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On a class of algebraic solutions to the Painlevé VI equation, its determinant formula and coalescence cascade. (English) Zbl 1151.34340
The author considers a certain class of algebraic solutions of the sixth Painlevé equation $P_{VI}$ (in Hamiltonian form), for which he presents a determinant formula. The entries of the determinant are essentially the Jacobi polynomials. The well known fact that each of the Painlevé equations can be obtained from $P_{VI}$ by a coalescence procedure is then used to obtain, from this family of algebraic solutions of $P_{VI}$, rational solutions of $P_{V}$, $P_{III}$ and $P_{II}$. Finally, the author considers the connection with the Umemura polynomials for $P_{VI}$.

34M55Painlevé and other special equations; classification, hierarchies
35F20General theory of first order nonlinear PDE
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