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Classical solutions of the third Painlevé equation. (English) Zbl 0846.34002

The author determines all “classical” solutions (in the sense of Painlevé and Umemura) of the third Painlevé equation. The equation is equivalent to

$\left({P}_{II{I}^{\text{'}}}\right)\phantom{\rule{85.35826pt}{0ex}}\frac{{d}^{2}q}{d{t}^{2}}=\frac{1}{q}{\left(\frac{dq}{dt}\right)}^{2}-\frac{1}{t}\frac{dq}{dt}+\frac{{q}^{2}}{4{t}^{2}}\left(\gamma q+\alpha \right)+\frac{\beta }{4t}+\frac{\delta }{4q}·$

After showing that ${P}_{II{I}^{\text{'}}}$ has a rational solution if and only if $\alpha =\gamma =0$ or $\beta =\delta =0$ and reviewing a result of V. I. Gromak for the case of $\gamma =0$, $\alpha \delta \ne 0$ (or $\delta =0$, $\beta \gamma \ne 0$), the author studies the case of $\gamma \delta \ne 0$ which is equivalent to the case of $\alpha =-4{\theta }_{\infty }$, $\beta =4\left({\theta }_{0}+1\right)$, $\gamma =4$, $\delta =-4$. The equation ${P}_{II{I}^{\text{'}}}$ with the above parameters is denoted by ${P}_{II{I}^{\text{'}}}\left({\theta }_{0},{\theta }_{\infty }\right)$.

The main results of this paper are stated as follows: (1) ${P}_{II{I}^{\text{'}}}\left({\theta }_{0},{\theta }_{\infty }\right)$ does not have rational solutions, (2) ${P}_{II{I}^{\text{'}}}\left({\theta }_{0},{\theta }_{\infty }\right)$ has algebraic solutions if and only if ${\theta }_{\infty }-{\theta }_{0}=1$ or ${\theta }_{\infty }+{\theta }_{0}+1$ is an even integer, (3) the number of algebraic solutions (in statement (2)) are one or two, and the latter occurs if and only if both ${\theta }_{\infty }-{\theta }_{0}-1$ and ${\theta }_{\infty }+{\theta }_{0}+1$ are even integers, (4) if ${\theta }_{\infty }+{\theta }_{0}$ (or ${\theta }_{\infty }-{\theta }_{0}\right)$ is an even integer, then ${P}_{II{I}^{\text{'}}}\left({\theta }_{0},{\theta }_{\infty }\right)$ has a one-parameter family of classical solutions which is a rational function of ${\theta }_{0}$, $t$ and a general solution of a Riccati equation $dq/dt=-{q}^{2}/t-{\theta }_{0}q/t+1$. Furthermore, the exact forms of algebraic solutions are given.

The author announces that the irreducibility of ${P}_{II{I}^{\text{'}}}$ except for the classical solutions in this paper will be proved in his forthcoming paper.

Reviewer: K.Takano (Kobe)
##### MSC:
 34A05 Methods of solution of ODE 33E30 Functions coming from differential, difference and integral equations