Generalized Umemura polynomials and Hirota-Miwa equations

Details Generalized Umemura polynomials and Hirota-Miwa equations Generalized Umemura polynomials and Hirota-Miwa equations

2001-06-05

arxiv.org/abs/math/0106025v1

Mathematics

Combinatorics

19 pages

Scientific paper

We introduce and study generalized Umemura polynomials $U_{n,m}^{(k)}(z,w;a,b)$ which are the natural generalization of the Umemura polynomials $U_n(z,w;a,b)$ related to the Painleve VI equation. We show that if either a=b, or a=0, or b=0, then polynomials $U_{n,m}^{(0)}(z,w;a,b)$ generate solutions to the Painleve VI equation. We give new proof of Noumi-Okada-Okamoto-Umemura conjecture, and describe connections between polynomials $U_{n,m}^{(0)}(z,w;a,0)$ and certain Umemura polynomials $U_k(z,w;\alpha,\beta)$. Finally we show that after appropriate rescaling, Umemura's polynomials $U_k(z,w;a,b)$ satisfy the Hirota-Miwa bilinear equations.

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