Recurrence coefficients for discrete orthonormal polynomials and the Painlevé equations

Journal of Physics A: Mathematical and Theoretical

Volumn 46, Issue 18, 2013, Pages

  Clarkson, Peter A ^a  

^a UNIVERSITY OF KENT   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84876580030     PISSN: 17518113     EISSN: 17518121     Source Type: Journal    
DOI: 10.1088/1751-8113/46/18/185205     Document Type: Article

References (63)

1. Álvarez-Nodarse R 2006 On characterizations of classical polynomials J. Comput. Appl. Math. 196 320-37 (Pubitemid 44040874)
2. Basor E and Chen Y 2009 Painlevé V and the distribution function of a discontinuous linear statistic in the Laguerre unitary ensembles J. Phys. A: Math. Theor. 42 035203
3. Basor E, Chen Y and Ehrhardt T 2010 Painlevé V and time-dependent Jacobi polynomials J. Phys. A: Math. Theor. 43 015204
4. Beals R and Wong R 2010 Special Functions: A Graduate Text (Cambridge Studies in Advanced Mathematics vol 126) (Cambridge: Cambridge University Press)
5. Boelen L, Filipuk G and van Assche W 2011 Recurrence coefficients of generalized Meixner polynomials and Painlevé equations J. Phys. A: Math. Theor. 44 035202
6. Boelen L, Filipuk G, Smet C, van Assche W and Zhang L 2013 The generalized Krawtchouk polynomials and the fifth Painlevé equation J. Difference Eqns Appl. at press (doi:10.1080/10236198.2012.755522)
7. Bochner S 1929 Über Sturm-Liouvillesche Polynomsysteme Math. Z. 29 730-6
8. Boelen L and van Assche W 2011 Discrete Painlevé equations for recurrence relations of semiclassical Laguerre polynomials Proc. Am. Math. Soc. 138 1317-31
9. Charlier C V L 1905-6 Über die Darstellung willkürlicher Funktionen Ark. Mat. Astr. Fys. 2 1-9
10. Chen Y and Dai D 2010 Painlevé V and a Pollaczek-Jacobi type orthogonal polynomials J. Approx. Theory 162 2149-67
11. Chen Y and Feigin M V 2006 Painlevé IV and degenerate Gaussian unitary ensembles J. Phys. A: Math. Gen. 39 12381-93 (Pubitemid 44437512)
12. Chen Y and Its A R 2010 Painlevé III and a singular linear statistics in Hermitian random matrix ensembles: I J. Approx. Theory 162 270-97
13. Chen Y and Pruessner G 2005 Orthogonal polynomials with discontinuous weights J. Phys. A: Math. Gen. 38 L191-8
14. Chen Y and Zhang L 2010 Painlevé VI and the unitary Jacobi ensembles Stud. Appl. Math. 125 91-112
15. Chihara T S 1978 An Introduction to Orthogonal Polynomials (New York: Gordon and Breach) (reprinted by Dover 2011)
16. Clarkson P A 2006 Painlevé equations - non-linear special functions Orthogonal Polynomials and Special Functions: Computation and Application (Lecture Notes in Mathematics vol 1883) ed F Marcellàn and W van Assche (Berlin: Springer) pp 331-411 (Pubitemid 44597579)
17. Clarkson P A and Jordaan K 2013 The relationship between semi-classical Laguerre polynomials and the fourth Painlevé equation arXiv:1301.4134 [nlin.SI]
18. Dai D and Zhang L 2010 Painlevé VI and Hankel determinants for the generalized Jacobi weight J. Phys. A: Math. Theor. 43 055207
19. Dominici D and Marcellàn F 2012 Discrete semiclassical orthogonal polynomials of class one arXiv:1211.2005 [math.CA]
20. Filipuk G and van Assche W 2011 Recurrence coefficients of a new generalization of the Meixner polynomials SIGMA 7 068
21. Filipuk G and van Assche W 2013 Recurrence coefficients of generalized Charlier polynomials and the fifth Painlevé equation Proc. Am. Math. Soc. 141 551-62
22. Filipuk G, van Assche W and Zhang L 2012 The recurrence coefficients of semi-classical Laguerre polynomials and the fourth Painlevé equation J. Phys. A: Math. Theor. 45 205201
23. Fokas A S, Its A R, Kapaev A A and Novokshenov V Yu 2006 Painlevé Transcendents: The Riemann-Hilbert Approach (Mathematical Surveys and Monographs vol 128) (Providence, RI: American Mathematical Society)
24. Fokas A S, Its A R and Kitaev A V 1991 Discrete Painlevé equations and their appearance in quantum-gravity Commun. Math. Phys. 142 313-44
25. Fokas A S, Its A R and Kitaev A V 1992 The isomonodromy approach to matrix models in 2D quantum-gravity Commun. Math. Phys. 147 395-430
26. Forrester P J 2010 Log-gases and random matrices (London Mathematical Society Monographs Series vol 34) (Princeton, NJ: Princeton University Press)
27. Forrester P J and Ormerod C M 2010 Differential equations for deformed Laguerre polynomials J. Approx. Theory 162 653-77
28. Forrester P J and Witte N S 2002 Application of the τ-function theory of Painlevé equations to random matrices: PV, PIII, the LUE, JUE and CUE Commun. Pure Appl. Math. 55 679-727 (Pubitemid 37244220)
29. Forrester P J and Witte N S 2007 The distribution of the first eigenvalue spacing at the hard edge of the Laguerre unitary ensemble Kyushu J. Math. 61 457-526
30. Freud G 1976 On the coefficients in the recursion formulae of orthogonal polynomials Proc. R. Ir. Acad. A 76 1-6
31. Gromak V I 1978 Single-parameter families of solutions of Painlevé's equations Differ. Eqns 14 1510-3
32. Gromak V I, Laine I and Shimomura S 2002 Painlevé Differential Equations in the Complex Plane (Studies in Mathematics vol 28) (Berlin: de Gruyter & Co)
33. Hendriksen E and van Rossum H 1985 Semi-classical orthogonal polynomials Polynômes Orthogonaux et Applications (Lecture Notes in Mathematics vol 1171) ed C Brezinski, A Draux, A P Magnus, P Maroni and A Ronveaux (Berlin: Springer) pp 354-61
34. Hounkonnou M N, Hounga C and Ronveaux A 2000 Discrete semi-classical orthogonal polynomials: generalized Charlier J. Comput. Appl. Math. 114 361-6 (Pubitemid 30541673)
35. Ince E L 1956 Ordinary Differential Equations (New York: Dover)
36. Ismail M E H 2005 Classical and Quantum Orthogonal Polynomials in One Variable (Encyclopedia of Mathematics and its Applications vol 98) (Cambridge: Cambridge University Press)
37. Iwasaki K, Kimura H, Shimomura S and Yoshida M 1991 From Gauss to Painlevé : A Modern Theory of Special Functions (Aspects of Mathematics E vol 16) (Braunschweig: Viewag)
38. Jimbo M and Miwa T 1981 Monodromy preserving deformations of linear ordinary differential equations with rational coefficients: II Physica D 2 407-48
39. Kajiwara K, Masuda T, Noumi M, Ohta Y and Yamada Y 2001 Determinant formulas for the Toda and discrete Toda equations Funkcial. Ekvac. 44 291-307
40. Krawtchouk M 1929 Sur une généralisation des polynomes d'Hermite C. R. Acad. Sci. 189 620-2
41. Magnus A 1995 Painlevé -type differential equations for the recurrence coefficients of semi-classical orthogonal polynomials J. Comput. Appl. Math. 57 215-37
42. Mansfield E L and Webster H N 1998 On one-parameter families of Painlevé III Stud. Appl. Math. 101 321-41
43. Maroni P 1987 Prolǵomènes à l'étude des polynômes orthogonaux semi-classiques Ann. Mat. Pura Appl. 149 165-84
44. Masuda T 2004 Classical transcendental solutions of the Painlevé equations and their degeneration Tohoku Math. J. 56 467-90 (Pubitemid 40083616)
45. Meixner J 1934 Orthogonale Polynomsysteme mit einer besonderen Gestalt der erzeugenden Funktion J. Lond. Math. Soc. s1-9 6-13
46. Milne A E, Clarkson P A and Bassom A P 1997 Bäcklund transformationsand solution hierarchies for the third Painlevé equation Stud. Appl. Math. 98 139-94 (Pubitemid 127034603)
47. Murata Y 1995 Classical solutions of the third Painlevé equations Nagoya Math. J. 139 37-65
48. Nakamura Y and Zhedanov A 2004 Special solutions of the Toda chain and combinatorial numbers J. Phys. A: Math. Gen. 37 5849-62 (Pubitemid 38799787)
49. Nikiforov A F, Suslov S K and Uvarov V B 1991 Classical Orthogonal Polynomials of a Discrete Variable (Springer Series in Computational Physics) (Berlin: Springer)
50. Okamoto K 1980 Polynomial Hamiltonians associated with Painlevé equations: I Proc. Japan Acad. A 56 264-8
51. Okamoto K 1980 Polynomial Hamiltonians associated with Painlevé equations: II Proc. Japan Acad. A 56 367-71
52. V Japan J. Math. 13 47-76
53. III Funkcial. Ekvac. 30 305-32
54. Olver F W J, Lozier D W, Boisvert R F and Clark C W (ed) 2010 NIST Handbook of Mathematical Functions (Cambridge: Cambridge University Press)
55. Osipov V Al and Kanzieper E 2010 Correlations of RMT characteristic polynomials and integrability: Hermitian matrices Ann. Phys. 325 2251-306
56. Shohat J 1939 A differential equation for orthogonal polynomials Duke Math. J. 5 401-17
57. Smet C and van Assche W 2012 Orthogonal polynomials on a bi-lattice Constr. Approx. 36 215-42
58. Sogo K 1993 Time-dependent orthogonal polynomials and theory of soliton - applications to matrix model, vertex model and level statistics J. Phys. Soc. Japan 62 1887-94
59. Szegö G 1975 Orthogonal Polynomials (AMS Colloquium Publications vol 23) (Providence RI: American Mathematical Society)
60. Umemura H and Watanabe H 1998 Solutions of the third Painlevé equation I Nagoya Math. J. 151 1-24
61. van Assche W et al 2007 Discrete Painleve equations for recurrence coefficients of orthogonal polynomials Difference Equations, Special Functions and Orthogonal Polynomials ed S Elaydi (Singapore: World Scientific) pp 687-725
62. van Assche W and Foupouagnigni M 2003 Analysis of non-linear recurrence relations for the recurrence coefficients of generalized Charlier polynomials J. Nonlinear Math. Phys. 10 231-7
63. Watanabe H 1995 Solutions of the fifth Painlevé equation: I Hokkaido Math. J. 24 231-67