We consider three special cases of the initial value problem of the first Painlevé (PI) equation. Our approach is based on the method of uniform asymptotics introduced by Bassom et al. A rigorous proof of a property of the PI solutions on the negative real axis, recently revealed by Bender and Komijani, is given by approximating the Stokes multipliers. Moreover, we build more precise relation between the large initial data of the PI solutions and their three different types of behavior as the independent variable tends to negative infinity. In addition, some limiting form connection formulas are obtained.

As a new application of the method of "uniform asymptotics" proposed by Bassom, Clarkson, Law and McLeod, we provide a simpler and more rigorous proof of the connection formulas of some special solutions of the fifth Painlevé equation, which have been established earlier by Andreev and Kitaev.

We derive uniform asymptotic expansions for polynomials orthogonal with respect to a class of weight functions that are real analytic and behave asymptotically like the Freud weight at infinity. Although the limiting zero distributions are the same as in the Freud cases, the asymptotic expansions are different due to the fact that the weight functions may have a finite or infinite number of zeros on the imaginary axis. To resolve the singularities caused by these zeros, an auxiliary function is introduced in the Riemann-Hilbert analysis. Asymptotic formulas are established in several regions covering the whole complex plane. We take the continuous dual Hahn polynomials as an example to illustrate our main results. Some numerical verifications are also given. K E Y W O R D S asymptotic approximation, asymptotic Freud-like weight, continuous dual Hahn polynomials, Riemann-Hilbert problem ∑ =0 ( ) ( 2 ) ( + ) − ( − + ) − !( − )! , Stud Appl Math. 2020;144:133-163. wileyonlinelibrary.com/journal/sapm

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We consider two special cases of the connection problem for the second Painlevé equation (PII) using the method of uniform asymptotics proposed by Bassom et al.. We give a classification of the real solutions of PII on the negative (positive) real axis with respect to their initial data. By product, a rigorous proof of a property associate with the nonlinear eigenvalue problem of PII on the real axis, recently revealed by Bender and Komijani, is given by deriving the asymptotic behavior of the Stokes multipliers.

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We study the asymptotic expansion for the Landau constants G n ,where N = n + 1, and γ is Euler's constant. We show that the signs of the coefficients α k demonstrate a periodic behavior such that (−1)for all n = 0, 1, 2 · · · , all p = 4s + 1, 4s + 2 and q = 4m, 4m + 3, with s = 0, 1, 2, · · · and m = 0, 1, 2, · · · .