Special functions, integral equations and Riemann-Hilbert problem

TL;DR

This paper introduces special functions defined via integral equations, proves their existence and uniqueness, and demonstrates their explicit connection to a Riemann-Hilbert problem, with initial asymptotic analysis and open questions for further study.

Contribution

It establishes the existence, uniqueness, and explicit construction of new special functions related to integral equations and Riemann-Hilbert problems, expanding the theoretical framework.

Findings

Existence and uniqueness of $u_\beta$ and $v_\beta$

Explicit solution construction for a Riemann-Hilbert problem

Preliminary asymptotic analysis of the functions

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 $G_\beta$.