomials are irrational. Furthermore, relations between the roots of these polynomials for consecutive degree are found by considering power series expansions of rational solutions of the second Painlevé equation.

Key words: second Painlevé equation; rational solutions; power series expansion; irrational roots; Yablonskii-Vorob'ev polynomials.

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References

  1. Yablonskii A.I., On rational solutions of the second Painlevé equation, Vesti AN BSSR, Ser. Fiz.-Tech. Nauk (1959), no. 3, 30-35 (in Russian).
  2. Vorob'ev A.P., On the rational solutions of the second Painlevé equations, Differ. Uravn. 1 (1965), 79-81 (in Russian).
  3. Gambier B., Sur les équations différentielles du second ordre et du premier degre dont l'intégrale est á points critiques fixes, Acta Math. 33 (1909), 1-55.
  4. Lukashevich N.A., The second Painlevé equation, Differ. Uravn. 7 (1971), 1124-1125 (in Russian).
  5. Noumi M., Painlevé equations through symmetry, Translations of Mathematical Monographs, Vol. 223, American Mathematical Society, Providence, RI, 2004.
  6. Airault H., McKean H.P., Moser J., Rational and elliptic solutions of the Korteweg-de Vries equation and related many-body problems, Pure Appl. Math. 30 (1977), 95-148.
  7. Clarkson P.A., Rational solutions of the Boussinesq equation, Anal. Appl. (Singap.) 6 (2008), 349-369.
  8. Clarkson P.A., Mansfield E.L., The second Painlevé equation, its hierarchy and associated special polynomials, Nonlinearity 16 (2003), R1-R26.
  9. Taneda M., Remarks on the Yablonskii-Vorob'ev polynomials, Nagoya Math. J. 159 (2000), 87-111.
  10. 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.
  11. Kaneko M., Ochiai H., On coefficients of Yablonskii-Vorob'ev polynomials, J. Math. Soc. Japan 55 (2003), 985-993, math.QA/0205178.
  12. Kametaka Y., On the irreducibility conjecture based on computer calculation for Yablonskii-Vorob'ev polynomials which give a rational solution of the Toda equation of Painlevé-II type, Japan J. Appl. Math. 2 (1985), 241-246.
  13. Kudryashov N.A., Demina M.V., Relations between zeros of special polynomials associated with the Painlevé equations, Phys. Lett. A 368 (2007), 227-234, nlin.SI/0610058.

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