Tohoku Mathematical Journal

Classical transcendental solutions of the Painlevé equations and their degeneration

Tetsu Masuda

Full-text: Open access

Abstract

We present a determinant expression for a family of classical transcendental solutions of the Painlevé V and the Painlevé VI equation. Degeneration of these solutions along the process of coalescence for the Painlevé equations is discussed.

Article information

Source
Tohoku Math. J. (2) Volume 56, Number 4 (2004), 467-490.

Dates
First available in Project Euclid: 11 April 2005

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1113246745

Digital Object Identifier
doi:10.2748/tmj/1113246745

Mathematical Reviews number (MathSciNet)
MR2097156

Zentralblatt MATH identifier
1087.34063

Subjects
Primary: 34M55: Painlevé and other special equations; classification, hierarchies;
Secondary: 33C05: Classical hypergeometric functions, $_2F_1$ 33C15: Confluent hypergeometric functions, Whittaker functions, $_1F_1$ 33E17: Painlevé-type functions

Citation

Masuda, Tetsu. Classical transcendental solutions of the Painlevé equations and their degeneration. Tohoku Math. J. (2) 56 (2004), no. 4, 467--490. doi:10.2748/tmj/1113246745. https://projecteuclid.org/euclid.tmj/1113246745


Export citation

References

  • M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover Publications, New York, 1964.
  • Y. Haraoka, Confluence of cycles for hypergeometric functions on $Z_2,n+1$, Trans. Amer. Math. Soc. 349 (1997), 675--712.
  • 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.
  • 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.
  • K. Kajiwara and Y. Ohta, Determinant structure of the rational solutions for the Painlevé IV equation, J. Phys. A 31 (1998), 2431--2446.
  • M. Noumi, Painlevé equations through symmetry, American Mathematical Society, 2004.
  • M. Noumi and Y. Yamada, Symmetries in the fourth Painlevé equation and Okamoto polynomials, Nagoya Math. J. 153 (1999), 53--86.
  • M. Noumi and Y. Yamada, Higher order Painlevé equations of type $A_l^(1)$, Funkcial. Ekvac. 41 (1998), 483--503.
  • M. Noumi and Y. Yamada, Affine Weyl groups, discrete dynamical systems and Painlevé equations, Comm. Math. Phys. 199 (1998), 281--295.
  • 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.
  • K. Okamoto, Studies on the Painlevé equations I, sixth Painlevé equation P$_\rm VI$, Ann. Mat. Pura Appl. (4) 146 (1987), 337--381.
  • K. Okamoto, Studies on the Painlevé equations II, fifth Painlevé equation P$_\rm V$, Japan J. Math. 13 (1987), 47--76.
  • 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.
  • K. Okamoto, Studies on the Painlevé equations IV, third Painlevé equation P$_\rm III$, Funkcial. Ekvac. 30 (1987), 305--332.
  • P. Painlevé, Sur les équations différentielles du second ordre à points critiques fixes, C. R. Acad. Sci. Paris 143 (1906), 1111--1117.