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