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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 \(P_{VI}\).

MSC:

34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
35F20 Nonlinear first-order PDEs
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