Special functions, integral equations and a Riemann-Hilbert problem
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- by R. Wong and Yu-Qiu Zhao PDF
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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 \begin{equation*} u(x)=1+\int ^\infty _0 \frac {K(t) u(t) dt}{t+x} ~~\mbox {and}~~v(x)=1-\int ^\infty _0 \frac { K(t) v(t) dt}{t+x},~~x\in [0, \infty ),\end{equation*} 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
- MR Author ID: 192744
- Yu-Qiu Zhao
- Affiliation: Department of Mathematics, Sun Yat-sen University, GuangZhou 510275, People’s Republic of China
- MR Author ID: 604554
- Email: stszyq@mail.sysu.edu.cn
- 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
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 4367-4380
- MSC (2010): Primary 33E30, 41A60, 45A05
- DOI: https://doi.org/10.1090/proc/13191
- MathSciNet review: 3531186