Generalized Umemura polynomials

Abstract

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 Painlev\'e $VI$ 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 Painlev\'e $VI$. We will describe a connection between polynomials $U_{n,m}^{(0)}(z,w;a,0)$ and certain Umemura polynomials $U_k(z,w;\alpha,\beta)$.