  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 $G_\beta$.
