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.
CITATION STYLE
Kirillov, A. N., & Taneda, M. (2002). Generalized Umemura Polynomials and the Hirota-Miwa Equation. In MathPhys Odyssey 2001 (pp. 313–331). Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-0087-1_11
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