One-Parameter Meromorphic Solution of the Degenerate Third Painlevé Equation with Formal Monodromy Parameter $a=\pm i/2$ Vanishing at the Origin

Abstract: We prove that there exists a one-parameter meromorphic solution $u(\tau)$ vanishing at $\tau=0$ of the degenerate third Painlev\'e equation, \begin{equation*} u^{\prime \prime}(\tau) ! = ! \frac{(u^{\prime}(\tau))^{2}}{u(\tau)} ! - ! \frac{u^{\prime}(\tau)}{\tau} ! + ! \frac{1}{\tau} ! \left(-8 \varepsilon (u(\tau))^{2} ! + ! 2ab \right) ! + ! \frac{b^{2}}{u(\tau)},\qquad \varepsilon=\pm1,\quad\varepsilon b>0, \end{equation*} for formal monodromy parameter $a=\pm i/2$. We study number-theoretic properties of the coefficients of the Taylor-series expansion of $u(\tau)$ at $\tau=0$ and its asymptotic behaviour as $\tau\to+\infty$. These asymptotics are visualized for generic initial data.