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Classical transcendental solutions of the Painlevé equations and their degeneration. (English) Zbl 1087.34063

The author constructs determinant formulae representing classical solutions to the fifth and sixth Painlevé equations and discusses the cascade of subsequent degenerations of the classical solutions of PVI to the classical solutions of PV, PIV, PIII and PII.

Basic ingredients of the construction are the so-called symmetric form of the Painlevé equation and the tau-functions as generating functions to the relevant Hamiltonians. Using the symmetric representation, it is not difficult to extract its Riccati specialization solvable in terms of the classical hypergeometric functions in PVI case or in terms of the confluent hypergeometric functions in PV case. Furthermore, in the symmetric representation, Bäcklund transformations for either Painlevé equation have quite simple form which allows one to construct shift operators in the corresponding parameter space. It is known that the sequence of the shifted tau-functions satisfies the bilinear Toda equation. Applying the Darboux’s formula to the latter equation with the appropriate initial conditions, the author finds the desired determinant formulae.

34M55Painlevé and other special equations; classification, hierarchies
33C05Classical hypergeometric functions, 2 F 1
33C15Confluent hypergeometric functions, Whittaker functions, 1 F 1