TY - JOUR
T1 - Special functions, integral equations and a Riemann-Hilbert problem
AU - Wong, R.
AU - Zhao, Yu-Qiu
PY - 2016/10
Y1 - 2016/10
N2 - 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.
AB - 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.
KW - Asymptotics
KW - Freud weight
KW - Integral equation
KW - Riemann-Hilbert problem
KW - Special function
KW - Stokes phenomenon
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UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-84982908380&origin=recordpage
U2 - 10.1090/proc/13191
DO - 10.1090/proc/13191
M3 - RGC 21 - Publication in refereed journal
VL - 144
SP - 4367
EP - 4380
JO - Proceedings of the American Mathematical Society
JF - Proceedings of the American Mathematical Society
SN - 0002-9939
IS - 10
ER -