The riemann-hilbert approach to global asymptotics of discrete orthogonal polynomials with infinite nodes

Chunhua Ou; R. Wong

doi:10.1142/9789814656054_0058

The riemann-hilbert approach to global asymptotics of discrete orthogonal polynomials with infinite nodes

Chunhua Ou
, R. Wong

Liu Bie Ju Centre for Mathematical Sciences

Research output: Chapters, Conference Papers, Creative and Literary Works › RGC 12 - Chapter in an edited book (Author) › peer-review

Abstract

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. 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 can be applied. Finally, we produce three Airy-type asymptotic expansions for formula 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, 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. Asymptotic formulas are also derived for large and small zeros of the Charlier polynomials. © 2016 by World Scientific Publishing Co. Ptc. Ltd.

Original language	English
Title of host publication	Selected Works Of Roderick S. C. Wong, The (In 3 Volumes)
Publisher	World Scientific Publishing Co. Pte Ltd
Pages	1297-1336
ISBN (Print)	9789814656054
DOIs	https://doi.org/10.1142/9789814656054_0058
Publication status	Published - 5 Aug 2015

Bibliographical note

Publication details (e.g. title, author(s), publication statuses and dates) are captured on an “AS IS” and “AS AVAILABLE” basis at the time of record harvesting from the data source. Suggestions for further amendments or supplementary information can be sent to [email protected].

Funding

The authors would like to thank Dr. Dan Dai for stimulating discussions on thisproblem, and Mr. Wang Xiangsheng for a careful reading of the original manuscript. The work of the ﬁrst-named author was partially supported by an NSERC discoverygrant of Canada.

Research Keywords

Airy function
Charlier polynomials
Discrete orthogonal polynomials
Global asymptotics
Riemann-hilbert problem

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abstract = "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. 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 can be applied. Finally, we produce three Airy-type asymptotic expansions for formula 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, 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. Asymptotic formulas are also derived for large and small zeros of the Charlier polynomials. {\textcopyright} 2016 by World Scientific Publishing Co. Ptc. Ltd.",

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The riemann-hilbert approach to global asymptotics of discrete orthogonal polynomials with infinite nodes. / Ou, Chunhua; Wong, R.
Selected Works Of Roderick S. C. Wong, The (In 3 Volumes). World Scientific Publishing Co. Pte Ltd, 2015. p. 1297-1336.

Research output: Chapters, Conference Papers, Creative and Literary Works › RGC 12 - Chapter in an edited book (Author) › peer-review

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N2 - 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. 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 can be applied. Finally, we produce three Airy-type asymptotic expansions for formula 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, 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. Asymptotic formulas are also derived for large and small zeros of the Charlier polynomials. © 2016 by World Scientific Publishing Co. Ptc. Ltd.

AB - 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. 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 can be applied. Finally, we produce three Airy-type asymptotic expansions for formula 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, 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. Asymptotic formulas are also derived for large and small zeros of the Charlier polynomials. © 2016 by World Scientific Publishing Co. Ptc. Ltd.

KW - Airy function

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KW - Global asymptotics

KW - Riemann-hilbert problem

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The riemann-hilbert approach to global asymptotics of discrete orthogonal polynomials with infinite nodes

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