The recurrence coefficients of semi-classical Laguerre polynomials and the fourth Painlevé equation

Journal of Physics A: Mathematical and Theoretical

Volumn 45, Issue 20, 2012, Pages

  Filipuk, Galina ^a   Van Assche, Walter ^b   Zhang, Lun ^b  

^a UNIVERSITY OF WARSAW   (Poland)

^b UNIVERSITY OF LEUVEN   (Belgium)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84860811761     PISSN: 17518113     EISSN: 17518121     Source Type: Journal    
DOI: 10.1088/1751-8113/45/20/205201     Document Type: Article

References (37)

1. Basor E, Chen Y and Ehrhardt T 2010 Painlevé V and time-dependent Jacobi polynomials J. Phys. A: Math. Theor. 43 015204
2. Boelen L and Van Assche W 2010 Discrete Painlevé equations for recurrence coefficients of semiclassical Laguerre polynomials Proc. Am. Math. Soc. 138 131731
3. 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
4. m), m an even positive integer J. Approx. Theory 46 40810
5. Bonan S and Nevai P 1984 Orthogonal polynomials and their derivatives: I J. Approx. Theory 40 13447
6. Bonan S, Lubinsky D S and Nevai P 1987 Orthogonal polynomials and their derivatives: II SIAM J. Math. Anal. 18 116376
7. Chen Y and Dai D 2010 Painlevé V and a PollaczekJacobi type polynomials J. Approx. Theory 162 214967
8. Chen Y and Feigin M V 2006 Painlevé IV and degenerate Gaussian unitary ensembles J. Phys. A: Math. Gen. 39 1238193 (Pubitemid 44437512)
9. Chen Y and Ismail M 2005 Jacobi polynomials from compatibility conditions Proc. Am. Math. Soc. 133 46572 (Pubitemid 40224412)
10. Chen Y and Its A 2010 Painlevé III and a singular linear statistics in Hermitian random matrix ensembles: I J. Approx. Theory 162 27097
11. Chen Y and Zhang L 2010 Painlevé VI and the unitary Jacobi ensembles Stud. Appl. Math. 125 91112
12. Chihara T S 1978 An Introduction to Orthogonal Polynomials (New York: Gordon and Breach)
13. Clarkson P A 2006 Painlevé equations - nonlinear special functions Orthogonal Polynomials and Special Functions: Computation and Applications (Lecture Notes in Mathematics vol 1883) (Berlin: Springer) pp 331411 (Pubitemid 44597579)
14. Dai D and Zhang L 2010 Painlevé VI and Hankel determinants for the generalized Jacobi weight J. Phys. A: Math. Theor. 43 055207
15. Filipuk G and Van Assche W 2011 Recurrence coefficients of a new generalization of the Meixner polynomials SIGMA 7 068
16. Filipuk G and Van Assche W 2012 Recurrence coefficients of generalized Charlier polynomials and the fifth Painlevé equation Proc. Am. Math. Soc. at press (arXiv:1106.2959)
17. Forrester P J and Ormerod C M 2010 Differential equations for deformed Laguerre polynomials J. Approx. Theory 162 65377
18. Forrester P J and Witte N S 2006 Random matrix theory and the sixth Painlevé equation J. Phys. A: Math. Gen. 39 1221133 (Pubitemid 44403716)
19. Freud G 1976 On the coefficients in the recursion formulae of orthogonal polynomials Proc. R. Irish Acad. Sect. A 76 16
20. Gromak V I, Laine I and Shimomura S 2002 Painlevé Differential Equations in the Complex Plane Studies in Mathematics (Berlin: de Gruyter)
21. Hendriksen E and Van Rossum H 1985 Semi-classical orthogonal polynomials Polynômes Orthogonaux et Applications (Lecture Notes in Mathematics vol 1171) (Berlin: Springer) pp 35461
22. Ismail M E H 2005 Classical and quantum orthogonal polynomials in one variable Encyclopedia of Mathematics and its Applications (Cambridge: Cambridge University Press)
23. Jimbo M and Miwa T 1981 Monodromy preserving deformations of linear ODEs with rational coefficients: II Physica D 2 40748
24. Joshi N, Kajiwara K and Mazzocco M 2006 Generating function associated with the determinant formula for the solutions of the Painlevé IV equation Funkcial. Ekvac. 49 45168
25. Kajiwara K and Ohta Y 1998 Determinant structure of the rational solutions for the Painlevé IV equation J. Phys. A: Math. Gen. 31 243146 (Pubitemid 128039547)
26. Magnus A P 1999 Freuds equations for orthogonal polynomials as discrete Painlevé equations Symmetries and Integrability of Difference Equations (London Mathematical Society Lecture Note Series vol 225) (Cambridge: Cambridge University Press) pp 22843
27. Magnus A P 1995 Painlevé type differential equations for the recurrence coefficients of semi-classical orthogonal polynomials J. Comput. Appl. Math. 57 21537
28. Maroni P 1987 Prolégomènes à létude des polynômes orthogonaux semi-classiques Ann. Mat. Pura Appl. 149 16584
29. Moser J 1975 Finitely many mass points on the line under the influence of an exponential potential - an integrable system Dynamical Systems, Theory and Applications (Lecture Notes in Physics vol 38) (Berlin: Springer) pp 46997
30. Murata Y 1985 Rational solutions of the second and the fourth Painlevé equations Funkcial. Ekvac. 28 132
31. Noumi M 2004 Painlevé Equations Through Symmetry (Translations of Mathematical Monographs vol 223) (Providence, RI: American Mathematical Society)
32. Noumi M and Okamoto K 1997 Irreducibility of the second and fourth Painlevé equations Funkcial. Ekvac. 40 13963
33. Noumi M and Yamada Y 1999 Symmetries in the fourth Painlevé equation and Okamoto polynomials Nagoya Math. J. 153 5386
34. Ohta Y, Kajiwara K and Satsuma J 1996 Bilinear structure and exact solutions of the discrete Painlevé I equation Symmetries and Integrability of Difference Equations (Proceedings and Lecture Notes vol 9) (Providence, RI: American Mathematical Society) pp 26568
35. Shohat J 1939 A differential equation for orthogonal polynomials Duke Math. J. 5 40117
36. Szeg G 1975 Orthogonal Polynomials (AMS Colloquium Publications vol 23) 4th edn (Providence, RI: American Mathematical Society)
37. Van Assche W 2007 Discrete Painlevé equations for recurrence coefficients of orthogonal polynomials Discrete Equations, Special Functions and Orthogonal Polynomials ed Elaydi S (Singapore: World Scientific) pp 687725