Large- $t$ asymptotic conjecture for the Painlev\e9 c3\mathrm{PIII}' transcendent

Let $\epsilon=\pm1$, $\tilde{\omega}=\omega/(2\pi)$, and let $v_\epsilon(t;\xi)$ be the particular $\sigma\operatorname{PIII}'$ transcendent satisfying the differential equation specified in the source subject to the boundary condition as $t\to0^+$. Define

$$\begin{aligned} A_\epsilon(t;\omega)={}&-i\tilde{\omega}\left(\sqrt t-\frac{\sin^2(2\sqrt t)}{4\sqrt t}\tilde{\omega}\right)+\frac12\tilde{\omega}^2\\ &-\frac{\epsilon(1-\tilde{\omega})}{4\pi}\left(\sin(\pi\tilde{\omega})\Gamma^2(1-\tilde{\omega})(4\sqrt t)^{2\tilde{\omega}}e^{-2i\sqrt t}+\overline{(\tilde{\omega}\mapsto-\tilde{\omega})}\right), \end{aligned}$$

and

$$B_\epsilon(t;\omega)=1+\frac{i\epsilon}{\pi}\left(\sin(\pi\tilde{\omega})\Gamma^2(1-\tilde{\omega})(4\sqrt t)^{2\tilde{\omega}-1}e^{-2i\sqrt t}+\overline{(\tilde{\omega}\mapsto-\tilde{\omega})}\right).$$

Painlevé asymptotic conjecture. For $0\leq\omega<\pi$, up to an order not yet determined,

$$v_\epsilon(t;\xi)\big|_{\xi=1-e^{i\omega}}\mathop{\sim}_{t\to\infty}\frac{A_\epsilon(t;\omega)}{B_\epsilon(t;\omega)}.$$

The proposed expansion is intended to provide the large-$t$ connection asymptotics needed to control the truncation error in the Painlevé c3\operatorname{PIII}' evaluation of the relevant quantities. The source says it was checked by numerical comparisons, but leaves the order of the asymptotic expansion and a rigorous justification for future work.