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.
Keywords:
second Painlevé equation; rational solutions; power series expansion; irrational roots; Yablonskii–Vorob'ev polynomials.
Received:November 13, 2010; in final form December 8, 2010; Published online December 14, 2010
Citation:
Pieter Roffelsen, “Irrationality of the Roots of the Yablonskii–Vorob'ev Polynomials and Relations between Them”, SIGMA, 6 (2010), 095, 11 pp.
\Bibitem{Rof10}
\by Pieter Roffelsen
\paper Irrationality of the Roots of the Yablonskii--Vorob'ev Polynomials and Relations between Them
\jour SIGMA
\yr 2010
\vol 6
\papernumber 095
\totalpages 11
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\crossref{https://doi.org/10.3842/SIGMA.2010.095}
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Linking options:
https://www.mathnet.ru/eng/sigma553
https://www.mathnet.ru/eng/sigma/v6/p95
This publication is cited in the following 4 articles:
Davide Masoero, Pieter Roffelsen, “Poles of Painlevé IV Rationals and their Distribution”, SIGMA, 14 (2018), 002, 49 pp.
Peter D. Miller, Yue Sheng, “Rational Solutions of the Painlevé-II Equation Revisited”, SIGMA, 13 (2017), 065, 29 pp.
Buckingham R.J. Miller P.D., “Large-Degree Asymptotics of Rational Painlevé-II Functions: Noncritical Behaviour”, Nonlinearity, 27:10 (2014), 2489–2577
Pieter Roffelsen, “On the Number of Real Roots of the Yablonskii–Vorob'ev Polynomials”, SIGMA, 8 (2012), 099, 9 pp.
Citation in format AMSBIB
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