We describe the pole-free regions of the one-parameter family of special solutions of P<inf>II</inf>, the second Painlevé equation, constructed from the Airy functions. This is achieved by exploiting the connection between these solutions and the recurrence coefficients of orthogonal polynomials that appear in the analysis of the ensemble of random matrices corresponding to the cubic potential.

We obtain asymptotics of polynomials satisfying the orthogonality relations (Formula presented.) where the complex parameter (Formula presented.) is in the so-called two-cut region. As an application, we deduce asymptotic formulas for certain families of solutions of Painlevé-IV which are indexed by a nonnegative integer and can be written in terms of parabolic cylinder functions. The proofs are based on the characterization of orthogonal polynomials in terms of a Riemann–Hilbert problem and the Deift–Zhou nonlinear steepest descent method.

The third Painlevé equation in its generic form, often referred to as Painlevé-III(D6), is given by [Formula Presented] Starting from a generic initial solution u0 (x) corresponding to parameters α, β, denoted as the triple (u0 (x), α, β), we apply an explicit Bäcklund transformation to generate a family of solutions (un (x), α + 4n, β + 4n) indexed by n ∈ N. We study the large n behavior of the solutions (un (x), α + 4n, β + 4n) under the scaling x = z/n in two different ways: (a) analyzing the convergence properties of series solutions to the equation, and (b) using a Riemann–Hilbert representation of the solution un (z/n). Our main result is a proof that the limit of solutions un (z/n) exists and is given by a solution of the degenerate Painlevé-III equation, known as Painlevé-III(D8), [Formula Presented]. z A notable application of our result is to rational solutions of Painlevé-III(D6), which are constructed using the seed solution (1, 4m, −4m) where m ∈ C\(Z +12) and can be written as a particular ratio of Umemura polynomials. We identify the limiting solution in terms of both its initial condition at z = 0 when it is well defined, and by its monodromy data in the general case. Furthermore, as a consequence of our analysis, we deduce the asymptotic behavior of generic solutions of Painlevé-III, both D6 and D8 at z = 0. We also deduce the large n behavior of the Umemura polynomials in a neighborhood of z = 0.