THE RIEMANN–HILBERT APPROACH TO GLOBAL ASYMPTOTICS OF DISCRETE ORTHOGONAL POLYNOMIALS WITH INFINITE NODES

2015 ◽  
pp. 1297-1336
Author(s):  
CHUNHUA OU ◽  
R. WONG
Keyword(s):  
Orthogonal Polynomials ◽  
Discrete Orthogonal Polynomials ◽  
Discrete Orthogonal ◽  
Global Asymptotics

THE RIEMANN–HILBERT APPROACH TO GLOBAL ASYMPTOTICS OF DISCRETE ORTHOGONAL POLYNOMIALS WITH INFINITE NODES

2010 ◽  
Vol 08 (03) ◽  
pp. 247-286 ◽  
Author(s):  
CHUNHUA OU ◽  
R. WONG
Keyword(s):  
Orthogonal Polynomials ◽  
Hilbert Problem ◽  
Descent Method ◽  
Steepest Descent Method ◽  
Asymptotic Formulas ◽  
Charlier Polynomials ◽  
Discrete Orthogonal Polynomials ◽  
Discrete Orthogonal ◽  
Global Asymptotics ◽  
Riemann Hilbert Problem

In this paper, we develop the Riemann–Hilbert approach to study the global asymptotics of discrete orthogonal polynomials with infinite nodes. We illustrate our method by concentrating on the Charlier polynomials [Formula: see text]. We first construct a Riemann–Hilbert problem Y associated with these polynomials and then establish some technical results to transform Y into a continuous Riemann–Hilbert problem so that the steepest descent method of Deift and Zhou ([8]) can be applied. Finally, we produce three Airy-type asymptotic expansions for [Formula: see text] in three different but overlapping regions whose union is the entire complex z-plane. When z is real, our results agree with the ones given in the literature. Although our approach is similar to that used by Baik, Kriecherbauer, McLaughlin and Miller ([3]), there are crucial differences in the details. For instance, our expansions hold in much bigger regions. Our results are completely new, and one of them answers a question raised in Bo and Wong ([4]). Asymptotic formulas are also derived for large and small zeros of the Charlier polynomials.

Stieltjes’ theorem for classical discrete orthogonal polynomials

2020 ◽  
Vol 61 (10) ◽  
pp. 103505
Author(s):  
K. Castillo ◽  
F. R. Rafaeli ◽  
A. Suzuki
Keyword(s):  
Orthogonal Polynomials ◽  
Discrete Orthogonal Polynomials ◽  
Discrete Orthogonal
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