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Zbl 0846.34002
Murata, Yoshihiro
Classical solutions of the third Painlevé equation. (English)
[J] Nagoya Math. J. 139, 37-65 (1995). ISSN 0027-7630

The author determines all ``classical'' solutions (in the sense of Painlevé and Umemura) of the third Painlevé equation. The equation is equivalent to $$(P_{III'}) \hskip3cm {{d^2 q} \over {dt^2}}= {1\over q} \Biggl( {{dq} \over {dt}} \Biggr)^2- {1\over t} {{dq} \over {dt}}+ {{q^2} \over {4t^2}} (\gamma q+ \alpha)+ {\beta \over {4t}}+ {\delta \over {4q}}.$$ After showing that $P_{III'}$ 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 $\alpha=- 4\theta_\infty$, $\beta= 4(\theta_0+ 1)$, $\gamma= 4$, $\delta= -4$. The equation $P_{III'}$ with the above parameters is denoted by $P_{III'} (\theta_0, \theta_\infty)$. \par The main results of this paper are stated as follows: (1) $P_{III'} (\theta_0, \theta_\infty)$ does not have rational solutions, (2) $P_{III'} (\theta_0, \theta_\infty)$ 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 $\theta_\infty- \theta_0)$ is an even integer, then $P_{III'} (\theta_0, \theta_\infty)$ 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. \par The author announces that the irreducibility of $P_{III'}$ except for the classical solutions in this paper will be proved in his forthcoming paper.
[K.Takano (Kobe)]

MSC 2000:: *34A05 Methods of solution of ODE
33E30 Functions coming from diff., difference and integral equations

Keywords: classical solutions; third Painlevé equation; irreducibility

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