Notes on Umemura polynomials

Noumi, Masatoshi

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume à la mémoire de Hiroshi Umemura : “Équations de Painlevé et théories de Galois différentielles”, Volume 29 (2020) no. 5, pp. 1091-1118.

We give a survey on special polynomials associated with algebraic solutions of the sixth Painlevé equation, and formulate a conjecture regarding a combinatorial formula for Umemura polynomials associated with a class of algebraic solutions of $P_{\mathrm{VI}}$ with two discrete parameters.

Nous donnons un survol des polynômes spéciaux associés aux solutions algébriques de la sixième équation de Painlevé, et formulons une conjecture concernant une formule combinatoire pour les polynômes d’Umemura associés à une classe de solutions algébriques de $P_{\mathrm{VI}}$ avec deux paramètres discrets.

Classification: 33C45, 05E35, 34M55
Mots-clés : Umemura polynomial, Painlevé equation, Toda equation, special polynomial

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