The Riemann – Hilbert approach to strong asymptotics for orthogonal polynomials on 1⁄2 1 ; 1

We consider polynomials that are orthogonal on 1⁄2 1; 1 with respect to a modified Jacobi weight ð1 xÞð1þ xÞhðxÞ; with a; b4 1 and h real analytic and strictly positive on 1⁄2 1; 1 : We obtain full asymptotic expansions for the monic and orthonormal polynomials outside the interval 1⁄2 1; 1 ; for the recurrence coefficients and for the leading coefficients of the orthonormal polynomials. We also deduce asymptotic behavior for the Hankel determinants and for the monic orthogonal polynomials on the interval 1⁄2 1; 1 : For the asymptotic analysis we use the steepest descent technique for Riemann–Hilbert problems developed by Deift and Zhou, and applied to orthogonal polynomials on the real line by Deift, Kriecherbauer, McLaughlin, Venakides, and Zhou. In the steepest descent method we will use the Szeg + o function associated with the weight and for the local analysis ARTICLE IN PRESS Corresponding author. E-mail addresses: arno@wis.kuleuven.ac.be (A.B.J. Kuijlaars), mcl@amath.unc.edu (K.T.-R. McLaughlin), walter@wis.kuleuven.ac.be (W. Van Assche), maarten.vanlessen@wis.kuleuven. ac.be (M. Vanlessen). Supported by FWO Research Projects G.0176.02 and G.0455.04, and by INTAS Research Network 03-51-6637. Supported by NSF Grants DMS-9970328 and DMS-0200749. Supported by FWO Research Projects G.0184.01 and G.0455.04, and by INTAS Research Network 03-51-6637. Research Assistant of the Fund for Scientific Research—Flanders (Belgium). 0001-8708/$ see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.aim.2003.08.015 around the endpoints71 we use Bessel functions of appropriate order, whereas Deift et al. use Airy functions. r 2003 Elsevier Inc. All rights reserved. MSC: Primary 42C05; 30E25; 33C10; 35Q15

[1]  Rene F. Swarttouw,et al.  Orthogonal Polynomials , 2005, Series and Products in the Development of Mathematics.

[2]  M. Stephanov,et al.  Random Matrices , 2005, hep-ph/0509286.

[3]  A. S. Fokas,et al.  The Isomonodromy Approach to Matrix Models in 2 D Quantum Gravity , 2004 .

[4]  M. Vanlessen.  Universal Behavior for Averages of Characteristic Polynomials at the Origin of the Spectrum , 2003, math-ph/0306078.

[5]  A. Kuijlaars,et al.  Universality for Eigenvalue Correlations at the Origin of the Spectrum , 2003, math-ph/0305044.

[6]  Z. Wang,et al.  Asymptotic expansions for second-order linear difference equations with a turning point , 2003, Numerische Mathematik.

[7]  A. Kuijlaars,et al.  Quadratic Hermite–Padé Approximation to the Exponential Function: A Riemann–Hilbert Approach , 2003, math/0302357.

[8]  Maarten Vanlessen,et al.  Strong asymptotics of the recurrence coefficients of orthogonal polynomials associated to the generalized Jacobi weight , 2002, J. Approx. Theory.

[9]  Arno B. J. Kuijlaars,et al.  Riemann-Hilbert Analysis for Orthogonal Polynomials , 2003 .

[10]  J. Baik,et al.  Uniform Asymptotics for Polynomials Orthogonal With Respect to a General Class of Discrete Weights and Universality Results for Associated Ensembles , 2002, math/0212149.

[11]  K. Mclaughlin,et al.  Asymptotics of the partition function for random matrices via Riemann-Hilbert techniques and applications to graphical enumeration , 2002, math-ph/0211022.

[12]  Y. Fyodorov,et al.  Universal Results for Correlations of Characteristic Polynomials: Riemann-Hilbert Approach , 2002, math-ph/0210010.

[13]  P. Deift,et al.  A priori $L^p$ estimates for solutions of Riemann-Hilbert Problems , 2002, math/0206224.

[14]  A. Kuijlaars,et al.  Asymptotic Zero Behavior of Laguerre Polynomials with Negative Parameter , 2002, math/0205175.

[15]  A. Kuijlaars,et al.  Universality for eigenvalue correlations from the modified Jacobi unitary ensemble , 2002, math-ph/0204006.

[16]  Pavel Bleher,et al.  Double scaling limit in the random matrix model: The Riemann‐Hilbert approach , 2002, math-ph/0201003.

[17]  P. Deift,et al.  Perturbation theory for infinite-dimensional integrable systems on the line. A case study , 2002 .

[18]  A. Kuijlaars,et al.  Riemann-Hilbert Analysis for Laguerre Polynomials with Large Negative Parameter , 2002, math/0204248.

[19]  P. Miller.  Asymptotics of semiclassical soliton ensembles: rigorous justification of the WKB approximation , 2001, nlin/0108052.

[20]  Stephanos Venakides,et al.  A Riemann-Hilbert approach to asymptotic questions for orthogonal polynomials , 2001 .

[21]  Doron S. Lubinsky,et al.  Orthogonal Polynomials for Exponential Weights , 2001 .

[22]  Laurent Baratchart,et al.  Asymptotic Error Estimates for L2 Best Rational Approximants to Markov Functions , 2001, J. Approx. Theory.

[23]  Peter D. Miller,et al.  Semiclassical soliton ensembles for the focusing nonlinear Schrödinger equation , 2000, nlin/0012034.

[24]  P. Deift.  Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach , 2000 .

[25]  D. Lubinsky.  Asymptotics of Orthogonal Polynomials: Some Old, Some New, Some Identities , 2000 .

[26]  W.-Y. QIU,et al.  Uniform Asymptotic Formula for Orthogonal Polynomials with Exponential Weight , 2000, SIAM J. Math. Anal..

[27]  R. Wong,et al.  On the Asymptotics of the Meixner—Pollaczek Polynomials and Their Zeros , 2000 .

[28]  Stephanos Venakides,et al.  Strong asymptotics of orthogonal polynomials with respect to exponential weights , 1999 .

[29]  Stephanos Venakides,et al.  UNIFORM ASYMPTOTICS FOR POLYNOMIALS ORTHOGONAL WITH RESPECT TO VARYING EXPONENTIAL WEIGHTS AND APPLICATIONS TO UNIVERSALITY QUESTIONS IN RANDOM MATRIX THEORY , 1999 .

[30]  Pavel Bleher,et al.  Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model , 1999, math-ph/9907025.

[31]  T. Kriecherbauer,et al.  Strong asymptotics of polynomials orthogonal with respect to Freud weights , 1999 .

[32]  J. Baik,et al.  On the distribution of the length of the longest increasing subsequence of random permutations , 1998, math/9810105.

[33]  Laurent Baratchart,et al.  Asymptotic Uniqueness of Best Rational Approximants of Given Degree to Markov Functions in L 2 of the Circle , 1998 .

[34]  Kurt Johansson,et al.  ON RANDOM MATRICES FROM THE COMPACT CLASSICAL GROUPS , 1997 .

[35]  A. Fokas,et al.  Complex Variables: Introduction and Applications , 1997 .

[36]  Stephanos Venakides,et al.  New results in small dispersion KdV by an extension of the steepest descent method for Riemann-Hilbert problems , 1997 .

[37]  P. Deift,et al.  Asymptotics for the painlevé II equation , 1995 .

[38]  C. Tracy,et al.  Level spacing distributions and the Bessel kernel , 1993, hep-th/9304063.

[39]  D. Lubinsky.  An update on orthogonal polynomials and weighted approximation on the real line , 1993 .

[40]  T. Nagao,et al.  Laguerre ensembles of random matrices: Nonuniversal correlation functions , 1993 .

[41]  P. Deift,et al.  A steepest descent method for oscillatory Riemann-Hilbert problems , 1992, math/9201261.

[42]  D. Lubinsky.  A survey of general orthogonal polynomials for weights on finite and infinite intervals , 1987, Acta Applicandae Mathematicae.

[43]  Walter Van Assche,et al.  Asymptotics for Orthogonal Polynomials , 1987 .

[44]  Paul Neval,et al.  Ge´za Freud, orthogonal polynomials and Christoffel functions. A case study , 1986 .

[45]  Leon M. Hall,et al.  Special Functions , 1998 .