Generalized Umemura polynomials and Hirota-Miwa equations

الأصل · EN

We introduce and study generalized Umemura polynomials Uₙ,ₘ⁽ᵏ⁾(z,w;a,b) which are the natural generalization of the Umemura polynomials Uₙ(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 Uₙ,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 Uₙ,ₘ⁽⁰⁾(z,w;a,0) and certain Umemura polynomials Uₖ(z,w;α,β). Finally we show that after appropriate rescaling, Umemura's polynomials Uₖ(z,w;a,b) satisfy the Hirota-Miwa bilinear equations.