Riemann-Hilbert problem for bi-orthogonal polynomials

Journal of Physics A: Mathematical and General

Volumn 36, Issue 16, 2003, Pages 4629-4640

  Kapaev, Andrei A ^a  

^a STEKLOV MATHEMATICAL INSTITUTE   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0038407689     PISSN: 03054470     EISSN: None     Source Type: Journal    
DOI: 10.1088/0305-4470/36/16/312     Document Type: Article

References (29)

1. n Comment. Math. Helv. 1 227-54
2. Szegö G 1939 Orthogonal Polynomials (Providence, RI: American Mathematical Society)
3. Freud G 1976 On the coefficients in the recursion formulae of orthogonal polynomials Proc. R. Ir. Acad. A 76 1-6
4. Nevai P 1986 Géza Freud, orthogonal polynomials and Christoffel functions. A case study J. Approx. Theory 48 3-167
5. 6/6) J. Approx. Theory 50 232-93
6. Magnus A P 1995 Painlevé-type differential equations for the recurrence coefficients of semi-classical orthogonal polynomials J. Comput. Appl. Math. 57 215-37
7. Fokas A S, Its A R and Kitaev A V 1990 Isomonodromic approach in the theory of two-dimensional quantum gravity Usp. Mat. Nauk 45 135-6 Fokas A S, Its A R and Kitaev A V 1991 Discrete Painlevé equations and their appearance in quantum gravity Comment Math. Phys. 142 313-44 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
8. Fokas A S, Its A R and Kitaev A V 1990 Isomonodromic approach in the theory of two-dimensional quantum gravity Usp. Mat. Nauk 45 135-6 Fokas A S, Its A R and Kitaev A V 1991 Discrete Painlevé equations and their appearance in quantum gravity Comment Math. Phys. 142 313-44 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
9. Fokas A S, Its A R and Kitaev A V 1990 Isomonodromic approach in the theory of two-dimensional quantum gravity Usp. Mat. Nauk 45 135-6 Fokas A S, Its A R and Kitaev A V 1991 Discrete Painlevé equations and their appearance in quantum gravity Comment Math. Phys. 142 313-44 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
10. Deift P A and Zhou X 1993 A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation Ann. Math. 137 295-368
11. Bleher P M and Its A R 1999 Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model Ann. Math. 150 185-266 Bleher P M and Its A R 2002 Double scaling limit in the random matrix model: Riemann-Hilbert approach Preprint math-ph/0201003
12. Bleher P M and Its A R 1999 Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model Ann. Math. 150 185-266 Bleher P M and Its A R 2002 Double scaling limit in the random matrix model: Riemann-Hilbert approach Preprint math-ph/0201003
13. Deift P, Kriecherbauer T, McLaughlin K T-R, Venakides S and Zhou X 1999 Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory Commun. Pure Appl. Math. 52 1335-425 Deift P, Kriecherbauer T, McLaughlin K T-R, Venakides S and Zhou X 1999 Strong asymptotics of orthogonal polynomials with respect to exponential weights Commun. Pure Appl. Math. 52 1491-552
14. Deift P, Kriecherbauer T, McLaughlin K T-R, Venakides S and Zhou X 1999 Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory Commun. Pure Appl. Math. 52 1335-425 Deift P, Kriecherbauer T, McLaughlin K T-R, Venakides S and Zhou X 1999 Strong asymptotics of orthogonal polynomials with respect to exponential weights Commun. Pure Appl. Math. 52 1491-552
15. Baik J, Deift P and Johansson K 1999 On the distribution of the length of the longest increasing subsequence of random permutations J. Am. Math. Soc. 12 1119-78 (Preprint math.CO/9810105)
16. Adler M and van Moerbeke P 2000 Generalized orthogonal polynomials, discrete KP and Riemann-Hilbert problems Preprint nlin.SI/0009002
17. van Moerbeke P 2001 Integrable Lattices: Random Matrices and Random Permutations (MSRI Publications No 40) (Cambridge: Cambridge University Press) (Preprint math.CO/0010135)
18. Ercolani N M and McLaughlin K T-R 2001 Asymptotics and integrable structures for biorthogonal polynomials associated to a random two-matrix model Physica D 152-153 232-68
19. Bertola M, Eynard B and Harnad J 2001 Duality, biorthogonal polynomials and multi-matrix models Preprint nlin.SI/0108049
20. Zinn-Justin P 2002 HCIZ integral and 2D Toda lattice hierarchy Preprint math-ph/0202045
21. Bertola M 2002 Bilinear semi-classical moment functionals and their integral representation Preprint math.CA/0205160
22. Kapaev A A 2002 The Riemann-Hilbert problem for the bi-orthogonal polynomials Preprint nlin.SI/0207036
23. Bertola M, Eynard B and Harnad J 2002 Differential systems appearing in 2-matrix models and the associated Riemann-Hilbert problem Preprint nlin. SI/0208002
24. Its A R and Kapaev A A 2002 The nonlinear steepest descent approach to the asymptotics of the second Painleve transcendent in the complex domain MathPhys Odyssey 2001, Integrable Models and Beyond (in Honour of Barry M McCoy) ed M Kashiwara and T Miwa (Progress in Mathematical Physics vol 23) (Boston, MA: Birkhäuser) pp 273-311 (Preprint nlin.SI/0108054)
25. Wasow W 1965 Asymptotic Expansions for Ordinary Differential Equations (New York: Interscience/Wiley)
26. Miwa T 1981 Painlevé property of monodromy preserving equations and the analyticity of τ-functions Publ. Res. Inst. Math. Sci. 17 703-21
27. Malgrange B 1991 Differential Equations with Polynomial Coefficients (Progress in Mathematics vol 96) (Boston, MA: Birkhäuser)
28. Bolibrukh A, Its A R and Kapaev A A On the Riemann-Hilbert-Birkhoff inverse monodromy problem and the Painlevé equations In preparation
29. Kapaev A A 2002 Monodromy approach to the scaling limits in the isomonodromy systems Preprint nlin.SI/0211022