Special functions, integral equations and Riemann-Hilbert problem

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 $G_{\beta }$.

14 pages, 1 figure

References in corpus (1)

• Asymptotic behavior and zero distribution of polynomials orthogonal with respect to Bessel functions