Makoto Taneda | Searcharxiv

arXiv2001

Generalized Umemura polynomials and Hirota-Miwa equations

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 $U_k(z,w;α,β)$. Finally we show that after appropriate rescaling, Umemura's polynomials $U_k(z,w;a,b)$ satisfy the Hirota-Miwa bilinear equations.

math.CO↗

arXiv2000

Generalized Umemura polynomials

We introduce and study generalized Umemura polynomials $U_{n,m}^{(k)}(z,w;a,b)$ which are a natural generalization of the Umemura polynomials $U_n(z,w;a,b)$ related to Painlevé $VI$ equation. We will show that if $a=b$, or $a=0$, or $b=0$ then polynomials $U_{n,m}^{(0)}(z,w;a,b)$ generate solutions to Painlevé $VI$. We will describe a connection between polynomials $U_{n,m}^{(0)}(z,w;a,0)$ and certain Umemura polynomials $U_k(z,w;α,β)$.

math.CO↗