Generalized Umemura polynomials

Details Generalized Umemura polynomials Generalized Umemura polynomials

2000-10-28

arxiv.org/abs/math/0010279v1

Mathematics

Combinatorics

10 pages

Scientific paper

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 $U_k(z,w;\alpha,\beta)$.

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