Special functions, integral equations and a Riemann-Hilbert problem

We consider a pair of special functions, u_β and v_β, defined respectively as the solutions to the integral equations [Formula presented] where (Formula presented) for β ∈ (0, 1). In this note, we establish the existence and uniqueness of u_β and v_β which are bounded and continuous in [0,+∞). 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_β and v_β, and a related new special function G_β.