@article{e1b2b193dc4743e8b4d21f0a7c61281a, title = "Special functions, integral equations and a Riemann-Hilbert problem", abstract = "We consider a pair of special functions, uβ and vβ, 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 uβ and vβ 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 uβ and vβ, and a related new special function Gβ.", keywords = "Asymptotics, Freud weight, Integral equation, Riemann-Hilbert problem, Special function, Stokes phenomenon", author = "R. Wong and Yu-Qiu Zhao", year = "2016", month = oct, doi = "10.1090/proc/13191", language = "English", volume = "144", pages = "4367--4380", journal = "Proceedings of the American Mathematical Society", issn = "0002-9939", publisher = "American Mathematical Society", number = "10", }