Special polynomials associated with rational solutions of the fifth Painlevé equation

Journal of Computational and Applied Mathematics

Volumn 178, Issue 1-2 SPEC. ISS., 2005, Pages 111-129

  Clarkson, Peter A ^a  

^a UNIVERSITY OF KENT   (United Kingdom)

Author keywords

Painlev equations; Rational solutions; Toda equations

Indexed keywords

DIFFERENCE EQUATIONS; INTEGRATION; LINEAR ALGEBRA; LINEAR EQUATIONS; ORDINARY DIFFERENTIAL EQUATIONS; POLES AND ZEROS; THEOREM PROVING;

BILINEAR DIFFERENTIAL-DIFFERENCE EQUATIONS; PAINLEVE EQUATIONS; RATIONAL SOLUTIONS; TODA EQUATIONS;

POLYNOMIAL APPROXIMATION;

EID: 14844357950     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cam.2004.04.015     Document Type: Article

References (52)

1. M.J. Ablowitz, P.A. Clarkson, Solitons, Nonlinear evolution equations and inverse scattering, Vol. 149, L.M.S. Lecture Notes in Mathematics, C.U.P., Cambridge, 1991.
2. M.J. Ablowitz H. Segur Solitons and the Inverse Scattering Transform 1981 SIAM Philadelphia
3. H. Airault Rational solutions of Painlevé equations Stud. Appl. Math. 61 1979 31-53
4. A.P. Bassom P.A. Clarkson A.C. Hicks Bäcklund transformations and solution hierarchies for the fourth Painlevé equation Stud. Appl. Math. 95 1995 1-71
5. P.A. Clarkson The third Painlevé equation and associated special polynomials J. Phys. A: Math. Gen. 36 2003 9507-9532
6. P.A. Clarkson The fourth Painlevé equation and associated special polynomials J. Math. Phys. 44 2003 5350-5374
7. P.A. Clarkson Remarks on the Yablonskii-Vorob'ev polynomials Phys. Lett. A 319 2003 137-144
8. P.A. Clarkson E.L. Mansfield The second Painlevé equation, its hierarchy and associated special polynomials Nonlinearity 16 2003 R1-R26
9. S. Fukutani K. Okamoto H. Umemura Special polynomials and the Hirota bilinear relations of the second and fourth Painlevé equations Nagoya Math. J. 159 2000 179-200
10. V.I. Gromak Theory of Painlevé's equation Differential Equations 11 1975 285-287
11. V.I. Gromak Solutions of Painlevé's fifth equation Differential Equations 12 1976 519-521
12. V.I. Gromak Bäcklund transformations of Painlevé equations and their applications R. Conte The Painlevé Property One Century Later, CRM series in Mathematical Physics 1999 Springer Berlin 687-734
13. V.I. Gromak G.V. Filipuk Functional relations between solutions of the fifth Painlevé equation Differential Equations 37 2001 614-620
14. V.I. Gromak G.V. Filipuk The Bäcklund transformations of the fifth Painlevé equation and their applications Math. Model. Anal. 6 2001 221-230
15. V.I. Gromak, I. Laine, S. Shimomura, Painlevé differential equations in the complex plane, Vol. 28, Studies in Mathematics, de Gruyter, Berlin, New York, 2002.
16. E.L. Ince Ordinary Differential Equations 1956 Dover New York
17. A.R. Its, V.Yu. Novokshënov, The isomonodromic deformation method in the theory of Painlevé equations, Vol. 1191, Lecture Notes in Mathematics, Springer, Berlin, 1986.
18. K. Iwasaki, H. Kimura, S. Shimomura, M. Yoshida, From Gauss to Painlevé: A modern theory of special functions, Vol. 16, Aspects of Mathematics E, Viewag, Braunschweig, Germany, 1991.
19. M. Jimbo T. Miwa Y. Mori M. Sato Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent Physica, D 1 1980 80-158
20. K. Kajiwara T. Masuda A generalization of determinant formulae for the solutions of Painlevé II and XXXIV equations J. Phys. A: Math. Gen. 32 1999 3763-3778
21. K. Kajiwara T. Masuda On the Umemura polynomials for the Painlevé III equation Phys. Lett. A 260 1999 462-467
22. K. Kajiwara Y. Ohta Determinantal structure of the rational solutions for the Painlevé II equation J. Math. Phys. 37 1996 4393-4704
23. K. Kajiwara Y. Ohta Determinant structure of the rational solutions for the Painlevé IV equation J. Phys. A: Math. Gen. 31 1998 2431-2446
24. Y. Kametaka M. Noda Y. Fukui S. Hirano A numerical approach to Toda equation and Painlevé II equation Mem. Fac. Eng. Ehime Univ. 9 1986 1-24
25. A.N. Kirillov M. Taneda Generalized Umemura polynomials and Hirota-Miwa equations M. Kashiwara T. Miwa MathPhys Odyssey is Prog. Math. Phys. Vol. 23 2002 Birkhauser-Boston Boston, MA 313-331
26. A.N. Kirillov M. Taneda Generalized Umemura polynomials Rocky Mount. J. Math. 32 2002 691-702
27. A.V. Kitaev C.K. Law J.B. McLeod Rational solutions of the fifth Painlevé equation Diff. Int. Eqns. 7 1994 967-1000
28. T. Masuda On a class of algebraic solutions to Painlevé VI equation, its determinant formula and coalescence cascade Funkcial. Ekvac. 46 2003 121-171
29. T. Masuda, Classical transcendental solutions of the Painlevé equations and their degeneration, preprint arXiv:nLin.SI/0302026, 2003.
30. T. Masuda Y. Ohta K. Kajiwara A determinant formula for a class of rational solutions of Painlevé V equation Nagoya Math. J. 168 2002 1-25
31. M. Mazzocco Rational solutions of the Painlevé VI equation J. Phys. A: Math. Gen. 34 2001 2281-2294
32. B.M. McCoy J.H.H. Perk R.E. Shrock Correlation functions of the transverse Ising chain at the critical field for large temporal and spatial separations Nucl. Phys. B 220 [FS 8] 1983 269-282
33. B.M. McCoy S. Tang Connection formulae for Painlevé functions. I Physica 19 D 1986 42-72
34. A.E. Milne P.A. Clarkson A.P. Bassom Bäcklund transformations and solution hierarchies for the third Painlevé equation Stud. Appl. Math. 98 1997 139-194
35. Y. Murata Rational solutions of the second and the fourth Painlevé equations Funkcial. Ekvac. 28 1985 1-32
36. Y. Murata Classical solutions of the third Painlevé equations Nagoya Math. J. 139 1995 37-65
37. M. Noumi S. Okada K. Okamoto H. Umemura Special polynomials associated with the Painlevé equations II M.-H. Saito Y. Shimizu R. Ueno Integrable Systems and Algebraic Geometry 1998 World Scientific Singapore 349-372
38. M. Noumi Y. Yamada Umemura polynomials for the Painlevé V equation Phys. Lett. A 247 1998 65-69
39. M. Noumi Y. Yamada Symmetries in the fourth Painlevé equation and Okamoto polynomials Nagoya Math. J. 153 1999 53-86
40. Y. Ohyama, On the third Painlevé equations of type D 7, preprint, Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University (http://www.math.sci.osaka-u.ac.jp/~ohyama/ohyama-home.html) 2001.
41. IV Math. Ann. 275 1986 221-255
42. M. Taneda Remarks on the Yablonskii-Vorob'ev polynomials Nagoya Math. J. 159 2000 87-111
43. M. Taneda Polynomials associated with an algebraic solution of the sixth Painlevé equation Japan. J. Math. 27 2001 257-274
44. M. Taneda Representation of Umemura polynomials for the sixth Painlevé equation by the generalized Jacobi polynomials A.N. Kirillov A. Tsuchiya H. Umemura Physics and Combinatorics 2001 World Scientific Singapore 366-376
45. C.A. Tracy H. Widom Random unitary matrices, permutations and Painlevé Commun. Math. Phys. 207 1999 665-685
46. H. Umemura, Special polynomials associated with the Painlevé equations. I, preprint; unpublished proceedings of Theory of Nonlinear Special Functions: The Painlevé Transcendents, Montreal, Canada, 1996.
47. H. Umemura Painlevé equations and classical functions Sugaku Expositions 11 1998 77-100
48. H. Umemura Painlevé equations in the past 100 Years A.M.S. Translations 204 2001 81-110
49. A.P. Vorob'ev On rational solutions of the second Painlevé equation Differential Equations 1 1965 58-59
50. A.I. Yablonskii On rational solutions of the second Painlevé equation Vesti Akad. Navuk. BSSR Ser. Fiz. Tkh. Nauk. 3 1959 30-35
51. Y. Yamada, Special polynomials and generalized Painlevé equations, in: K. Koike, M. Kashiwara, S. Okada, I. Terada, H.F. Yamada (Eds.), Combinatorial Methods in Representation Theory, Adv. Stud. Pure Math. 28 Kinokuniya, Tokyo, Japan, 2000, pp. 391-400.
52. W.-J. Yuan Y.-Z. Li Rational solutions of Painlevé equations Canad. J. Math. 54 2002 648-670