Asymptotic zero behavior of laguerre polynomials with negative parameter

Constructive Approximation

Volumn 20, Issue 4, 2004, Pages 497-523

  Kuijlaars, A B J ^a   McLaughlin, K T R ^b,c  

^a UNIVERSITY OF LEUVEN   (Belgium)

^b University of North Carolina at Chapel Hill   (United States)

^c University of Arizona   (United States)

Author keywords

Nonlinear steepest descent; Riemann Hilbert problems; Sensitivity to parameter

Indexed keywords

EID: 4644337591     PISSN: 01764276     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00365-003-0536-3     Document Type: Article

References (33)

1. A. I. APTEKAREV (2002): Sharp constants for rational approximations of analytic functions. Mat. Sb., 193:3-72.
2. Z. D. BAI (1999): Methodologies in spectral analysis of large-dimensional random matrices. A review. Statist. Sinica, 9:611-677.
3. J. BAIK, P. DEIFT, K. MCLAUOHLIN, P. MILLER, X. ZHOU (2001): Optimal tail estimates for directed last passage site percolation with geometric random variables. Adv. in Theoret. Math. Phys., 5:1207-1250.
4. P. BLEHER, A. I. ITS (1999): Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model. Ann. of Math., 150:185-266.
5. P. DEIFT (1999): Orthogonal polynomials and random matrices: A Riemann-Hilbert approach. Courant Lecture Notes 3, Courant Institute.
6. P. DEIFT, T. KRIECHERBAUER, K. T-R MCLAUGHLIN, S. VENAKIDES, X. ZHOU (1999): Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Comm. Pure Appl. Math., 52:1335-1425.
7. P. DEIFT, T. KRIECHERBAUER, K. T-R MCLAUGHLIN, S. VENAKIDES, X. ZHOU (1999): Strong asymptotics of orthogonal polynomials with respect to exponential weights. Comm. Pure Appl. Math., 52:1491-1552.
8. P. DEIFT, S. VENAKIDES, X. ZHOU (1997): New results in small dispersion KdV by an extension of the steepest descent method for Riemann-Hilbert problems. Internat. Math. Res. Notices, 1997:286-299.
9. P. DEIFT, X. ZHOU (1993): A steepest descent method for oscillatory Riemann-Hilbert problems, asymptotics for the mKDV equation. Ann. of Math., 137:295-368.
10. H. DETTE (2002): Strong approximation of eigenvalues of large dimensional Wishart matrices by roots of generalized Laguerre polynomials. J. Approx. Theory, 118:290-304.
11. H. DETTE, W. J. STUDDEN (1995): New asymptotic properties for the zeros of Jacobi, Laguerre, and Hermite polynomials. Constr. Approx., 11:227-238.
12. K. DRIVER, P. DUREN (2000): Zeros of the hypergeometric polynomials F(-n, b; 2b; z). Indag. Math., 11:43-51.
13. K. DRIVER, P. DUREN (2001): Trajectories of the zeros of hypergeometric polynomials F(-n, b; 2b; z) for b < -1/2. Constr. Approx., 17:169-179.
14. T. DUNSTER (2001): Uniform asymptotic expansions for the reverse generalized Bessel polynomials, and related functions. SIAM J. Math. Anal., 32:987-1013.
15. A. S. FOKAS, A. R. ITS, A. V. KITAEV (1992): The isomonodromy approach to matrix models in 2D quantum gravity. Comm. Math. Phys., 147:395-430.
16. W. GAWRONSKI (1987): On the asymptotic distribution of the zeros of Hermite, Laguerre, and Jonquière polynomials. J. Approx. Theory, 50:214-231.
17. (an+α). Analysis, 13:29-67.
18. F. HIAI, D. PETZ (2000): The Semicircle Law, Free Random Variables and Entropy. Mathematical Surveys and Monographs, Vol. 77. Providence, RI: American Mathematical Society.
19. S. KAMVISSIS, K. T-R MCLAUGHLIN, P. MILLER (to appear): Semiclassical soliton ensembles for the focusing nonlinear Schrödinger equation. Ann. of Math., Studies Series (preprint nlin.SI/0012034).
20. T. KRIECHERBAUER, K. T-R MCLAUGHLIN (1999): Strong asymptotics of polynomials orthogonal with respect to Freud weights. Internat. Math. Res. Notices, 1999:299-333.
21. A. B. J. KUIJLAARS, K. T-R MCLAUGHLIN (2001): Riemann-Hilbert analysis for Laguerre polynomials with large negative parameter. Comput. Methods Function Theory, 1:205-233.
22. A. B. J. KUIJLAARS, K. T-R MCLAUGHLIN, W. VAN ASSCHE, M. VANLESSEN (2001): The Riemann-Hilbert approach to strong asymptotics for orthogonal polynomials on [-1, 1]. Manuscript (preprint math.CA/0111252).
23. A. B. J. KUIJLAARS, W. VAN ASSCHE (1999): The asymptotic zero distribution of orthogonal polynomials with varying recurrence coefficients, J. Approx. Theory, 99:167-197.
24. V. A. MARCHENKO, L. A. PASTUR (1967): Distribution of eigenvalues in certain sets of random matrices. Mat. Sb., 72:507-536; English transl., Math. USSR-Sb, 1:457-483.
25. V. A. MARCHENKO, L. A. PASTUR (1967): Distribution of eigenvalues in certain sets of random matrices. Mat. Sb., 72:507-536; English transl., Math. USSR-Sb, 1:457-483.
26. A. MARTÍNEZ-FINKELSHTEIN, P. MARTÍNEZ-GONZÁLEZ, R. DRIVE (2001): On asymptotic zero distribution of Laguerre and generalized Bessel polynomials with varying parameters. J. Comput. Appl. Math., 127:255-266.
27. A. MARTÍNEZ-FINKELSHTEIN, P. MARTÍNEZ-GONZÁLEZ, R. DRIVE (2000): Zeros of Jacobi polynomials with varying non-classical parameters. In: Proceedings of the International Workshop on Special Functions: Asymptotics, Harmonic Analysis, and Mathematical Physics, City University of Hong Kong, June 21-25, 1999. Singapore: World Scientific, pp. 98-113.
28. H. N. MHASKAR, E. B. SAFF (1991): The distribution of zeros of asymptotically extremal polynomials. J. Approx. Theory, 65:279-300.
29. E. B. SAFF, V. TOTIK (1997): Logarithmic Potentials with External Fields. Berlin: Springer-Verlag.
30. z, III. Numer. Math., 30:241-266.
31. K. STREBEL (1984): Quadratic Differentials. Berlin: Springer-Verlag.
32. G. SZEGO′ (1939): Orthogonal Polynomials. AMS Colloq. Publ., Vol. 23. Providence, RI: American Mathematical Society.
33. R. WONG, J.-M. ZHANG (1997): Asymptotic expansions of the generalized Bessel polynomials. J. Comput. Appl. Math., 85:87-112.