Special polynomials associated with the Painlevé equations I
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 29 (2020) no. 5, pp. 1063-1089.

The Painlevé equations have rational or algebraic solutions on special parameters. We can find rational or algebraic solutions of the Painlevé equations as fixed points of the Bäcklund transformations. The τ function of the rational or algebraic solution can be written as the product of a special polynomial and an exponential factor. Since a series of τ functions satisfies the Toda equation, we obtain a recursive relation of the special polynomials. The coefficients of the special polynomials for the sixth Painlevé equation are described by the Young diagram.

(The original manuscript by the author was submitted to the proceeding of the Montreal conference in 1996, which were not published. The abstract was not part of the original manuscript and has not been written by the author.)

Pour certaines valeurs spéciales des paramètres, les équations de Painlevé ont des solutions algébriques ou rationnelles. Elles sont associées aux points fixes des transformations de Bäcklund. La fonction τ de la solution rationnelle ou algébrique peut alors être écrite comme un produit de polynômes spéciaux et d’un facteur exponentiel. Puisque une série de fonctions τ satisfait l’équation de Toda, nous obtenons une relation de récurrence pour les polynômes spéciaux. Pour la sixième équation de Painlevé les coefficients des polynômes spéciaux sont décrits à l’aide de diagrammes de Young.

(Le manuscrit original a été soumis aux comptes-rendus de la conférence de Montréal en 1996, qui n’ont pas été publiés. Le résumé ne faisait pas partie du manuscrit original et il n’a pas été rédigé par l’auteur.)

Published online:
DOI: 10.5802/afst.1657
Umemura, Hiroshi 1

1 Graduate School of Polymathematics, Nagoya University
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Umemura, Hiroshi. Special polynomials associated with the Painlevé equations I. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 29 (2020) no. 5, pp. 1063-1089. doi : 10.5802/afst.1657. http://archive.numdam.org/articles/10.5802/afst.1657/

[1] Noumi, Masatoshi; Okada, Soichi; Okamoto, Kazuo; Umemura, Hiroshi Special polynomials associated with the Painlevé equations. II, Integrable systems and algebraic geometry (Kobe/Kyoto, 1997), World Scientific, 1998, pp. 349-372

[2] Noumi, Masatoshi; Okamoto, Kazuo On the Umemura polynomials (in preparation)

[3] Okamoto, Kazuo Studies on the Painlevé equations III, Math. Ann., Volume 275 (1986), pp. 221-255

[4] Okamoto, Kazuo Studies on the Painlevé equations I, Ann. Mat. Pura Appl., Volume 146 (1987), pp. 337-381

[5] Okamoto, Kazuo Studies on the Painlevé equations II, Jap. J. Math., Volume 13 (1987), pp. 47-76

[6] Okamoto, Kazuo Studies on the Painlevé equations IV, Funk. Ekv., Volume 30 (1987), pp. 305-332

[7] Umemura, Hiroshi Irreducibility of the Painlevé equations - Evolution in the past 100 years “in this volume”. published as “100 years of the Painlevé equation”, Sûgaku 51 (1999), no. 4, 395–420

[8] Umemura, Hiroshi Special polynomials associated with the Painlevé equations II (in preparation)

[9] Umemura, Hiroshi On the irreducibility of the first differential equation of Painlevé, Algebraic geometry and Commutative algebra in honor of Masayoshi Nagata, Konokuniya Company Ltd, 1988, pp. 101-119 | Zbl

[10] Umemura, Hiroshi; Watanabe, Humihiko Solutions of the second and fourth Painlevé equations I, Nagoya Math. J., Volume 148 (1997), pp. 151-198

[11] Vorobʼev, A. P. On rational solutions of the second Painlevé equation, Differ. Uravn, Volume 1 (1965), pp. 58-59

[12] Whittaker, Edmund T.; Watson, George N. A course of modern analysis, Cambridge University Press, 1935

[13] Yablonskii, A. I. On rational solutions of the second Painlevé equation, Vesti A.N. BSSR., Ser. Fiz-Tekh. Nauk., Volume 3 (1959), pp. 30-35 (in Russian)

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