Global asymptotics of the Meixner polynomials

Using the steepest descent method for oscillatory Riemann-Hilbert problems introduced by Deift and Zhou (Ann. Math. 137 (1993), 295-368), we derive asymptotic formulas for the Meixner polynomials in two regions of the complex plane separated by the boundary of a rectangle. The asymptotic formula on the boundary of the rectangle is obtained by taking limits from either inside or outside. Our results agree with the ones obtained earlier for z on the positive real line by using the steepest descent method for integrals (Constr. Approx. 14 (1998), 113-150).

[1]  Walter Van Assche,et al.  Probabilistic proofs of asymptotic formulas for some classical polynomials , 1985 .

[2]  A. Erdélyi,et al.  Higher Transcendental Functions , 1954 .

[3]  P. Deift,et al.  A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the MKdV equation , 1993 .

[4]  橋本 英典,et al.  A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi ; Higher Transcendental Functions, Vols. I, II, III. McGraw-Hill, New York-Toronto-London, 1953, 1953, 1955. xxvi+302, xvii+396, xvii+292頁. 16×23.5cm. $6.50, $7.50, $6.50. , 1955 .

[5]  Chunhua Ou,et al.  THE RIEMANN–HILBERT APPROACH TO GLOBAL ASYMPTOTICS OF DISCRETE ORTHOGONAL POLYNOMIALS WITH INFINITE NODES , 2010 .

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

[7]  Lun Zhang,et al.  Global asymptotics of Hermite polynomials via Riemann-Hilbert approach , 2007 .

[8]  Peter D. Miller,et al.  Discrete orthogonal polynomials: Asymptotics and applications , 2007 .

[9]  Roderick Wong,et al.  Global Asymptotics of Krawtchouk Polynomials – a Riemann-Hilbert Approach* , 2007 .

[10]  Jinho Baik,et al.  Discrete Orthogonal Polynomials , 2007, Encyclopedia of Special Functions: The Askey-Bateman Project.

[11]  F. Olver.  Asymptotics and Special Functions , 1974 .

[12]  P. Deift,et al.  A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the MKdV equation , 1992, math/9201261.

[13]  Roderick Wong,et al.  Asymptotic Formulas for the Zeros of the Meixner Polynomials , 1999 .

[14]  R. Wong,et al.  Uniform Asymptotic Expansions for Meixner Polynomials , 1998 .

[15]  Roderick Wong,et al.  Uniform Asymptotics for Orthogonal Polynomials with Exponential Weights—the Riemann–Hilbert Approach , 2005 .

[16]  K. Johansson.  Shape Fluctuations and Random Matrices , 1999, math/9903134.

[17]  Roderick Wong,et al.  Uniform asymptotics for Jacobi polynomials with varying large negative parameters— a Riemann-Hilbert approach , 2006 .

[18]  P. D. Miller,et al.  Uniform Asymptotics for Polynomials Orthogonal With Respect to a General Class of Discrete Weights and Universality Results for Associated Ensembles , 2002 .

[19]  Roderick Wong,et al.  Asymptotic approximations of integrals , 1989, Classics in applied mathematics.

[20]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[21]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[22]  Roderick Wong,et al.  ASYMPTOTIC EXPANSIONS FOR RIEMANN–HILBERT PROBLEMS , 2008 .