Generalized Umemura polynomials

الأصل · EN

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