SPECIAL FUNCTIONS, INTEGRAL EQUATIONS AND A RIEMANN-HILBERT PROBLEM

R. WONG; Yu-Qiu ZHAO

doi:10.1142/9789811290862_0016

SPECIAL FUNCTIONS, INTEGRAL EQUATIONS AND A RIEMANN-HILBERT PROBLEM

R. WONG
, Yu-Qiu ZHAO^*

^*Corresponding author for this work

Department of Mathematics

Research output: Chapters, Conference Papers, Creative and Literary Works › Reprint in book › peer-review

Abstract

We consider a pair of special functions, u_β and v_β, defined respectively as the solutions to the integral equations [Formula Presented], where [Formula Presented] for β ∈ (0, 1). In this note, we establish the existence and uniqueness of u_β and v_β which are bounded and continuous in [0,+∞). Also, we show that a solution to a model Riemann-Hilbert problem in Kriecherbauer and McLaughlin [Int. Math. Res. Not., 1999] can be constructed explicitly in terms of these functions. A preliminary asymptotic study is carried out on the Stokes phenomena of these functions by making use of their connection formulas.
Several open questions are also proposed for a thorough investigation of the analytic and asymptotic properties of the functions u_β and v_β, and a related new special function G_β.
© 2016 American Mathematical Society.

Original language	English
Title of host publication	The Selected Works of Roderick S. C. Wong
Editors	Dan Dai, Ding-Xuan Zhou
Publisher	World Scientific Publishing Co. Pte Ltd
Pages	1810-1823
Volume	4
ISBN (Electronic)	978-981-12-9086-2, 978-981-12-9087-9
ISBN (Print)	978-981-12-9085-5
DOIs	https://doi.org/10.1142/9789811290862_0016
Publication status	Published - 2026

Funding

We would like to thank Dr. Xiang-Sheng Wang for drawing the graphs and helping us with the numerical simulation. The research of the second author was supported in part by the National Natural Science Foundation of China under grant numbers 10871212 and 11571375.

Research Keywords

asymptotics
Freud weight
integral equation
Riemann-Hilbert problem
Special function
Stokes phenomenon.

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title = "SPECIAL FUNCTIONS, INTEGRAL EQUATIONS AND A RIEMANN-HILBERT PROBLEM",

abstract = "We consider a pair of special functions, uβ and vβ, defined respectively as the solutions to the integral equations [Formula Presented], where [Formula Presented] for β ∈ (0, 1). In this note, we establish the existence and uniqueness of uβ and vβ which are bounded and continuous in [0,+∞). Also, we show that a solution to a model Riemann-Hilbert problem in Kriecherbauer and McLaughlin [Int. Math. Res. Not., 1999] can be constructed explicitly in terms of these functions. A preliminary asymptotic study is carried out on the Stokes phenomena of these functions by making use of their connection formulas. Several open questions are also proposed for a thorough investigation of the analytic and asymptotic properties of the functions uβ and vβ, and a related new special function Gβ. {\textcopyright} 2016 American Mathematical Society.",

keywords = "asymptotics, Freud weight, integral equation, Riemann-Hilbert problem, Special function, Stokes phenomenon.",

author = "R. WONG and Yu-Qiu ZHAO",

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T1 - SPECIAL FUNCTIONS, INTEGRAL EQUATIONS AND A RIEMANN-HILBERT PROBLEM

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AU - ZHAO, Yu-Qiu

PY - 2026

Y1 - 2026

N2 - We consider a pair of special functions, uβ and vβ, defined respectively as the solutions to the integral equations [Formula Presented], where [Formula Presented] for β ∈ (0, 1). In this note, we establish the existence and uniqueness of uβ and vβ which are bounded and continuous in [0,+∞). Also, we show that a solution to a model Riemann-Hilbert problem in Kriecherbauer and McLaughlin [Int. Math. Res. Not., 1999] can be constructed explicitly in terms of these functions. A preliminary asymptotic study is carried out on the Stokes phenomena of these functions by making use of their connection formulas. Several open questions are also proposed for a thorough investigation of the analytic and asymptotic properties of the functions uβ and vβ, and a related new special function Gβ. © 2016 American Mathematical Society.

AB - We consider a pair of special functions, uβ and vβ, defined respectively as the solutions to the integral equations [Formula Presented], where [Formula Presented] for β ∈ (0, 1). In this note, we establish the existence and uniqueness of uβ and vβ which are bounded and continuous in [0,+∞). Also, we show that a solution to a model Riemann-Hilbert problem in Kriecherbauer and McLaughlin [Int. Math. Res. Not., 1999] can be constructed explicitly in terms of these functions. A preliminary asymptotic study is carried out on the Stokes phenomena of these functions by making use of their connection formulas. Several open questions are also proposed for a thorough investigation of the analytic and asymptotic properties of the functions uβ and vβ, and a related new special function Gβ. © 2016 American Mathematical Society.

KW - asymptotics

KW - Freud weight

KW - integral equation

KW - Riemann-Hilbert problem

KW - Special function

KW - Stokes phenomenon.

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UR - https://www.scopus.com/pages/publications/105042774373#tab=impact

U2 - 10.1142/9789811290862_0016

DO - 10.1142/9789811290862_0016

M3 - Reprint in book

SN - 978-981-12-9085-5

VL - 4

SP - 1810

EP - 1823

BT - The Selected Works of Roderick S. C. Wong

A2 - Dai, Dan

A2 - Zhou, Ding-Xuan

PB - World Scientific Publishing Co. Pte Ltd

ER -

SPECIAL FUNCTIONS, INTEGRAL EQUATIONS AND A RIEMANN-HILBERT PROBLEM

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Research Keywords

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