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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.

##### MSC:
 34M55 Painlevé and other special equations; classification, hierarchies 33C05 Classical hypergeometric functions, ${}_2F_1$ 33C15 Confluent hypergeometric functions, Whittaker functions, ${}_1F_1$
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##### References:
 [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover Publications, New York, 1964. · Zbl 0171.38503 [2] Y. Haraoka, Confluence of cycles for hypergeometric functions on $Z_2,n+1$, Trans. Amer. Math. Soc. 349 (1997), 675--712. · Zbl 0885.33008 · doi:10.1090/S0002-9947-97-01471-2 [3] K. Iwasaki, H. Kimura, S. Shimomura and M. Yoshida, From Gauss to Painlevé -- A Modern Theory of Special Functions, Aspects of Mathematics E16, Vieweg, 1991. · Zbl 0743.34014 [4] K. Kajiwara, T. Masuda, M. Noumi, Y. Ohta and Y. Yamada, Determinant formulas for the Toda and discrete Toda equations, Funkcial. Ekvac. 44 (2001), 291--307. · Zbl 1145.37327 [5] K. Kajiwara and Y. Ohta, Determinant structure of the rational solutions for the Painlevé IV equation, J. Phys. A 31 (1998), 2431--2446. · Zbl 0919.34008 · doi:10.1088/0305-4470/31/10/017 [6] M. Noumi, Painlevé equations through symmetry, American Mathematical Society, 2004. · Zbl 1077.34003 [7] M. Noumi and Y. Yamada, Symmetries in the fourth Painlevé equation and Okamoto polynomials, Nagoya Math. J. 153 (1999), 53--86. · Zbl 0932.34088 [8] M. Noumi and Y. Yamada, Higher order Painlevé equations of type $A_l^(1)$, Funkcial. Ekvac. 41 (1998), 483--503. · Zbl 1140.34303 [9] M. Noumi and Y. Yamada, Affine Weyl groups, discrete dynamical systems and Painlevé equations, Comm. Math. Phys. 199 (1998), 281--295. · Zbl 0952.37031 · doi:10.1007/s002200050502 [10] M. Noumi and Y. Yamada, A new Lax pair for the sixth Painlevé equation associated with $\wh\mathfrakso(8)$, In: Microlocal Analysis and Complex Fourier Analysis (Eds. T. Kawai and K. Fujita), 238--252, World Scientific, 2002. · Zbl 1047.34105 [11] K. Okamoto, Studies on the Painlevé equations I, sixth Painlevé equation P$_\rm VI$, Ann. Mat. Pura Appl. (4) 146 (1987), 337--381. · Zbl 0637.34019 · doi:10.1007/BF01762370 [12] K. Okamoto, Studies on the Painlevé equations II, fifth Painlevé equation P$_\rm V$, Japan J. Math. 13 (1987), 47--76. · Zbl 0694.34005 [13] K. Okamoto, Studies on the Painlevé equations III, second and fourth Painlevé equations, P$_\rm II$ and P$_\rm IV$, Math. Ann. 275 (1986), 222--254. · Zbl 0589.58008 · doi:10.1007/BF01458459 · eudml:164145 [14] K. Okamoto, Studies on the Painlevé equations IV, third Painlevé equation P$_\rm III$, Funkcial. Ekvac. 30 (1987), 305--332. · Zbl 0639.58013 [15] P. Painlevé, Sur les équations différentielles du second ordre à points critiques fixes, C. R. Acad. Sci. Paris 143 (1906), 1111--1117. · Zbl 37.0341.04