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