Large-Degree Asymptotics of Rational Painlevé-IV Functions Associated to Generalized Hermite Polynomials

The Painlevé-IV equation has three families of rational solutions generated by the generalized Hermite polynomials. Each family is indexed by two positive integers m and n. These functions have applications to nonlinear wave equations, random matrices, fluid dynamics, and quantum mechanics. Numerical studies suggest the zeros and poles form a deformed n× m rectangular grid. Properly scaled, the zeros and poles appear to densely fill certain curvilinear rectangles as m,n→ ∞ with $r:=m/n$ a fixed positive real number. Generalizing a method of Bertola and Bothner [2] used to study rational Painlevé-II functions, we express the generalized Hermite rational Painlevé-IV functions in terms of certain non-Hermitian orthogonal polynomials. Using the Deift–Zhou nonlinear steepest-descent method, we asymptotically analyze the associated Riemann–Hilbert problem in the limit n→ ∞ with m=r· n for r fixed. We obtain an explicit characterization of the boundary curve and determine the leading-order asymptotic expansion of the rational Painlevé-IV functions associated to generalized Hermite polynomials in the pole-free region.