Geodesic
On the Number of Real Roots of the Yablonskii–Vorob'ev Polynomials
Symmetry, integrability and geometry: methods and applications, Tome 8 (2012) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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 $n$th Yablonskii–Vorob'ev polynomial equals $\left[\frac{n+1}{2}\right]$. 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 $n$th Yablonskii–Vorob'ev polynomial.
Keywords: second Painlevé equation; rational solutions; real roots; interlacing of roots; Yablonskii–Vorob'ev polynomials. 
@article{SIGMA_2012_8_a98,
     author = {Pieter Roffelsen},
     title = {On the {Number} of {Real} {Roots} of the {Yablonskii{\textendash}Vorob'ev} {Polynomials}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2012},
     volume = {8},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2012_8_a98/}
}
TY  - JOUR
AU  - Pieter Roffelsen
TI  - On the Number of Real Roots of the Yablonskii–Vorob'ev Polynomials
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2012
VL  - 8
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2012_8_a98/
LA  - en
ID  - SIGMA_2012_8_a98
ER  - 
%0 Journal Article
%A Pieter Roffelsen
%T On the Number of Real Roots of the Yablonskii–Vorob'ev Polynomials
%J Symmetry, integrability and geometry: methods and applications
%D 2012
%V 8
%U http://geodesic.mathdoc.fr/item/SIGMA_2012_8_a98/
%G en
%F SIGMA_2012_8_a98
Pieter Roffelsen. On the Number of Real Roots of the Yablonskii–Vorob'ev Polynomials. Symmetry, integrability and geometry: methods and applications, Tome 8 (2012). http://geodesic.mathdoc.fr/item/SIGMA_2012_8_a98/

[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 | MR | Zbl

[2] Clarkson P. A., Mansfield E. L., “The second Painlevé equation, its hierarchy and associated special polynomials”, Nonlinearity, 16 (2003), R1–R26 | DOI | MR | Zbl

[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 | MR | Zbl

[4] Kaneko M., Ochiai H., “On coefficients of Yablonskii–Vorob'ev polynomials”, J. Math. Soc. Japan, 55 (2003), 985–993, arXiv: math.QA/0205178 | DOI | MR | Zbl

[5] Roffelsen P., “Irrationality of the roots of the Yablonskii–Vorob'ev polynomials and relations between them”, SIGMA, 6 (2010), 095, 11 pp., arXiv: arXiv:1012.2933[math.CA] | DOI | MR | Zbl

[6] Taneda M., “Remarks on the Yablonskii–Vorob'ev polynomials”, Nagoya Math. J., 159 (2000), 87–111 | MR | Zbl

[7] Vorob'ev A. P., “On the rational solutions of the second Painlevé equation”, Differ. Uravn., 1 (1965), 79–81 | MR

[8] Yablonskii A. I., “On rational solutions of the second Painlevé equation”, Vesti AN BSSR, Ser. Fiz.-Tech. Nauk, 1959, no. 3, 30–35