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Special functions, integral equations and a Riemann-Hilbert problem


Authors: R. Wong and Yu-Qiu Zhao
Journal: Proc. Amer. Math. Soc. 144 (2016), 4367-4380
MSC (2010): Primary 33E30, 41A60, 45A05
DOI: https://doi.org/10.1090/proc/13191
Published electronically: June 3, 2016
MathSciNet review: 3531186
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Abstract: We consider a pair of special functions, $ u_\beta $ and $ v_\beta $, defined respectively as the solutions to the integral equations

$\displaystyle u(x)=1+\int ^\infty _0 \frac {K(t) u(t) dt}{t+x} ~~$$\displaystyle \mbox {and}~~v(x)=1-\int ^\infty _0 \frac { K(t) v(t) dt}{t+x},~~x\in [0, \infty ),$    

where $ K(t)= \frac {1} \pi \exp \left (- t^\beta \sin \frac {\pi \beta } 2\right )\sin \left ( t^\beta \cos \frac {\pi \beta } 2 \right )$ for $ \beta \in (0, 1)$. In this note, we establish the existence and uniqueness of $ u_\beta $ and $ v_\beta $ which are bounded and continuous in $ [0, +\infty )$. 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_\beta $ and $ v_\beta $, and a related new special function $ G_\beta $.


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Additional Information

R. Wong
Affiliation: Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong

Yu-Qiu Zhao
Affiliation: Department of Mathematics, Sun Yat-sen University, GuangZhou 510275, People’s Republic of China
Email: stszyq@mail.sysu.edu.cn

DOI: https://doi.org/10.1090/proc/13191
Keywords: Special function, Freud weight, integral equation, Riemann-Hilbert problem, asymptotics, Stokes phenomenon
Received by editor(s): September 5, 2015
Received by editor(s) in revised form: December 17, 2015
Published electronically: June 3, 2016
Communicated by: Walter Van Assche
Article copyright: © Copyright 2016 American Mathematical Society